Non-Hyperoctahedral Categories of Two-Colored Partitions, Part I: New Categories
Alexander Mang Moritz Weber
math.CO math.QA 05A18 (Primary), 20G42 (Secondary)

Non-Hyperoctahedral Categories, Part I]Non-Hyperoctahedral Categories 
of Two-Colored Partitions
 Part I: New Categories 


Saarland University, Fachbereich Mathematik, 
	66041 Saarbrücken, Germany
s9almang@stud.uni-saarland.de, weber@math.uni-sb.de
The second author was supported by the ERC Advanced Grant NCDFP, held
by Roland Spei­cher, by the SFB-TRR 195, and by the DFG project Quantenautomorphismen von Graphen. This work was part of the first author's Master's thesis.


[2010]05A18 (Primary),  20G42 (Secondary)



Compact quantum groups can be studied by investigating their co-re­pre­sen­ta­tion ca­te­go­ries in analogy to the Schur-Weyl/Tannaka-Krein  approach. For the special class of (unitary) easy quantum groups these categories arise from a combinatorial structure: Rows of two-colored points form the objects, partitions of two such rows the morphisms; vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes 𝒪, ℬ, 𝒮 and ℋ of such categories (inspired respectively by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups) we treat the first three – the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions.  This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. The article is purely combinatorial in nature; The quantum group aspects are left out.

[
    Moritz Weber
    December 30, 2023
=====================



§ INTRODUCTION
 


In Woronowicz's approach (<cit.>, <cit.>, <cit.>), (compact) quantum groups are understood as certain non-commutative spaces, the formal duals of C^*-algebras, carrying a special Hopf algebra structure, for which a non-commutative version of Pontryagin duality can be proven. Reminiscent of the theorems of Tannaka-Krein and Schur-Weyl, a duality exists between the class of compact quantum groups and a particular class of involutive monoidal linear categories. The finite-dimensional unitary co-representations of a given compact quantum group form such a category, and, conversely, a unique maximal compact quantum group can be reconstructed from any such tensor category (<cit.>). 
Banica and Speicher (<cit.>) showed that sets of points as objects and partitions of finite sets as morphisms with vertical concatenation as composition, horizontal concatenation as monoidal product and reflection as involution provide concrete, combinatorial initial data for such representation categories. Their construction yields compact quantum subgroups of the free orthogonal quantum group O_n^+ introduced by Wang (<cit.>) as a non-commutative counterpart of the classical group  O_n of orthogonal real-valued matrices. All examples of compact quantum groups arising in this fashion, the so-called easy quantum groups, have since been classified (<cit.>, <cit.>, <cit.>,<cit.>, and <cit.>).
As Freslon, Tarrago and the second author demonstrated (<cit.>, <cit.>, <cit.>), Banica and Speicher's approach can be generalized to categories of partitions of sets of two-colored points.  In contrast to the uncolored case, here, vertical concatenation of partitions, i.e. the composition of morphisms, is restricted to such partitions with matching colorings of their points. This construction yields combinatorial compact quantum subgroups of the free unitary quantum group U_n^+, a quantum analogue of the classical unitary group U_n also introduced by Wang (<cit.>). A collective endeavour to find all such unitary easy quantum groups was initiated by Tarrago and the second author in <cit.> and has since been advanced by Gromada in <cit.> as well as by the authors in <cit.> and <cit.>.
The classification of all unitary easy quantum groups has been approached from several different angles of attack.
Tarrago and the second author classified in <cit.> all non-crossing categories C⊆ of two-colored partitions, i.e., C⊆NC^∘∙, and all categories C of two-colored partitions with ∈ C, the so-called group case. In contrast, in <cit.> Gromada determined all categories C with the property of being globally colorized, meaning ∈ C. Lastly, the authors of the present article found all categories C with ⟨∅⟩⊆ C ⊆⟨⟩, i.e. categories of neutral pair partitions, corresponding to easy compact quantum groups G with U_n^+⊇ G⊇ U_n, the unitary half-liberations, in <cit.> and <cit.>.  
The present article is concerned with non-hy­per­oc­ta­hed­ral categories of two-colored partitions, i.e. categories C⊆ with ∈ C or ∉ C. We define explicitly certain sets of partitions and show that each of them constitutes a non-hyperoctahedral category. See the next section for an overview.
This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. In subsequent articles it will be shown that the categories found in the present article are  pairwise distinct and actually constitute all possible non-hyperoctahedral categories. Furthermore, a set of generating partitions for each non-hyperoctahedral category will be determined.
About hyperoctahedral categories of two-colored partitions very little is known for the moment. Note that in the uncolored case categories C⊆P with ∉ C and ∈ C give rise to quantum subgroups of the free hyperoctahedral quantum group H^+_n (<cit.>), hence the name. Amongst others, Laura Maaßen is currently doing research on the hyperoctahedral case.


§ MAIN RESULT


Many new examples of non-hyperoctahedral categories of two-colored partitions are provided. Roughly, they are determined by combinations of constraints on the


  * sizes of blocks,

  * coloring of the points,

  * allowed crossings between blocks

of their partitions.
More precisely: The coloring of any two-colored partition p∈ induces on the set of points a measure-like structure, the color sum σ_p, and a metric-like one, the color distance δ_p. Measuring the set of all points yields the total color sum (p).
Let now 𝒮⊆ be an arbitrary set of two-colored partitions and consider the following data:


  * The set of block sizes: 
    F( S){  |B| | p∈ S,  B block of  p}.


  * The set of block color sums: 
    V( S){ σ_p(B)| p∈ S,  B block of p}.


  * The set of total color sums: 
    ( S){ (p)| p∈ S}.


  * The set of color distances between any two subsequent legs of the same block having the same normalized color:
  rCl
              L(S){ δ_p(α_1,α_2)   |    p∈S,   B block of p,  α_1,α_2∈B,  α_1≠α_2,

         ]α_1,α_2[_p∩B=∅,  σ_p({α_1,α_2})≠0}.


       
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    whp/.style=circle, inner sep=0pt, text width=, draw=black, fill=white
    blp/.style=circle, inner sep=0pt, text width=, draw=black, fill=black
    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+1*,0+0*) ;
    [whp] (l2) at (0+4*,0+0*) ;    
    
    (l1) –++(0,) -| (l2) ((l1)+(0,)) –++(-0.5*,0) ((l2)+(0,)) –++(0.5*,0);
    [densely dashed] ((l1)+(-0.5*,)) – ++ (-0.75*,0) ((l2)+(0.5*,)) – ++ (0.75*,0);
    [draw=gray, pattern= north west lines, pattern color = gray] ((l1)+(,)) – ((l1)+(,-)) –((l2)+(,-)) |- cycle;
    
    at (-,0.5*) p;
    at (3*,0.75*) B;    
    [below = 0.35* of l1] α_1;
    [below = 0.35* of l2] α_2;    
  


  * The set of color distances between any two subsequent legs of the same block having different normalized colors:
   rCl
              K(S){ δ_p(α_1,α_2)   |    p∈S,   B block of p,  α_1,α_2∈B,  α_1≠α_2,

         ]α_1,α_2[_p∩B=∅,  σ_p({α_1,α_2})= 0}.


       
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    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+1*,0+0*) ;
    [whp] (l2) at (0+4*,0+0*) ;    
    
    (l1) –++(0,) -| (l2) ((l1)+(0,)) –++(-0.5*,0) ((l2)+(0,)) –++(0.5*,0);
    [densely dashed] ((l1)+(-0.5*,)) – ++ (-0.75*,0) ((l2)+(0.5*,)) – ++ (0.75*,0);
    [draw=gray, pattern= north west lines, pattern color = gray] ((l1)+(,)) – ((l1)+(,-)) –((l2)+(-,-)) |- cycle;
    
    at (-,0.5*) p;
    at (3*,0.75*) B;    
    [below = 0.35* of l1] α_1;
    [below = 0.35* of l2] α_2;    
  


  * The set of color distances between any two legs belonging to two crossing blocks
   rCl
                          X(S){ δ_p(α_1,α_2)    |    p∈S,  B_1,B_2 blocks of p,   B_1 crosses  B_2,

         α_1∈B_1, α_2∈B_2}.


       
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    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+1*,0+0*) ;
    [whp] (l2) at (0+3*,0+0*) ;
    [blp] (l3) at (0+6*,0+0*) ;
    [whp] (l4) at (0+8*,0+0*) ;        
    
    (l1) –++(0,) -| (l3) ((l1)+(0,)) –++(-0.5*,0) ((l3)+(0,)) –++(0.5*,0);
    [densely dashed] ((l1)+(-0.5*,)) – ++ (-0.75*,0) ((l3)+(0.5*,)) – ++ (0.75*,0);
    
    (l2) –++(0,2*) -| (l4) ((l2)+(0,2*)) –++(-0.5*,0) ((l4)+(0,2*)) –++(0.5*,0);
    [densely dashed] ((l2)+(-0.5*,2*)) – ++ (-0.75*,0) ((l4)+(0.5*,2*)) – ++ (0.75*,0);
    
    [draw=gray, pattern= north west lines, pattern color = gray] ((l2)+(,)) – ((l2)+(,-)) –((l3)+(-,-)) |- cycle;
    
    at (-,0.5*) p;
    [above = 0.85* of l1] B_1;
    [above = 1.85* of l4] B_2;        
    [below = 0.35* of l2] α_1;
    [below = 0.35* of l3] α_2;    
  


It is a subtle question which combinations of conditions on these six quantities define categories of partitions. We clarify it and we obtain a huge variety of new categories of partitions.


[Theorem <ref>] For every u∈{0}∪, m∈, D⊆{0,…,⌊m/2⌋}, E⊆{0}∪ and every subsemigroup N of (,+) each parameter tuple (f,v,s,l,k,x) listed below defines a non-hyperoctahedral category of two-colored partitions
  rCCCl
    R_(f,v,s,l,k,x){p∈   |   F({p})⊆f,    V({p})⊆v,    ({p})⊆s,
        L({p})⊆l,    K({p})⊆k,    X({p})⊆x }.
  

                    
    f   v   s    l    k    x  
    
    {2}   ±{0, 2}    2um    m    m   
    {2}   ±{0, 2}    2um    m+2m    2m   
    {2}   ±{0, 2}    2um    m+2m    2m   \ m
    {2}   {0}   {0}   ∅    m   
    {2}   ±{0, 2}   {0}   {0}   {0}   \± N
    {2}   {0}   {0}   ∅   {0}   \± N
    {2}   {0}   {0}   ∅   {0}   \{0}\± N
    {1,2}   ±{0, 1, 2}    um    m    m   \ (± D+m)
    {1,2}   ±{0, 1, 2}    2um    m+2m    2m   \ (± D+m)
    {1,2}   ±{0, 1}    um   ∅    m   \ (± D+m) 
    {1,2}   ±{0, 1, 2}   {0}   {0}   {0}   \± E
    {1,2}   ±{0,  1}   {0}   ∅   {0}   \± E
           um    m    m   \ (± D+m)
          {0}   {0}   {0}   \± E

                    Here,    ± S S∪(-S) for any set S⊆.
                  
                  




§ BASIC DEFINITIONS: PARTITIONS


Details on, examples and illustrations of two-colored partitions and their categories can be found in <cit.>. However, we quickly recall the basics. In addition, certain definitions from <cit.> and <cit.> are given here in greater generality.
After revisiting the fundamental definition of two-colored partitions, specialized language is introduced to describe them, in particular the concepts of orientation,   normalized color, color sum and color distance.


 §.§ Two-Colored Partitions

  A (two-colored) partition is specified by the following three data: 1) two disjoint (possibly empty) finite totally ordered sets R_L, the lower row comprising the lower points, and R_U, the upper row comprising the upper points, 2) an exhaustive decomposition of the union R_L∪ R_U, the set of points, into disjoint subsets, the blocks, and 3) a two-valued mapping on R_L∪ R_U, the coloration, assigning to every point its color, either ∙ (black) or ∘ (white). We represent partitions pictorially as follows:
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+0*,0+0*) ;
    
    [->] (l1) –++(0,);
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    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+0*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    
    (l1) – (u1);
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    
    [blp] (u1) at (0+0*,0+1*) ;
    
    (l1) – (u1);
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    
    (l1) –++ (0,) -| (l2);
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+0*,0+0*) ;
    [blp] (l2) at (0+1*,0+0*) ;
    
    (l1) –++ (0,) -| (l2);
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [blp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    (u1) –++(0,-) -|(u3);
    (u2) –++(0,-) -|(u3);
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    
    (l2) – (u1);    
    (l1) – (u2);
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [whp] (l3) at (0+2*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    [->] (l1) –++ (0,);
    (l2) – (u2);
    (u1) –++(0,-) -|(u3);
    (l3) –++ (0,) -| (l4);
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    [whp] (u4) at (0+3*,0+1*) ;    
    
    (u1) –++(0,-) -| (u3);
    (u2) –++(0,-2*) -| (u4);    
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [blp] (l3) at (0+2*,0+0*) ;
    [whp] (l4) at (0+3*,0+0*) ;
    
    [lk, yshift=2*] at (l1) ;
    [lk, yshift=2*] at (l2) ;
    [lk, yshift=2*] at (l4) ;    
    
    [->] (l3) –++ (0,);
    (l1) –++ (0,2*) -| (l2);    
    (l2) –++ (0,2*) -| (l4);
    
  
  Moreover, we say that the two colors ∙ and ∘ are inverse to each other. A point α which is less than another point β with respect to the total order of that row is said to lie left of β. And, if so, β lies right of α.
  The elements of a block are called its legs. If a block contains points from both R_L and R_U, we speak of a through block. If on the other hand a block is contained in one row, either R_L or R_U, it is said to be non-through, speaking of lower and upper non-through blocks respectively. 
We call a block with just one leg a singleton. Blocks with exactly two legs are pairs.  The number of legs of a block is its size. Likewise, the total number of points of R_L∪ R_U is the size of the partition. The number of lower points is the length of the lower row and we adopt the same terminology  for the upper row. A partition with only pairs is called a pair partition. 
We say that the upper and the lower rows are opposite each other. To address points in a specified row we sometimes use the rank of the point in the total order of that row, which we count up from the least element, which ranks 1. To refer to the lower point of rank i, we write i, and, similarly, i for the upper point of rank i. The points i and i are said to be counterparts of each other. Likewise, a set of points of one given row is said to be the counterpart of the set comprising exactly the points of the same ranks of the opposite row. A through block B is called straight if L∩ B and L∩ U are counterparts of each other. 
The set of all two-colored partitions is denoted .  The empty partition ∅ is the unique element with empty rows. We denote by  the set of all pair partitions and by _≤ 2 the set of all partitions all of whose blocks have sizes one or two.
Whenever we deal with multiple partitions at a time, we identify all their points via the total orderings to the extent that this is possible. For example, given p,p'∈ and a point i in p, we do not distinguish between the point i in p and the point i in p' (if p' has at least i lower points). And the same goes for subsets of points. Here is an overview of the graphical representation of partitions:
    [1em]
    
    0.666
    *0.075cm
    *0.25cm
    
    *0.5cm
    *1cm
    *0.3cm
    12*    
    12*
    *1cm
    *0.3cm
    3*
    
    whp/.style=circle, inner sep=0pt, text width=, draw=black, fill=white
    blp/.style=circle, inner sep=0pt, text width=, draw=black, fill=black
    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+0*,0+0*) ;    
    [whp] (l2) at (0+1*,0+0*) ;    
    [whp] (l3) at (0+2*,0+0*) ;
    [whp] (l4) at (0+3*,0+0*) ;
    [blp] (l5) at (0+4*,0+0*) ;
    [whp] (l8) at (0+7*,0+0*) ;
    [whp] (l10) at (0+9*,0+0*) ;
    [whp] (l11) at (0+10*,0+0*) ;
    [whp] (l12) at (0+11*,0+0*) ;
    [whp] (l13) at (0+12*,0+0*) ;        
    
    [whp] (u3) at (0+2*,0+1*) ;
    [blp] (u4) at (0+3*,0+1*) ;
    [blp] (u5) at (0+4*,0+1*) ;
    [blp] (u6) at (0+5*,0+1*) ;
    [whp] (u7) at (0+6*,0+1*) ;
    [whp] (u8) at (0+7*,0+1*) ;
    [whp] (u9) at (0+8*,0+1*) ;
    [whp] (u10) at (0+9*,0+1*) ;
    [whp] (u11) at (0+10*,0+1*) ;
    [whp] (u12) at (0+11*,0+1*) ;
    [whp] (u13) at (0+12*,0+1*) ;        
    
    [lk, yshift=-] at (u5) ;
    [lk, yshift=-] at (u7) ;
    [lk, yshift=-] at (u8) ;    
    [lk, yshift=-] at (u9) ;
    [lk, yshift=-] at (u11) ;        
    [lk, yshift=-] at (u12) ;
    [lk, yshift=] at (l10) ;        
    [lk, yshift=] at (l13) ;            
    
    [->] (l1) –++ (0,);
    (l3) – (u3);
    (l5) – (u5);
    (l8) – (u8);
    (l10) – (u10);
    (l11) – (u11);
    (l12) – (u12);
    (l13) – (u13);    
    (u4) –++(0,-) -|(u6);
    (u7) –++(0,-) -|(u9);    
    (l2) –++ (0,) -| (l4);
    (u11) –++(0,-) -|(u12);
    (l10) –++ (0,) -| (l13);
    (lab1) at (-1,-1) singleton;
    [densely dotted,->, shorten >=5pt] (lab1.north) – (l1.south);
    [text width=5em] (lab2) at (-1.5,1.5) two pairs
crossing; 
    [densely dotted,->, shorten >=5pt] (lab2.south) – ((l3)+(0,));
    [below = 4pt of l1] 1;
    [below = 4pt of l2] 2;
    [below = 4pt of l3] 3;
    [below = 4pt of l4] 4;
    [below = 4pt of l5] 5;
    [below = 4pt of l8] 6;
    [below = 4pt of l10] 7;
    [below = 4pt of l11] 8;
    [below = 4pt of l12] 9;
    [below = 4pt of l13] 10;
    [above = 4pt of u3] 1;
    [above = 4pt of u4] 2;
    [above = 4pt of u5] 3;
    [above = 4pt of u6] 4;
    [above = 4pt of u7] 5;
    [above = 4pt of u8] 6;
    [above = 4pt of u9] 7;
    [above = 4pt of u10] 8;
    [above = 4pt of u11] 9;
    [above = 4pt of u12] 10;
    [above = 4pt of u13] 11;
    [text width=5em] (lab3) at (4,-1.5) four-legged
blocks each;
    [densely dotted,->, shorten >=5pt] (lab3.north) – ((u5)+(0,-));
    [densely dotted,->, shorten >=5pt] (lab3.north) – ((u8)+(0,-));
    [text width = 5em] (lab4) at (6,3.5) just for
readability; 
    [densely dotted,->, shorten >=5pt] (lab4.south) – ((u7)+(0,-));
    [densely dotted,->, shorten >=5pt] (lab4.south) – ((u9)+(0,-));
    [text width = 6em] (lab5) at (10,0) two separate
blocks; 
    [densely dotted,->, shorten >=2pt] (lab5.north) – ((l12)+(0,0.5*));
    [densely dotted,->, shorten >=2pt] (lab5.north) – ((u13)+(0,-));        
  



    

 §.§ Orientation and Intervals


While any p∈ already comes with total orders ≤_L on its lower row R_L and ≤ _U on its upper row R_U, in many situations it is more advantageous to consider a cyclic order on the entire set R_L∪ R_U of its points. This orientation is uniquely determined by the following four conditions: It concurs with ≤_L on R_L, but with the inverse of ≤_U on R_U, and the maximum of ≤_U succeeds the maximum of ≤_L and likewise the minimum of ≤_L succeeds the minimum of ≤_U. 

  [0.5em]
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    7*
    0.5
    [dotted] (-,+) –  (+4*+,+);
    [dotted] (-,) –  (+6*+,);
    [scale=0.4, circle, draw=black, fill=black] (a0) at (+0*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a1) at (+1*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a2) at (+2*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a3) at (+3*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a4) at (+4*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a5) at (+5*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a6) at (+6*,) ;
    [scale=0.4, circle, draw=black, fill=black] (b0) at (+0*,+) ;
    [scale=0.4, circle, draw=black, fill=white] (b1) at (+1*,+) ;
    [scale=0.4, circle, draw=black, fill=black] (b2) at (+2*,+) ;
    [scale=0.4, circle, draw=black, fill=white] (b3) at (+3*,+) ;
    [scale=0.4, circle, draw=black, fill=black] (b4) at (+4*,+) ;
    [lightgray] (a0) – ++(0,) -| (a1);
    [lightgray] (a3) – ++(0,) -| (a5);
    [lightgray] (a4) – ++(0,2*) -| (a6);
    [lightgray] (a2) – (b2);
    [lightgray] (b0) – ++(0,-2*) -| (b3);
    [lightgray] (b1) – ++(0,-1*) -| (b4);
    	[->] (b4) to [out=135, in=45] (b3);
	[->] (b3) to [out=135, in=45] (b2);
	[->] (b2) to [out=135, in=45] (b1);
	[->] (b1) to [out=135, in=45] (b0);
	[->] (b0) to [out=225, in=135] (a0);
	[->] (a0) to [out=-45, in=225] (a1);
	[->] (a1) to [out=-45, in=225] (a2);
	[->] (a2) to [out=-45, in=225] (a3);
	[->] (a3) to [out=-45, in=225] (a4);
	[->] (a4) to [out=-45, in=225] (a5);
	[->] (a5) to [out=-45, in=225] (a6);
	[->] (a6) to [out=45, in=0] (b4);
        at (0.45,3.5) R_U;
        at (0.45,0.5) R_L;
        at (-0.5,2) p;
        [->] (3.5,2) arc (0:330:0.55);        
      

That means p carries the counter-clockwise orientation.The terms successor, predecessor and neighbor always refer to the cyclic order.   If necessary to avoid
con­fu­sion between this cyclic order and the total orders ≤_L and ≤_U, the latter two will be referred to as the native orderings.
With respect to the cyclic order it makes sense to speak of intervals ]α,β[_p, ]α,β]_p, [α,β[_p and [α,β]_p for any two points α,β∈ R_L∪ R_U of p with α≠β. Note that intervals of any kind are only defined for distinct limits.
  [0.25em]
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    [dotted] (-,+) –  (+4*+,+);
    [dotted] (-,) –  (+6*+,);
    [scale=0.4, circle, draw=lightgray, fill=lightgray] (a0) at (+0*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a1) at (+1*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a2) at (+2*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a3) at (+3*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a4) at (+4*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a5) at (+5*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a6) at (+6*,) ;
    [scale=0.4, circle, draw=lightgray, fill=lightgray] (b0) at (+0*,+) ;
    [scale=0.4, circle, draw=lightgray, fill=white] (b1) at (+1*,+) ;
    [scale=0.4, circle, draw=lightgray, fill=lightgray] (b2) at (+2*,+) ;
    [scale=0.4, circle, draw=black, fill=white] (b3) at (+3*,+) ;
    [scale=0.4, circle, draw=black, fill=black] (b4) at (+4*,+) ;
    [lightgray] (a0) – ++(0,) -| (a1);
    [lightgray] (a3) – ++(0,) -| (a5);
    [lightgray] (a4) – ++(0,2*) -| (a6);
    [lightgray] (a2) – (b2);
    [lightgray] (b0) – ++(0,-2*) -| (b3);
    [lightgray] (b1) – ++(0,-1*) -| (b4);
    [yshift=-0.5cm] at (a0) α;
    [yshift=0.5cm] at (b3) β;
    [dashed] (+6*+,-)  – (+-,-) – (+-,+) – (+6*+,+);
    [dashed] (+4*+,+-)  – (+3*-,+-) – (+3*-,++) – (+4*+,++);
    (lab) at (++1.5*,+0.7*) ]α,β]_p;
      [densely dotted] (lab.west) to (+5*,+);
      [densely dotted] (lab.west) to (+7*,);      
    
    
    We call a set S of points consecutive if S is empty, an interval or of the forms {α} or (R_L∪ R_U)\{α} for some point α.
    

 

 §.§ Ordered Tuples and Crossings

 We can extend the notion of intervals to tuples of more than two points:
 For n≥ 3 pairwise distinct points α_1,…,α_n in p∈ we say that the tuple (α_1,…,α_n) is ordered in p if for all i,j,k∈ with i,j,k≤ n and i<j the set ]α_i,α_j[_p contains α_k if and only if i<k<j. In fact, we can even talk about tuples of pairwise disjoint consecutive sets being ordered.
 We say that two distinct blocks B and B'  of p cross each other if there exist points α,β,α',β' in p such that the tuple (α,α',β,β') is ordered and such that α,β∈ B and α',β'∈ B'. 
 [0.25em]
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    7*
    0.5
    0.3
    0.3    
    [dotted] (-,) –  (+3*+,);
    [scale=0.4, circle, draw=black, fill=white] (a0) at (+0*,) ;
    [scale=0.4, circle, draw=darkgray, fill=darkgray] (a1) at (+1*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a2) at (+2*,) ;
    [scale=0.4, circle, draw=darkgray, fill=white] (a3) at (+3*,) ;
    (a0) – ++(0,) -| (a2);
    [gray] (a1) – ++(0,2*) -| (a3);
    [yshift=-0.5cm] at (a0) α;
    [yshift=-0.5cm] at (a1) α';
    [yshift=-0.5cm] at (a2) β;
    [yshift=-0.5cm] at (a3) β';
    [yshift=0.5cm,xshift=-0.5cm] at (a0) B;
    [yshift=1cm,xshift=0.5cm] at (a3) B';            


If p has no crossing blocks, we call p a non-crossing partition. The set of all non-crossing partitions is denoted by NC^∘∙.
    



 §.§ Normalized Color and Color Sum

Just like the cyclic order is often more convenient than the total orderings of the rows, it is useful to consider besides the original, native coloring of the points a second one:
By the normalized color of a point α in p∈ we mean simply its native color in case α is a lower point, but the inverse of its native color if α is an upper point.We call the signed measure σ_p on the set P_p of all points of p which assigns 1 to normalized ∘ and -1 to normalized ∙ the color sum of p.  
  Null sets of σ_p are also referred to as neutral.The color sum (p)σ_p(P_p) of the set P_p of all points of p is called the total color sum of p.



 §.§ Color Distance

Besides the color sum, a measure-like structure on the points of a partition, the coloring of the partition also induces a metric-like one: Given two points α and β  in p∈ we call
   
    δ_p(α,β)(p)   if α= β,
    σ_p(]α,β[_p)   if α≠β and α, β have different normalized colors,
    σ_p(]α,β]_p)   if α≠β and α, β have the same normalized color,

   the color distance from α to β in p. The map δ_p indeed has properties of a distance.

  
   Let  α, β and γ be points in p∈.
   
   
  *   δ_p(α,α)≡ 0 (p).
   
  *   δ_p(α,β)≡ -δ_p(β,α)(p).
   
  *   δ_p(α,γ)≡δ_p(α,β)+δ_p(β,γ)(p).
   
 
 
   
   
  * The first claim is part of the definition of δ_p.
   
  * We can assume α≠β. Rewrite the definition of δ_p as

    δ_p(α,β)=σ_p(]α,β]_p)+1/2(σ_p({α})-σ_p({β})).

Using σ_p(]α,β]_p)≡ -σ_p(]β,α]_p)(p) now proves the claim.

  * Again, we can suppose α,β and γ are pairwise different. Otherwise Parts <ref> and <ref> already prove the claim. Now compute, employing the formula for δ_p from the proof of Claim <ref>,
  
    δ_p(α,β)+δ_p(β,γ)   =σ_p(]α,β]_p)+σ_p(]β,γ]_p)
       =+1/2(σ_p({α})-σ_p({β}))+1/2(σ_p({β})-σ_p({γ})).

  From σ_p(]α,β]_p)+σ_p(]β,γ]_p)≡σ_p(]α,γ]_p)(p) now follows the claim.
 

 

§ BASIC DEFINITIONS: CATEGORIES OF PARTITIONS

 The definition of two-colored partitions recalled, we recapitulate the definitions of operations for partitions and of categories. Again, see <cit.> for more.


 §.§ Fundamental Operations on Partitions
 
  For all p,p'∈ we call the partition which is obtained by appending the rows of p' (left to right) to the right of the respective rows of p the tensor product p⊗ p' of (p,p'). Especially, we can write tensor powers like p^⊗ n given by p⊗…⊗ p with n factors. And we define p^⊗ 0∅.
    [1em]
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    [dotted] (-,) –  (+1*+,);
    [dotted] (-,+) –  (+1*+,+);    
    [scale=0.4, circle, draw=black, fill=black] (a0) at (+0*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a1) at (+1*,) ;
    [black] (a0) – ++(0,) -| (a1);
    at (0.5*,-) p_1;
  
  ⊗
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    [dotted] (-,+) –  (+4*+,+);
    [dotted] (-,) –  (+4*+,);
    [scale=0.4, circle, draw=black, fill=white] (a2) at (+2*,) ;
    [scale=0.4, circle, draw=black, fill=black] (b0) at (+0*,+) ;
    [scale=0.4, circle, draw=black, fill=white] (b1) at (+1*,+) ;
    [scale=0.4, circle, draw=black, fill=black] (b2) at (+2*,+) ;
    [scale=0.4, circle, draw=black, fill=white] (b3) at (+3*,+) ;
    [scale=0.4, circle, draw=black, fill=black] (b4) at (+4*,+) ;
    [black] (a2) – (b2);
    [black] (b0) – ++(0,-2*) -| (b3);
    [black] (b1) – ++(0,-1*) -| (b4);
    at (0.5*,-) p_2;    
  
  ⊗
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    0.5
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    [dotted] (-,+) –  (+3*+,+);
    [dotted] (-,) –  (+3*+,);
    [scale=0.4, circle, draw=black, fill=white] (a3) at (+0*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a4) at (+1*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a5) at (+2*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a6) at (+3*,) ;
    [black] (a3) – ++(0,) -| (a5);
    [black] (a4) – ++(0,2*) -| (a6);
        at (0.5*,-) p_3;
  
  =
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    0.5
    0.3
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    [dotted] (-,+) –  (+4*+,+);
    [dotted] (-,) –  (+6*+,);
    [scale=0.4, circle, draw=black, fill=black] (a0) at (+0*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a1) at (+1*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a2) at (+2*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a3) at (+3*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a4) at (+4*,) ;
    [scale=0.4, circle, draw=black, fill=black] (a5) at (+5*,) ;
    [scale=0.4, circle, draw=black, fill=white] (a6) at (+6*,) ;
    [scale=0.4, circle, draw=black, fill=black] (b0) at (+0*,+) ;
    [scale=0.4, circle, draw=black, fill=white] (b1) at (+1*,+) ;
    [scale=0.4, circle, draw=black, fill=black] (b2) at (+2*,+) ;
    [scale=0.4, circle, draw=black, fill=white] (b3) at (+3*,+) ;
    [scale=0.4, circle, draw=black, fill=black] (b4) at (+4*,+) ;
    [black] (a0) – ++(0,) -| (a1);
    [black] (a3) – ++(0,) -| (a5);
    [black] (a4) – ++(0,2*) -| (a6);
    [black] (a2) – (b2);
    [black] (b0) – ++(0,-2*) -| (b3);
    [black] (b1) – ++(0,-1*) -| (b4);
        at (0.5*,-) p_1⊗ p_2⊗ p_3;
  

The partition which is obtained from p by switching the roles of upper and lower row is called the involution p^* of p.
  [1em]
        [baseline=0.666*1.5cm]
    0.666
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    *0.25cm
    
    *0.5cm
    *1cm
    *0.3cm
    2*    
    3*
    *1cm
    *0.3cm
    3*
    
    whp/.style=circle, inner sep=0pt, text width=, draw=black, fill=white
    blp/.style=circle, inner sep=0pt, text width=, draw=black, fill=black
    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [whp] (l3) at (0+2*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    [->] (l1) –++ (0,);
    (l2) – (u2);
    (u1) –++(0,-) -|(u3);
    (l3) –++ (0,) -| (l4);
    at (-,0.5*) p;
  
    ↦  [scale=1,yscale=-1, baseline=-0.666*1.5cm]
    0.666
    *0.075cm
    *0.25cm
    
    *0.5cm
    *1cm
    *0.3cm
    2*    
    3*
    *1cm
    *0.3cm
    3*
    
    whp/.style=circle, inner sep=0pt, text width=, draw=black, fill=white
    blp/.style=circle, inner sep=0pt, text width=, draw=black, fill=black
    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [whp] (l3) at (0+2*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=] at (u1) ;
    [lk, yshift=] at (u2) ;
    [lk, yshift=] at (u3) ;    
    
    [->] (l1) –++ (0,);
    (l2) – (u2);
    (u1) –++(0,-) -|(u3);
    (l3) –++ (0,) -| (l4);
    at (+,0.5*) p^∗;    
  
  
  We say that the pairing (p,p') is composable if the upper row of p and the lower row of p' agree in size and coloration if compared according to their total orders both left to right.
  For all x∈{p,p'} let R_x,U denote the upper and R_x,L the lower row of x. If (p,p') is composable, then the composition pp' of (p,p') is defined as follows: Its lower row is given by R_p,L and its upper row by R_p',U. The blocks of p contained in R_p,L and those of p' contained in R_p',U are also blocks of pp'.  Once we identify the points of R_p,U and R_p',L we can form the join s of the partitions induced there by p and p'. For every block D of s the set
    
    (R_p,L∩⋃_B  block of  p
    B∩ D≠∅B)∪( R_p',U∩⋃_B'  block of  p'
    B'∩ D≠∅B')

    is a block of pp' unless it is empty.
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    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    [dotted] (0-,+) – (+,+);    
    
    [whp] (b1) at (0+0*,0+0*) ;
    [whp] (b2) at (0+1*,0+0*) ;
    [blp] (b3) at (0+2*,0+0*) ;
    [whp] (b4) at (0+3*,0+0*) ;
    [blp] (b5) at (0+4*,0+0*) ;
    [whp] (b6) at (0+5*,0+0*) ;    
    
    [blp] (m1) at (0+0*,0+1*) ;
    [whp] (m2) at (0+1*,0+1*) ;
    [whp] (m3) at (0+2*,0+1*) ;
    [blp] (m4) at (0+3*,0+1*) ;
    [whp] (m5) at (0+4*,0+1*) ;
    [blp] (m6) at (0+5*,0+1*) ;
    [blp] (m7) at (0+6*,0+1*) ;
    [whp] (m8) at (0+7*,0+1*) ;
    
    [blp] (t1) at (0+0*,0+1*+1*) ;
    [whp] (t2) at (0+1*,0+1*+1*) ;
    [whp] (t3) at (0+2*,0+1*+1*) ;
    [whp] (t4) at (0+3*,0+1*+1*) ;
    [blp] (t5) at (0+4*,0+1*+1*) ;
    
    [lk, yshift=-] at (m1) ;
    [lk, yshift=-] at (m3) ;
    [lk, yshift=-] at (m4) ;
    [lk, yshift=-] at (t1) ;
    [lk, yshift=-] at (t2) ;
    [lk, yshift=-] at (t3) ;        
    
    [->] (b4) –++ (0,);
    [->] (m1) –++ (0,);
    [->] (t4) –++ (0,-);        
    (b2) – (m2);
    (b5) – (m5);
    (b6) – (m6);
    (m5) – (t5);    
    (b1) –++(0,) -| (b3);
    (m1) –++(0,-) -| (m3);
    (m3) –++(0,-) -| (m4);
    (m7) –++(0,-) -| (m8);        
    (m2) –++(0,) -| (m4);
    (m3) –++(0,2*) -| (m6);
    (m7) –++(0,) -| (m8);
    (t1) –++(0,-) -| (t2);
    (t2) –++(0,-) -| (t3);
    at (-,+0.5*) p';
    at (-,0.5*) p;    
  
  =
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    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (b1) at (0+0*,0+0*) ;
    [whp] (b2) at (0+1*,0+0*) ;
    [blp] (b3) at (0+2*,0+0*) ;
    [whp] (b4) at (0+3*,0+0*) ;
    [blp] (b5) at (0+4*,0+0*) ;
    [whp] (b6) at (0+5*,0+0*) ;    
    
    [blp] (m1) at (0+0*,0+1*) ;
    [whp] (m2) at (0+1*,0+1*) ;
    [whp] (m3) at (0+2*,0+1*) ;
    [blp] (m4) at (0+3*,0+1*) ;
    [whp] (m5) at (0+4*,0+1*) ;
    
    [lk, yshift=-] at (m1) ;
    [lk, yshift=-] at (m2) ;
    [lk, yshift=-] at (m3) ;
    
    [->] (b4) –++ (0,);
    [->] (m4) –++ (0,-);        
    (b5) – (m5);
    (b1) –++(0,) -| (b3);
    (b2) –++(0,2*) -| (b6);
    (m1) –++(0,-) -| (m2);
    (m2) –++(0,-) -| (m3);
    at (+,0.5*) pp';
  
  




 §.§ Categories of Two-Colored Partitions

We call a set C⊆ a category if it contains the partitions ∅, , ,  and  and is closed under tensor products, involution and composition of composable pairings (<cit.>, building on <cit.>). The set of all categories of two-colored partitions is denoted by .
For every set G⊆ we  write ⟨ G⟩ for the smallest category (with respect to ⊆) which contains G. We say that G generates ⟨ G⟩. If G={p} for some p∈, we slightly abuse notation by writing ⟨ p⟩ instead of ⟨{p}⟩ for ⟨ G⟩. Also, we mix the two, writing, ⟨ G,q⟩ for q∈ instead of ⟨ G∪{p}⟩.

<cit.>
  
  Let C⊆  be a category. We say  C is
  
  
  * …case O if ∉ C and ∉ C,
  
  * …case B if ∈ C and ∉ C,
      
  * …case S if ∈ C and ∈ C,
  
  * …case H if ∉ C and ∈ C.
  

In this series of articles, we are only interested in the first three cases; Compare with the classification in the uncolored case, where also the corresponding Case H, i.e. ∉ C or ∈ C for categories C⊆P, is  most complex (<cit.> and <cit.>).

   Let C⊆ be a category.
   
     
  
  *   Equivalently to saying C is case H, we also call C hy­per­oc­ta­he­dral.
  
  *  Hence, if instead C is case O, B or S, we say that C is non-hy­per­oc­ta­he­dral.
    
  *  The set of all non-hyperoctahedral categories is denoted by .
  





 §.§ Composite Category Operations

The basic category operations can be utilized to construct further generic transformations.
Categories are closed under four basic rotations: We obtain the partition p^ from p∈ by removing the leftmost upper point α and adding a new point β to the left of the leftmost lower point, assigning to β the inverse color of α and also replacing α by β as far as the blocks are concerned. We say that α is rotated down. Similarly, we can rotate down the rightmost upper point by inverting its color and appending it to the lower row, producing p^. And the rotations p^ and p^ result form reversing these procedures and rotating up the leftmost respectively rightmost point. We call p^↻ (p^)^  the clockwise and  p^↺ (p^)^ the counter-clockwise cyclic rotation of p.
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    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [blp] (l3) at (0+2*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    [->] (l1) –++ (0,);
    (l2) – (u2);
    (u1) –++(0,-) -|(u3);
    (l3) –++ (0,) -| (l4);
    
    at (-,0.5*) p;
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    [dotted] (--,0) – (+,0);
    [dotted] (--,) – (+,);
    
    [blp] (l0) at (0-1*,0+0*) ;    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [blp] (l3) at (0+2*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=2*] at (l0) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    [->] (l1) –++ (0,);
    (l2) – (u2);
    (l0) –++(0,2*) -|(u3);
    (l3) –++ (0,) -| (l4);
    
    at (+,0.5*) p^;
    
  
From p we obtain the reflection p̂ by reversing the native total orders on both rows. The color inversion p of p is constructed by inverting the native coloring.
And the verticolor reflection p̃ is the color inversion of the reflection of p. Categories are closed under verticolor reflection but generally neither under reflection nor color inversion. 
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [blp] (l3) at (0+2*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    [->] (l1) –++ (0,);
    (l2) – (u2);
    (u1) –++(0,-) -|(u3);
    (l3) –++ (0,) -| (l4);
    
    at (-,0.5*) p;    
  
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [whp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [blp] (l3) at (0+2*,0+0*) ;
    [whp] (l4) at (0+3*,0+0*) ;
    
    [blp] (u1) at (0+1*,0+1*) ;
    [whp] (u2) at (0+2*,0+1*) ;
    [blp] (u3) at (0+3*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    [->] (l4) –++ (0,);
    (l3) – (u2);
    (u1) –++(0,-) -|(u3);
    (l1) –++ (0,) -| (l2);
    
    at (+,0.5*) p̃;    
  
  

 Lastly, categories are closed under erasing turns: A turn is a consecutive neutral set of size two. The erasing of a set S of points from p is the partition E(p,S) obtained by removing S and combining all the points whose blocks had a non-empty intersection with S into one block.[0.5em]
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [whp] (l2) at (0+1*,0+0*) ;
    [blp] (l3) at (0+2*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;    
    
    [->] (l1) –++ (0,);
    (l2) – (u2);
    (u1) –++(0,-) -|(u3);
    (l3) –++ (0,) -| (l4);
    
    [densely dashed] ((l3)+(,)) |- ((l2)+(-,-)) |- cycle;
    at (0.5*,-) S;
    at (-,0.5*) p;    
  
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    
    [blp] (l1) at (0+0*,0+0*) ;
    [blp] (l4) at (0+3*,0+0*) ;
    
    [whp] (u1) at (0+0*,0+1*) ;
    [blp] (u2) at (0+1*,0+1*) ;
    [whp] (u3) at (0+2*,0+1*) ;
    
    [lk, yshift=-] at (u1) ;
    [lk, yshift=-] at (u2) ;
    [lk, yshift=-] at (u3) ;
    [lk, yshift=2*] at (l4) ;        
    
    [->] (l1) –++ (0,);
    (u1) –++(0,-) -|(u2);
    (u2) –++(0,-) -|(u3);
    (u3) –++(0,-) -|(l4);    
    
    at (+2*,0.5*) E(p,S);    
    
  





 §.§ Alternative Characterization of Categories

Using the composite operations, we can give a helpful characterization of categories based on the idea in the proof of <cit.>.
 

  
  A set C⊆ is a category of two-colored partitions if and only if
  
  
  *  ∈ C and
  
  *  C is closed under tensor products, basic rotations, verticolor reflection and  erasing turns.
  
   
    
    
      Suppose C satisfies <ref> and  <ref>.
      Since ^=, since ^= and since ^= and because C is closed under rotations, we find ,,∈ C. Erasing the only turn in ∈ C, under which C is invariant, produces  ∅∈ C.
      For every partition p∈ C with k upper and m lower points, the identity p^∗= ((p̃)^ k)^ l and the assumptions that C is stable under rotations and verticolor reflection proves that p^∗∈ C. Hence, C is also involution-invariant.
      Lastly, suppose that (p,q) is a composable pairing from C. We want to show  r pq∈ C.  Let q have k upper and l lower and let p have l upper and m lower points. Since C is closed under rotations, it suffices to prove  r^ m∈ C. Let (c_1,…,c_l) for c_1,…,c_l∈ be the coloring of the lower row of q left to right. Let s be the tensor product of partitions from {,} with lower row of coloring (c_1,…,c_l). Then, (p,s) and (s,q) are composable and psq=pq=r. The diagram below illustrates that the pairing (s^ l,q⊗ ((p^ l)^ m)) is composable as well and that its composition yields the partition r^ m.
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    [dotted] (0-,-) – (+,-);
    
    (u1) at (0*,1*);
    (u2) at (2*,1*);
    (m1) at (0*,0*);
    (m2) at (2*,0*);
    (m3) at (3*,0*);
    (l1) at (0*,-1*);
    (l2) at (4*,-1*);    
    
    [fill=white] ((u1)+(-,)) – ((m1)+(-,-)) – ((m3)+(,-)) –++ (0,0.5*) -| ((u2)+(,)) – cycle;
    [fill=gray] ((m1)+(-,)) – ((l1)+(-,-)) – ((l2)+(,-)) –++ (0,0.5*) -| ((m3)+(,)) – cycle;
    [fill=lightgray] ((m1)+(-,-)) rectangle ((m3)+(,));
    
    (m1)– node[pos=0.5] …  (m2);
    
    at (m1) c_1;
    at (m2) c_l-1;
    at (m3) c_l;
    
    at (1*,0.5*) q;
    at (2*,-0.5*) p;    
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    [dotted] (0-,-) – (+,-);
    [dotted] (0-,-2*) – (+,-2*);    
    
    (u1) at (0*,1*);
    (u2) at (2*,1*);
    (m1) at (0*,0*);
    (m2) at (2*,0*);
    (m3) at (3*,0*);
    (l1) at (0*,-1*);
    (l2) at (2*,-1*);
    (l3) at (3*,-1*);
    (b1) at (0*,-2*);
    (b2) at (4*,-2*);    
    
    
    (l1.north) – (m1.south);
    (l2.north) – (m2.south);
    (l3.north) – (m3.south);
    
    [fill=white] ((u1)+(-,)) – ((m1)+(-,-)) – ((m3)+(,-)) –++ (0,0.5*) -| ((u2)+(,)) – cycle;
    [fill=gray] ((l1)+(-,)) – ((b1)+(-,-)) – ((b2)+(,-)) –++ (0,0.5*) -| ((l3)+(,)) – cycle;
    
    (m1)– node[pos=0.5] …  (m2);
    (l1)– node[pos=0.5] …  (l2);
    ((l1)+(0,0.333*)) – node[pos=0.5] …  ((l2)+(0,0.333*));
    
    at (m1) c_1;
    at (m2) c_l-1;
    at (m3) c_l;
    at (l1) c_1;
    at (l2) c_l-1;
    at (l3) c_l;    
    
    at (1*,0.5*) q;
    at (2*,-1.5*) p;
    at (1*,-0.333*) s;
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    [dotted] (0-,0) – (+,0);
    [dotted] (0-,) – (+,);
    [dotted] (0-,-) – (+,-);
    
    (u1) at (0*,1*);
    (u2) at (2*,1*);
    (u3) at (7*,1*);
    (m1) at (0*,0*);
    (m2) at (2*,0*);
    (m3) at (3*,0*);
    (m4) at (4*,0*);    
    (m5) at (5*,0*);
    (m6) at (7*,0*);        
    
    
    (m1.south) – ++ (0,--2*) -| (m6.south);
    (m2.south) – ++ (0,--1*) -| (m5.south);
    (m3.south) – ++ (0,--0.5*) -| (m4.south);
    
    [fill=white] ((u1)+(-,)) – ((m1)+(-,-)) – ((m3)+(,-)) – ++ (0,0.5*) -| ((u2)+(,)) – cycle;
    [fill=gray] ((m4)+(-,-)) – ((m6)+(,-)) – ((u3)+(,)) – ((u2)+(-,)) – ++ (0,-0.5*) -| cycle;
    
    (m1)– node[pos=0.5] …  (m2);
    (m5)– node[pos=0.5] …  (m6);
    
    at (m1) c_1;
    at (m2) c_l-1;
    at (m3) c_l;
    at (m4) c_l;
    at (m5) c_l-1;
    at (m6) c_1;    
    
    at (1*,0.5*) q;
    at (5.5*,0.5*) (p^ l)^ m;
    at (1*,-1*) s^ l;
    (0.5*,--)– node[pos=0.5] ⋮  (0.5*,--2*);    
    


      Our assumptions guarantee e_0 q⊗ ((p^ l)^ m)∈ C. Because C is assumed invariant under erasing  turns, if we define the turn T_0{l,(l+1)} in e_0∈ C and then for every j=1,…,l-1 the turn T_j{(l-j),(l-j+1)} in e_j E(e_j-1, T_j-1)∈ C, then the partition e_l E(e_l-1, T_l-1)∈ C is identical with r^ m as the diagram below shows.
      [1em]
              [baseline=0.644*-1cm]
    0.644
    *0.075cm
    *0.25cm
    
    *0.5cm
    *1cm
    *0.3cm
    10*    
    10*
    *1cm
    *0.3cm
    1.5*
    
    whp/.style=circle, inner sep=0pt, text width=, draw=black, fill=white
    blp/.style=circle, inner sep=0pt, text width=, draw=black, fill=black
    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (+0-,2*) – (++,2*);
    [dotted] (+0-,1*) – (++,1*);    
    [dotted] (+0-,0*) – (++,0*);
    [dotted] (+0-,-1*) – (++,-1*);
    [dotted] (+0-,-2*) – (++,-2*);
    [dotted] (+0-,-3*) – (++,-3*);   
    
    (t1) at (1*,2*);
    (t2) at (2*,2*);
    (t3) at (3.5*,2*);
    (t4) at (4.5*,2*);
    (t5) at (5.5*,2*);
    (t6) at (6.5*,2*);
    (t7) at (7.5*,2*);
    (t8) at (8.5*,2*);
    (t9) at (10*,2*);
    (t10) at (11*,2*);
    (u1) at (1*,1*);
    (u2) at (2*,1*);
    (u3) at (3.5*,1*);
    (u4) at (4.5*,1*);
    (u5) at (5.5*,1*);
    (u6) at (6.5*,1*);
    (u7) at (7.5*,1*);
    (u8) at (8.5*,1*);
    (u9) at (10*,1*);
    (u10) at (11*,1*);
    (m1) at (1*,0*);
    (m2) at (2*,0*);
    (m3) at (3.5*,0*);
    (m4) at (4.5*,0*);
    (m5) at (5.5*,0*);
    (m6) at (6.5*,0*);
    (m7) at (7.5*,0*);
    (m8) at (8.5*,0*);
    (m9) at (10*,0*);
    (m10) at (11*,0*);
    (l1) at (1*,-1*);
    (l2) at (2*,-1*);
    (l3) at (3.5*,-1*);
    (l4) at (4.5*,-1*);
    (l5) at (5.5*,-1*);
    (l6) at (6.5*,-1*);
    (l7) at (7.5*,-1*);
    (l8) at (8.5*,-1*);
    (l9) at (10*,-1*);
    (l10) at (11*,-1*);
    (b1) at (1*,-2*);
    (b2) at (2*,-2*);
    (b3) at (3.5*,-2*);
    (b4) at (4.5*,-2*);
    (b5) at (5.5*,-2*);
    (b6) at (6.5*,-2*);
    (b7) at (7.5*,-2*);
    (b8) at (8.5*,-2*);
    (b9) at (10*,-2*);
    (b10) at (11*,-2*);
    
    ((t2)+(0,-0.5*))– node[pos=0.5] … ((t3)+(0,-0.5*));
    ((u2)+(0,-0.5*))– node[pos=0.5] … ((u3)+(0,-0.5*));
    ((t8)+(0,-0.5*))– node[pos=0.5] … ((t9)+(0,-0.5*));
    ((u8)+(0,-0.5*))– node[pos=0.5] … ((u9)+(0,-0.5*));
    (+0.5*,0*) – node[pos=0.5] ⋮ (+0.5*,-1*);
    
    (b1) – (l1);
    (b10) – (l10);
    (m1) – (t1);
    (m10) – (t10);    
    (m1) – (t1);
    (m2) – (t2);
    (m3) – (t3);
    (m8) – (t8);    
    (m9) – (t9);    
    (m10) – (t10);
    (u4) – (t4);
    (u7) – (t7);
    (t5) –++(0,-) -| (t6);            
    (u4) –++(0,-) -| (u7);        
    (l2) –++(0,-) -| (l9);    
    (b1) –++(0,-) -| (b10);
    
    [fill=white, inner sep=0cm] at (t1) c_1;
    [fill=white, inner sep=0cm] at (t2) c_2;
    [fill=white, inner sep=0cm] at (t3) c_l-2;
    [fill=white, inner sep=0cm] at (t4) c_l-1;
    [fill=white, inner sep=0cm] at (t5) c_l;
    [fill=white, inner sep=0cm] at (t10) c_1;
    [fill=white, inner sep=0cm] at (t9) c_2;
    [fill=white, inner sep=0cm] at (t8) c_l-2;
    [fill=white, inner sep=0cm] at (t7) c_l-1;
    [fill=white, inner sep=0cm] at (t6) c_l;
    [fill=white, inner sep=0cm] at (u1) c_1;
    [fill=white, inner sep=0cm] at (u2) c_2;
    [fill=white, inner sep=0cm] at (u3) c_l-2;
    [fill=white, inner sep=0cm] at (u4) c_l-1;
    [fill=white, inner sep=0cm] at (u10) c_1;
    [fill=white, inner sep=0cm] at (u9) c_2;
    [fill=white, inner sep=0cm] at (u8) c_l-2;
    [fill=white, inner sep=0cm] at (u7) c_l-1;
    [fill=white, inner sep=0cm] at (m1) c_1;
    [fill=white, inner sep=0cm] at (m2) c_2;
    [fill=white, inner sep=0cm] at (m3) c_l-2;
    [fill=white, inner sep=0cm] at (m10) c_1;
    [fill=white, inner sep=0cm] at (m9) c_2;
    [fill=white, inner sep=0cm] at (m8) c_l-2;
    [fill=white, inner sep=0cm] at (l1) c_1;
    [fill=white, inner sep=0cm] at (l2) c_2;
    [fill=white, inner sep=0cm] at (l10) c_1;
    [fill=white, inner sep=0cm] at (l9) c_2;
    [fill=white, inner sep=0cm] at (b1) c_1;
    [fill=white, inner sep=0cm] at (b10) c_1;
  
  =
              [baseline=0.644*0cm]
    0.644
    *0.075cm
    *0.25cm
    
    *0.5cm
    *1cm
    *0.3cm
    10*    
    10*
    *1cm
    *0.3cm
    2*
    
    whp/.style=circle, inner sep=0pt, text width=, draw=black, fill=white
    blp/.style=circle, inner sep=0pt, text width=, draw=black, fill=black
    lk/.style=regular polygon, regular polygon sides=4, inner sep=0pt, text width=, draw=black, fill=black
    
    
    [dotted] (+0-,2*) – (++,2*);
    [dotted] (+0-,0*) – (++,0*);
    
    (t1) at (1*,2*);
    (t2) at (2*,2*);
    (t3) at (3.5*,2*);
    (t4) at (4.5*,2*);
    (t5) at (5.5*,2*);
    (t6) at (6.5*,2*);
    (t7) at (7.5*,2*);
    (t8) at (8.5*,2*);
    (t9) at (10*,2*);
    (t10) at (11*,2*);            
    
    ((t2)+(0,-0.5*))– node[pos=0.5] … ((t3)+(0,-0.5*));
    ((t8)+(0,-0.5*))– node[pos=0.5] … ((t9)+(0,-0.5*));
    (+0.5*,2*) – node[pos=0.5] ⋮ (+0.5*,0*);
    
    (t5) –++(0,-1*0.5*) -| (t6);
    (t4) –++(0,-2*0.5*) -| (t7);
    (t3) –++(0,-3*0.5*) -| (t8);
    (t2) –++(0,-5*0.5*) -| (t9);
    (t1) –++(0,-6*0.5*) -| (t10);    
    
    [fill=white, inner sep=0cm] at (t1) c_1;
    [fill=white, inner sep=0cm] at (t2) c_2;
    [fill=white, inner sep=0cm] at (t3) c_l-2;
    [fill=white, inner sep=0cm] at (t4) c_l-1;
    [fill=white, inner sep=0cm] at (t5) c_l;
    [fill=white, inner sep=0cm] at (t10) c_1;
    [fill=white, inner sep=0cm] at (t9) c_2;
    [fill=white, inner sep=0cm] at (t8) c_l-2;
    [fill=white, inner sep=0cm] at (t7) c_l-1;
    [fill=white, inner sep=0cm] at (t6) c_l;
  
  
      Thus we have deduced r^ m∈ C as claimed. This proves that C is closed under composition of composable pairs and thus a category. 
    



§ THE SETS R(Q)



In the following, we will define in several steps an index set 𝖰 and for each Q∈𝖰  a set R_Q⊆ of partitions. The aim will be to show that each of these constitutes a non-hyperoctahedral category (see Theorem <ref>, the main result of this article). Auxiliary objects 𝖫 and Z aid in defining 𝖰 and ( R_Q)_Q∈𝖰.
        
          
For every set S denote its power set by 𝔓(S).
        
        
          
          We define the parameter domain 𝖫 as the six-fold Cartesian product of 𝔓(): 
    𝖫𝔓()×𝔓()×𝔓()×𝔓()×𝔓()×𝔓().

        
        
          
  Define the analyzer  Z:  𝔓()→𝖫 by
        
    Z (  F,  V, ,   L,   K,   X )

where, for all S⊆,


  * F( S){  |B| | p∈ S,  B block of  p} is the set of block sizes,

  * V( S){ σ_p(B)| p∈ S,  B block of p} is the set of block color sums,

  * ( S){ (p)| p∈ S} is the set of total color sums,

  * [t]
                L( S){ δ_p(α_1,α_2)|     p∈ S,   B block of p, α_1,α_2∈ B, α_1≠α_2,
     ]α_1,α_2[_p∩ B=∅, σ_p({α_1,α_2})≠ 0} 

is the set of color distances between any two subsequent legs of the same block having the same normalized color,

  * [t]
    K( S){ δ_p(α_1,α_2)|     p∈ S,   B block of p, α_1,α_2∈ B, α_1≠α_2,
     ]α_1,α_2[_p∩ B=∅, σ_p({α_1,α_2})= 0}

is the set of color distances between any two subsequent legs of the same block having different normalized colors and

  * [t]
    X( S){ δ_p(α_1,α_2) |     p∈ S,  B_1,B_2 blocks of p,   B_1 crosses  B_2,
    α_1∈ B_1, α_2∈ B_2} 

is the set of color distances between any two legs belonging to two crossing blocks.

        
        The parameter domain 𝖫 and the analyzer Z allow us to now define the following map which will later induce the announced family ( R_Q)_Q∈𝖰.

        
Given a family (S_i)_i∈ I of sets, we write ≤ for the product order on the Cartesian product  _i∈ I𝔓(S_i) induced by the partial orders ⊆ on the factors.
        

  
        Define the parametrization as the mapping
        
    R: 𝖫→𝔓(),  L ↦ R_L{p∈| Z({p})≤ L}.

      
      With R we can single out sets of partitions by placing restrictions  on the six aggregated combinatorial features of partitions listed above.
      
     
     
            
  *  For all x,y∈ and A,B⊆ write
         
    xA+yB{xa+yb| a∈ A,  b∈ B}.

         Moreover, put then xA-yB xA+(-y)B. Per A={1} expressions like x+yB  are defined as well, and per x=1 so are such like A+yB. 
       
  *  Let  ± S S∪(-S) for all sets S⊆.
       
  *  For all m∈ and D⊆ define
         
    D_m (D∪(m- D))+m  and   D_m' (D∪(m-D)∪{0})+m.

                
  *    Use the abbreviations 0∅ and k{1,…,k} for all k∈.
     
   

   
     
    Define the parameter range 𝖰 as the subset of 𝖫 comprising exactly all tuples
    
    (f,v,s,l,k,x)

    listed below, where  u∈{0}∪, m∈, D⊆{0}∪⌊m/2⌋, where E⊆{0}∪ and where N is a subsemigroup of (,+):
    f   v   s   l   k    x  
    
    {2}   ±{0, 2}    2um    m    m   
    {2}   ±{0, 2}    2um    m+2m    2m   
    {2}   ±{0, 2}    2um    m+2m    2m   \ m
    {2}   {0}   {0}   ∅    m   
    {2}   ±{0, 2}   {0}   {0}   {0}   \ N_0 
    {2}   {0}   {0}   ∅   {0}   \ N_0 
    {2}   {0}   {0}   ∅   {0}   \ N_0' 
    {1,2}   ±{0, 1, 2}    um    m    m   \ D_m
    {1,2}   ±{0, 1, 2}    2um    m+2m    2m   \ D_m
    {1,2}   ±{0, 1}    um   ∅    m   \ D_m 
    {1,2}   ±{0, 1, 2}   {0}   {0}   {0}   \ E_0 
    {1,2}   ±{0,  1}   {0}   ∅   {0}   \ E_0
           um    m    m   \ D_m
          {0}   {0}   {0}   \ E_0

                  
With 𝖰 and R defined, so has been the family ( R_Q)_Q∈𝖰. The characterizing conditions Z({ · })≤ Q of the sets R_Q, Q∈𝖰, will be successively explained in Section <ref> in the process of proving their invariance under the category operations.





§ INVARIANCE OF R(Q) UNDER CATEGORY OPERATIONS


The strategy for proving that the sets defined in the preceding Section <ref> are actually categories of partitions is the following: We choose the most convenient elements Q of 𝖰, the ones for which it is easiest to prove that R_Q is a category. Once we have verified that these sets R_Q are categories, we show that every other Q∈𝖰 can be written as a meet of a suitable family 𝖰'⊆𝖰 of convenient ones  in the complete lattice 𝖫. Then, the following Lemma <ref> will allow us to conclude that R_Q is a category for every Q∈𝖰.

 Given a family (S_i)_i∈ I of sets, we use the symbols ⋂_× for the meet and ⋃_× for the join operator of the product order ≤ with respect to ⊆ on  _i∈ I𝔓(S_i).

      
        
        The mapping R: 𝖫→𝔓() is monotonic and preserves meets.
      

      
 For L,L'∈𝖫 with L≤ L' and p∈ the condition p∈ R_L, i.e. Z({p})≤ L, implies Z({p})≤ L' and thus p∈ R_L'. Hence, R is monotonic. And for all subsets 𝖫'⊆𝖫 holds
        
    ⋂{ R_L| L∈𝖫'}   =⋂{{p∈| Z({p})≤ L}| L∈𝖫'}
       = {p∈|∀ L∈𝖫': Z({p})≤ L}
       = {p∈|  Z({p})≤⋂_×𝖫'}
       = R_⋂_×𝖫',

        where we have only used the definition of R. 

      

 §.§ Behavior of Z under Category Operations

      In order to show that R_Q is a category for the convenient values Q ∈𝖰, we will use the Alternative Characterization (Lemma <ref>). It  pays to consider abstractly how Z behaves under the (alternative) category operations beforehand.
      

  §.§.§ Behavior of Z under Rotation


      
  For all r∈{,,,} and p∈ C holds
  
    Z({p^r})=Z({p}).



  Let ρ be the map that rotates the points of p to those of p^r.
  Step 1: Relating σ_p^r to σ_p.  Because the normalized color is defined specially to compensate for the color change involved in rotations,
  
    σ_p^r(S)=σ_p(ρ^-1(S))

  for all sets S of points of p^r.
  Step 2: Relating δ_p^r to δ_p. The cyclic order is also defined exactly to be respected by rotations:
  For all points α and β of p^r, the identity  ]α,β]_p^r=ρ(]ρ^-1(α),ρ^-1(β)]_p), Equation (<ref>) and the definition of δ_p imply
  
    δ_p^r(α,β)   =σ_p^r(]α,β]_p^r)+1/2(σ_p^r({α})-σ_p^r({β}))
       =σ_p(]ρ^-1(α),ρ^-1(β)]_p)+1/2(σ_p(ρ^-1({α}))-σ_p(ρ^-1{β}))
       =δ_p(ρ^-1(α),ρ^-1(β)).

  
  
  
  
  
  Step 3: Implications for Z. By definition, the blocks of p^r are precisely the sets ρ(B) for blocks B of p. Hence, F({p^r})=F({p}) as ρ is bijective. But also therefore, V({p^r})=V({p}) by Equation (<ref>). The same equation also shows (p^r)=(p) by choosing for S the total set of points.
  Not only does ρ map blocks to blocks but also subsequent legs to subsequent legs because rotation respects the cyclic order. And this is how all pairs of subsequent legs in p^r arise. And Equation (<ref>) shows that two subsequent legs α and β in p^r satisfy σ_p^r({α,β})=0 if and only if their preimages, subsequent legs in p, satisfy σ_p({ρ^-1(α),ρ^-1(β)})=0. Hence, Equation (<ref>) proves  (L,K)({p^r})=(L,K)({p}).
  Since ρ maps crossing blocks to crossing blocks and since all crossings of p^r come from crossings in p, Equation (<ref>) also verifies X({p^r})=X({p}). Thus, Z({p^r})=Z({p}) as asserted.


      

  §.§.§ Behavior of Z under Verticolor Reflection


  
  For all p∈ C holds
  
    Z({p̃})=(F,-V,-,-L,-K,-X)({p}).



  Same procedure as in Lemma <ref>. Let κ be the map that reflects the points of p.
  Step 1: Relating σ_p̃ to σ_p.  Reflection only moves points and does not change their native colors. As it sends upper points to upper points and lower points to lower points, the normalized colors are unaffected too. Verticolor reflection however involves not only reflection but also an inversion of (native and thus also normalized) color. We hence find for all sets of S of points of p̃:
  
    σ_p^r(S)=-σ_p̂(S)=-σ_p(κ^-1(S)).

  Step 2: Relating δ_p̃ to δ_p.
The reflection operation reverses the cyclic order. Let α and β be arbitrary points of p̃. Then, ]α,β]_p̃=]α,β]_p̂=κ([κ^-1(β),κ^-1(α)[_p). Note that the left-open interval has become a right-open one. Hence, by definition of δ_p and by Equation (<ref>),

    δ_p̃(α,β)   =σ_p̃(]α,β]_p̃)+1/2(σ_p̃({α})-σ_p̃({β}))
       =-σ_p([κ^-1(β),κ^-1(α)[_p)-1/2(σ_p({κ^-1(α)})-σ_p({κ^-1(β)}))
       =-σ_p(]κ^-1(β),κ^-1(α)]_p)-1/2(σ_p({κ^-1(β)})-σ_p({κ^-1(α)}))
       =-δ_p(κ^-1(β),κ^-1(α)).

  In short, δ_p̃ is is given by flipping the arguments in δ_p and inverting the sign.
  Step 3: Implications for Z. Still, the blocks of p̃ are exactly the sets κ(B) for blocks of B of p. Thus, F({p̃})=F({p}) is true. Equation (<ref>) hence shows  V({p̃})=-V({p}). It also proves (p̃)=-(p), again by choosing for S the total set of points.
  Given points α and β in p', Equation (<ref>) shows σ_p̃({α,β})=0 if and only if σ_p({κ^-1(α),κ^-1(β)})=0. Moreover, since κ inverts the cyclic order, if α and β belong to the same block B of p̃, then  ]α,β[_p̃∩ B=κ(]κ^-1(β),κ^-1(α)[_p∩κ^-1(B)), where κ^-1(α) and κ^-1(β) are now legs of the block κ^-1(B) of p. That means, κ maps subsequent legs to subsequent legs, it however reverses the order in which they are subsequent. And all pairs of subsequent legs arise in this way. Since the exchange of the order is compensated for by the flip of the arguments in Equation (<ref>), it follows (L,K)({p̃})=(-L,-K)({p}).
  And Equation (<ref>) also immediately proves X({p̃})=- X({p}). Here the flip is irrelevant since being crossing is a symmetric relation on the set of blocks. That is all we needed to see.


      

  §.§.§ Behavior of Z under Tensor Products


  
  Let p,p'∈ C be arbitrary.
  
  
  *  F({p⊗ p'})= F({p,p'}).
  
  *  V({p⊗ p'})= V({p,p'}).
  
  *    (p⊗ p')=(p)+(p').
  
  * 
    Y({p⊗ p'})⊆ Y({p,p'})+((p),(p')) for all Y∈{L,K,X}.
  


  One last time we proceed as in Lemmata <ref> and <ref>.
  Let S_p and S_p' be the sets of points in p⊗ p' stemming from p and p' respectively, and let τ denote the map translating the points of p' to their locations in S_p'.
  Step 1: Relating σ_p⊗ p' to σ_p and σ_p'. The tensor product leaves S_p untouched, it only involves a shift of τ^-1(S_p') to S_p'. The native colors are not affected. Since τ maps upper points to upper points and lower points to lower points, the normalized colors do not  change either. We will be able to confine our considerations to sets S of points with S⊆ S_p or S⊆ S_p', for which then holds in conclusion:
  
    σ_p⊗ p'(S)   =
                                 σ_p(S)   if  S⊆ S_p,
    σ_p'(τ^-1(S))   if S⊆ S_p'.
  
  Step 2: Relating δ_p⊗ p' to δ_p and δ_p'. 
  Again, we are only interested in the color distances of points α and β which both belong to S_p or both to S_p'. Then,
  
    ]α,β]_p⊗ p'   =
                                      
                                        ]α,β]_p∪ S_p'   if α,β∈ S_p and ]α,β[_p⊗ p'∩ S_p'≠∅,
    
                                        ]α,β]_p   if α,β∈ S_p and ]α,β[_p⊗ p'∩ S_p'=∅,
    τ(]τ^-1(α),τ^-1(β)]_p')∪ S_p   if α,β∈ S_p' and ]α,β[_p⊗ p'∩ S_p≠∅,
    τ(]τ^-1(α),τ^-1(β)]_p')   if α,β∈ S_p' and ]α,β[_p⊗ p'∩ S_p= ∅.

  Hence, (p)=σ_p(S_p) and (p')=σ_p'(τ^-1(S_p')), Equation (<ref>) and the definition of δ_p⊗ p' yield in these cases
  
    δ_p⊗ p'(α,β)   =σ_p⊗ p'(]α,β]_p⊗ p')+1/2(σ_p⊗ p'({α})-σ_p⊗ p'({β}))
       =                                                              δ_p(α,β)+(p')   if α,β∈ S_p and ]α,β[_p⊗ p'∩ S_p'≠∅,
    δ_p(α,β)   if α,β∈ S_p and ]α,β[_p⊗ p'∩ S_p'=∅,
    δ_p'(τ^-1(α),τ^-1(β))+(p)   if α,β∈ S_p' and ]α,β[_p⊗ p'∩ S_p≠∅,
    δ_p'(τ^-1(α),τ^-1(β))   if α,β∈ S_p' and ]α,β[_p⊗ p'∩ S_p= ∅.

  In particular, it follows
  
    δ_p⊗ p'(α,β)≡δ_p(α,β)   if α,β∈ S_p
    δ_p'(τ^-1(α),τ^-1(β))    if α,β∈ S_p'((p),(p')).

  Step 3: Implications for Z. The blocks of p⊗ p' are by definition precisely the blocks of p plus the sets τ(B') for blocks B' of p'. This proves Part <ref>. Especially, B⊆ S_p or B⊆ S_p' for all blocks B of p⊗ p'. Hence, Equation (<ref>) shows Part <ref>. The same equation shows (p⊗ p')=σ_p⊗ p'(S_p∪ S_p')=σ_p(S_p)+σ_p'(τ^-1(S_p'))=(p)+(p') and thus Claim <ref>.
  Since every block of B of p' satisfies B⊆ S_p or B⊆ S_p' any two subsequent legs of a block B of p' are also subsequent legs in the block of p or p' where they originate. Moreover, Equation (<ref>) shows that their combined color sum in p⊗ p' is identical to the one in p respectively p'. Thus, Equation (<ref>) proves Part <ref> for Y∈{L,K}.
  Lastly, if  B_1 and B_2 are crossing blocks of p⊗ p', then necessarily B_1,B_2⊆ S_p or B_1,B_2⊆ S_p' by definition of the tensor product: Indeed, if we suppose, e.g., B_1⊆ S_p and B_2⊆ S_p', then a crossing of B_1 and B_2 would mean a crossing between S_p and S_p', which is impossible since S_p and S_p' are disjoint consecutive sets.  Hence, Equation (<ref>) also proves Part <ref> for Y=X, which completes the proof.




  §.§.§ Behavior of Z under Erasing Turns

We have not understood yet how Z behaves under erasing turns. However, the effect of the erasing operation on Z is too complicated to treat well abstractly beyond the following simple (but important) observations. 
      
        
        Let p∈ be a partition with no upper points and let T be a turn comprising exactly the two rightmost lower points of p.
        
        
  *  Every point of  E(p,T) is also a point of p.
        
  *  For every set S of points in E(p,T) holds σ_E(p,T)(S)=σ_p(S).
          
  *  For all points γ_1 and γ_2 of E(p,T),
        
    δ_E(p,T)(γ_1,γ_2)=δ_p(γ_1,γ_2).

        
      
      
        Since the erasing procedure only affects the points in T, Parts <ref> and <ref> are clear. We prove Part <ref>.
        If γ_1 lies to the left of γ_2, then ]γ_1,γ_2]_E(p,T)=]γ_1,γ_2]_p and thus δ_E(p,T)(γ_1,γ_2)=δ_p(γ_1,γ_2). Otherwise, ]γ_1,γ_2]_E(p,T)=]γ_1,γ_2]_p∪ T, which is a disjoint union, implying δ_p(γ_1,γ_2)=δ_E(p,T)(γ_1,γ_2)+σ_p(T)=δ_E(p,T)(γ_1,γ_2) since σ_p(T)=0. Thus, in any case δ_E(p,T)(γ_1,γ_2)=δ_p(γ_1,γ_2).
      
      
      

 §.§ Total Color Sum
 The first family of elements Q∈𝖰 is chosen such that all components except the one for  are trivial. That means we restrict the allowed total color sums of partitions.
      
        
        For all m∈{0}∪, the set R_Q with Q=(,,m,,,) is exactly the set of all partitions whose total color sum is a multiple of m.
        No further constraints on p∈ are imposed by the other components of Q via Z({p})≤ Q since these conditions are satisfied by any partition.
      
<cit.>
  
      
    For every m∈{0}∪ the set R_Q, where 
    Q=(,,m,,,),
 is a category of two-colored partitions. 


  We use the Alternative Characterization (Lemma <ref>) to show that R_Q is a category. By Remark <ref> it suffices to show that ({ · })⊆ m is stable under the alternative category operations. 
  Rotation: According to Lemma <ref> for any r∈{,,,} and p∈ R_Q holds (p^r)=(p)⊆ m. Thus,  ({ · })⊆ m is stable under rotations.
  Verticolor Reflection: Lemma <ref> and -m=m show (p̃)=-(p̃)⊆ m for all p∈ R_Q, proving that  ({ · })⊆ m is preserved by verticolor reflection
  Tensor Product: Given p, p'∈ R_Q we can employ Lemma [lemma:Z-tensor-product-3]<ref>lemma:Z-tensor-product <ref>lemma:Z-tensor-product-3 to infer (p⊗ p')=(p)+(p') and thus (p⊗ p')∈ m since (p),(p')∈ m. In conclusion,  ({ · })⊆ m is invariant under tensor products.
  Erasing: Lastly, it remains to consider p∈ R_Q and a turn T in p and to show (E(p,T))∈ m. Because we already know ({ · })⊆ m is respected by rotations, we can assume that p has no upper points and that T comprises exactly the two rightmost lower points. Let P_p denote the set of all points of p. Since T is a turn, σ_p(T)=0. Hence, by Lemma [lemma:erasing-tricks-2]<ref>lemma:erasing-tricks <ref>lemma:erasing-tricks-2,
     
    (E(p,T))=σ_E(p,T)(P_p\ T)=σ_p(P_p\ T)=σ_p(P_p)-σ_p(T)=σ_p(P_p)=(p).

  That concludes the proof.
  


  

 §.§ Block Size and Color Sum
 A second subset of 𝖰 gathers elements where only the components for F and V are non-trivial. So, we only impose (interdependent) conditions on the sizes and color sums of blocks.

  
    
    
    
  * For Q=({1,2},±{0,1,2},,,,) (Part <ref> of the ensuing Lemma <ref>) the set R_Q is simply _≤ 2, the set of all partitions all of whose blocks have at most two legs.
      Only the F-condition in Z({ · })≤ Q is important; The apparent V-constraint is merely a reflection of the one for F: Any singleton block must have color sum  1 (∘) or -1  (∙), and a pair block can only ever be neutral (∘∙ or ∙∘) or have color sums 2 (∘∘) or  -2 (∙∙). And all the remaining conditions imposed by Z({ · })≤ Q are trivially satisfied by any partition.
    
  * If Q=({2},±{0,2},,,,) (Part <ref>), then R_Q=_2 is simply the set of all pair partitions. Again, the V-condition merely reflects the F-constraint.
      Likewise, the non-trivial value 2 of the -component of Q does not impose any restriction on R_Q not already placed by the F-constraints: For any partition p∈,
           
    (p)=σ_p(⋃_B block of pB)=∑_B block of pσ_p(B).

   Hence, (p)=2 is a necessary consequence of V({p})⊆±{0,2}.
   The conditions induced by the L-, K- and X-components of Q remain redundant.
   
  * The set R_Q with Q=({2},{0},{0},∅,,) (Part <ref>) is identical to , the set of all pair partitions with neutral blocks. (See <cit.> and <cit.> for a classification of all its subcategories.)
     In contrast to the two previous examples, the constraint on block color sums is not merely a consequence of the range of allowed block sizes. The F-constraint restricts R_Q to a subset of  and the V-constraint prohibits non-neutral blocks.
     However, the non-trivial value {0} of the -component of Q imposes  no additional restrictions: (p)=0 is, by Equation (<ref>), a necessary consequence of V({p})={0}  for any p∈.
   And also the non-trivial L-component  ∅ of Q provides no additional constraints beyond the ones induced by the F- and V-components: A pair block which is neutral, i.e. has normalized coloring ∘∙ or ∙∘, cannot have subsequent legs of the same normalized color. Thus, for every p∈ with V({p})={0} automatically holds L({p})=∅.
 
  * Lastly, the set R_Q where Q=({1,2},±{0,1},,∅,,) (Part <ref>) is given exactly by the set of all partitions all of whose blocks have at most two legs and all of whose pair blocks are neutral.
   As said before, restricting F to {1,2} reduces the allowed set of block color sums to ±{0,1,2} and the values -2 and 2 can then only stem from pair blocks. Excluding these two values then forces all pair blocks to be neutral. So, as in Part <ref>, the F- and V-components induce  constraints in their own right.
   However, the non-trivial value  ∅ of the L-component of Q is merely a reflection of the F- and V-constraints because neutral pair blocks cannot have subsequent legs of the same normalized color.
    
  
  
  
     
     The set R_Q is a category of partitions
   
   
  *  …if Q=({1,2},±{0,1,2},,,,),
   
  *  …if Q=({2},±{0,2},2,,,),
   
  *  …if Q=({2},{0},{0},∅,,), or               
   
  *  …if Q=({1,2},±{0,1},,∅,,).
        
 

 
   We treat the four claims largely simultaneously.
   Let Q be one of the four elements named in Claims <ref>–<ref> and let f_Q be the first and v_Q the second component of Q. Remark <ref> showed that, if (f_Q,v_Q)=({1,2},±{0,1,2}) (Case <ref>) or (f_Q,v_Q)=({2},±{0,2})  (Case <ref>), then
   
    R_Q={p∈| F({p})⊆ f_Q},

   and that, if (f_Q,v_Q)=({1,2},±{0,1}) (Case <ref>) or (f_Q,v_Q)=({2},{0}) (Case <ref>), then
   
    R_Q={p∈| (F,V)({p})≤ (f_Q,v_Q)}.

   We use the Alternative Characterization (Lemma <ref>) to show that R_Q is a category. As ∈ R_Q is clearly true, that means it suffices to prove that in Cases <ref> and <ref> the condition F({ · })⊆ f_Q and in Cases <ref> and <ref> the condition (F,V)({ · })≤ (f_Q,v_Q) are stable under rotations, tensor products, verticolor reflection and erasing turns.
   Rotation: For all p∈ R_Q and r∈{,,,} by Lemma <ref> holds (F,V)({p^r})=(F,V)({p})≤ (f_Q,v_Q). Thus, (F,V)({ · })≤ (f_Q,v_Q) is stable under rotations.
   Verticolor reflection: Given arbitrary p∈ R_Q we are guaranteed by Lemma <ref> that (F,V)({p})=(F,-V)({p})≤ (f_Q,-v_Q). Since -v_Q=v_Q this proves  that the constraint (F,V)({ · })≤ (f_Q,v_Q) is preserved by verticolor reflection.
   Tensor product: According to Lemma [lemma:Z-tensor-product-1]<ref>lemma:Z-tensor-product <ref>lemma:Z-tensor-product-1 and <ref>, for all partitions p,p'∈ R_Q holds (F,V)({p⊗ p'})=(F,V)({p,p'})≤ (f_Q,v_Q). Hence, tensor products respect (F,V)({ · })≤ (f_Q,v_Q). 
    Erasing:
    Lastly, let p∈ R_Q be arbitrary and let T be a turn in p. We have to show F({E(p,T)})⊆ f_Q  and, but just in Cases <ref> and <ref>, also V({E(p,T)})⊆ v_Q. Since we already know that (F,V)({ · })≤ (f_Q,v_Q) is invariant under rotations, we can assume that p has no upper points and that T comprises exactly the two rightmost lower points of p. 
    Let B be an arbitrary block of E(p,T). What we need to prove is |B|∈ f_Q as well as σ_E(p,T)(B)∈ v_Q, the latter however just in Cases <ref> and <ref>. If B is in fact a block of p, then |B|∈ f_Q and, by Lemma [lemma:erasing-tricks-2]<ref>lemma:erasing-tricks <ref>lemma:erasing-tricks-2, also σ_E(p,T)(B)=σ_p(B)∈ v_Q since we have assumed p∈ R_Q. Thus, suppose that B is not a block of p. Then, there are (not necessarily distinct) blocks B_1 and B_2 of p with B_1∩ T,B_2∩ T≠∅, with T⊆ B_1∪ B_2 and with B=(B_1∪ B_2)\ T≠∅. 
    Step 1: Block size. We prove |B|∈ f_Q.  From |T|=2 and from T⊆ B_1∪ B_2 follows
   
    |B|=|B_1∪ B_2|- |T|=
         
           |B_1|+|B_2|-2    if B_1≠ B_2,
    
           |B_1|-2   if B_1=B_2.

   By definition, f_Q={1,2} or f_Q={2}. We treat the two cases individually.
   Case 1.1: Parts <ref> and <ref>. If f_Q= {1,2}, then in particular |B_1|,|B_2|≤ 2. Hence, no matter whether B_1=B_2 or B_1≠ B_2, Equation (<ref>) implies |B|≤ 2, meaning |B|∈ f_Q, as was claimed.Case 1.2: Parts <ref> and <ref>. If we assume f_Q= {2} instead, then |B_1|=|B_2|=2. Now, B_1=B_2 is impossible since otherwise Equation (<ref>) would imply |B|=0,  con­tra­dic­ting B≠∅. Rather, B_1≠ B_2 must be true. Then, |B|=2∈ f_Q by Equation (<ref>).
  Step 2: Block color sum. We show σ_p(B)∈ v_Q in Cases <ref> and <ref>. Since σ_p(T)=0 and  T⊆ B_1∪ B_2, we infer by Lemma [lemma:erasing-tricks-2]<ref>lemma:erasing-tricks <ref>lemma:erasing-tricks-2:
   
    σ_E(p,T)(B)   =σ_p((B_1∪ B_2)\ T)
       =σ_p((B_1∪ B_2)\ T)+σ_p(T)
    
            =σ_p(B_1∪ B_2)
       =
           σ_p(B_1)+σ_p(B_2)    if B_1≠ B_2,
    σ_p(B_1)   if B_1=B_2.

   Cases <ref> and <ref> correspond to the possibilities (f_Q,v_Q)=({2},{0}) and  (f_Q,v_Q)=({1,2},±{0,1,2}). We treat them individually.
   Case 2.1: Part <ref>. If we assume (f_Q,v_Q)=({2},{0}), i.e., in particular σ_p(B_1)=σ_p(B_2)=0, then, irrespective of whether B_1=B_2 or B_1≠ B_2, Equation (<ref>) proves σ_E(p,T)(B)=0.
   Case 2.2: Part <ref>.
      Now, let (f_Q,v_Q)=({0,1,2},±{0,1}). Then, B_1 and B_2 are singletons or pair blocks and σ_p(B_1),σ_p(B_2)∈±{0,1}.  If both B_1 and B_2 were singletons, then B_1∩ T,B_2∩ T≠∅ and T⊆ B_1∪ B_2 would imply B_1∪ B_2=T and thus the contradiction B=(B_1∪ B_2)\ T=∅. Hence, we know that at least one of the blocks B_1 and B_2 must be a pair. The assumptions p∈ R_Q and v_Q=±{0,1} force the pair blocks of p to be neutral (since non-neutral pair blocks have color sums 2 or -2). It follows σ_p(B_1)=0 or σ_p(B_2)=0. Now, no matter whether B_1=B_2 or B_1≠ B_2, Equation (<ref>) proves σ_E(p,T)(B)∈±{0,1}=v_Q, which concludes the proof. 
  
 
  


 §.§ Color Distances between Legs of the Same Block

For our third family of elements of 𝖰 the L- and K-components are non-trivial, implying constraints on the color distances between subsequent legs of the same block. In part, this translates to a condition on the color distances between arbitrary – not just subsequent – legs.


  
  Let m∈{0}∪ and 
    Q=(,,m,m,m,)
 and let p∈ be arbitrary. Then, Z({p})≤ Q if and only if for all blocks B in p and any two legs α,β∈ B holds δ_p(α,β)∈ m. 


  Suppose  δ_p(α,β)∈ m for all blocks B of p and all α,β∈ B. Since we can especially choose α≠β and ]α,β[_p∩ B=∅, it follows (L,K)({p})≤ (m,m). On the other hand, picking α=β shows (p)=δ_p(α,α)∈ m, which lets us conclude ({p})⊆ m. As (F,V,X)({p})≤ (,,) is trivially true, that proves one implication.
  To show the converse, let Z({p})≤ Q and let α,β be legs of the same block B of p. If α=β, then δ_p(α,β)=(p)∈ m as ({p})⊆ m. Thus, we can assume α≠β.  We can further suppose that ]α,β[_p∩ B≠∅ since otherwise δ_p(α,β)∈ (L∪ K)({p})⊆ m. Thus, let ]α,β[_p∩ B≠∅. Then, p has at least three points. If so, then we find iteratively by moving from α in direction of β in accordance with the orientation a number n∈ of legs γ_1,…,γ_n of B such that, writing γ_0α and γ_n+1β, the points γ_0,…,γ_n+1 are pairwise distinct, such that (γ_0,…,γ_n+1) is ordered and such that ]α,β[_p∩ B={γ_1,…,γ_n}. Now, Lemma [lemma:color-distance-3]<ref>lemma:color-distance <ref>lemma:color-distance-3 implies δ_p(α,β)≡∑_k=0^nδ_p(γ_k,γ_k+1) m. Because for every k∈{0}∪n  holds  ]γ_k,γ_k+1[∩ B=∅ and thus δ_p(γ_k,γ_k+1)∈ (L∪ K)({p})⊆ m,  that proves the claim.


  
  
  
  * For every m∈{0}∪, if Q=(,,m,m,m,) (Part <ref> of Lemma <ref> below), then  R_Q is given by the set of all partitions such that the color distance between any two legs of the same block is a multiple of m, as seen in Lemma <ref>.
    Mostly, it is the L- and K-components of Q inducing the characteristic constraints via Z({ · })≤ Q. The F-, V- and X-conditions are redundant. However,  the -condition is not. And it is not implied by the L- and K-constraints, either.
  
  * If Q=({1,2},±{0,1,2},2m,m+2m,2m,) (Part <ref>) for some m∈, then the set R_Q is a subset of the set from Part <ref>: Still, a partition p∈ is  required to have color distances in m between any two legs of the same block if it is to be an element of R_Q. However, now, additionally, all blocks of p must have size one or two, the total color sum (p) must be a multiple of 2m (and not just m) and, most importantly, the color distances between subsequent legs of the same block must satisfy two different conditions, depending on their normalized colors. Since blocks can have size two at most, two legs of the same block are subsequent if and only if they are distinct.  Hence, R_Q is the set of all partitions such that
    
    
  * every block has at most two legs,
    
  * the total color sum is an even multiple of m,
    
  * the color distance between any two distinct legs of the same block is
      
      
  * an odd multiple of m if they have identical normalized colors and
              
  * an even multiple of m if they have different normalized colors.
      
    
    Also, note that, in contrast to Part <ref>, the parameter m=0 is not allowed.
    For points α and β in p∈, saying 
    
    δ_p(α,β)∈σ_p({α,β})/2m+2m.

    is a helpful technical way of expressing that the distance of the points is an odd multiple of m if they have the same normalized color and an even multiple otherwise.
    The constraints induced via Z({ ·})≤ Q by the F-, V- and -components are non-trivial and not implied by the L- and K-restrictions. 
    Of course, the X-condition is still redundant.
  



  
  For every m∈{0}∪ the set R_Q is a category of partitions
  
  
  *  …if Q=(,,m,m,m,), or
  
  *  …if Q=({1,2},±{0,1,2},2m,m+2m,2m,).
  



  The two claims can be verified largely simultaneously. Let Q be one of the tuples given in Claims <ref> and <ref> and abbreviate Q=(f_Q,{0}∪(± f_Q),k_Q,l_Q,k_Q,).
  We prove that R_Q is a category by means of the Alternative Characterization (Lemma <ref>). Clearly, ∈ R_Q. By Lemma <ref> the condition ({ · })⊆ k_Q is stable under rotation, tensor products, verticolor reflection and erasing turns. The constraint (F,V)({ · })≤ (f_Q, {0}∪(± f_Q)) is trivially preserved in Case <ref>, and it is preserved in Case <ref>  as well according to Lemma [lemma:categories-2-1]<ref>lemma:categories-2 <ref>lemma:categories-2-1. Hence, it is sufficient to show that (L,K)({ · })≤ (l_Q,k_Q) is invariant under the alternative category operations.
  Rotation: By Lemma <ref> holds (L,K)({p^r})=(L,K)({p})≤ (l_Q,k_Q) for all p∈ R_Q and  r∈{,,,}. Consequently, (L,K)({ · })≤ (l_Q,k_Q) is stable under rotations.
  Verticolor reflection: Given p∈ R_Q,  Lemma <ref> lets us infer that (L,K)({p̃})=(-L,-K)({p̃})≤ (-l_Q,-k_Q). Because -l_Q=l_Q and -k_Q=k_Q, we have thus proven that verticolor reflection preserves (L,K)({ · })≤ (l_Q,k_Q).
  Tensor product: Let p,p'∈ R_Q be arbitrary. Then, in particular (p),(p')∈ k_Q. Since k_Q is a subgroup of  we conclude ((p),(p'))⊆  k_Q. Therefore, Lemma [lemma:Z-tensor-product-4]<ref>lemma:Z-tensor-product <ref>lemma:Z-tensor-product-4 implies L({p⊗ p'})⊆ L({p,p'})+((p),(p'))⊆ l_Q+k_Q and K({p⊗ p'})⊆ K({p,p'})+((p),(p'))⊆ k_Q+k_Q. Because l_Q+k_Q⊆ l_Q and k_Q+k_Q⊆ k_Q by definition of l_Q and k_Q, this proves (L,K)({p⊗ p'})≤ (l_Q,k_Q). In conclusion, (L,K)({ · })≤ (l_Q,k_Q) is respected by tensor products.
  Erasing: Lastly, we let p∈ R_Q have a turn T and show (L,K)({E(p,T)})≤ (l_Q,k_Q). Because we have already shown invariance under rotation, we can suppose that p has no upper points and that T comprises exactly the rightmost two lower points of p.
  Let α and β be legs of the same block B of E(p,T) such that α≠β and such that ]α,β[_E(p,T)∩ B=∅. What we then have to prove is that δ_E(p,T)(α,β)∈ l_Q if σ_E(p,T)({α,β})≠ 0 and that δ_E(p,T)(α,β)∈ k_Q if σ_E(p,T)({α,β})=0. By Lemma [lemma:erasing-tricks-2]<ref>lemma:erasing-tricks <ref>lemma:erasing-tricks-2 and <ref> it suffices to show
  
    δ_p(α,β)!∈
          l_Q   if σ_p({α,β})≠ 0,
    
          k_Q   otherwise.

  To prove (<ref>) we now distinguish the two Cases <ref> and <ref>.
  Case 1: Part <ref>.
  Since l_Q=k_Q=m, Assertion (<ref>) simplifies to the claim
    
    δ_p(α,β)!∈ m.

    Because Q is as required by Lemma <ref>, we can immediately  deduce (<ref>) from p∈ R_Q if α and β belong to the same block in p. Thus, it remains to  assume that the blocks B_1 of α and B_2 of β in p are distinct and prove (<ref>) under this premise.
    Step 1.1: Decomposing δ_p(α,β). Because B_1≠ B_2, the definition of E(p,T) requires the existence of α'∈ B_1∩ T and β'∈ B_2∩ T.
Lemma [lemma:color-distance-3]<ref>lemma:color-distance <ref>lemma:color-distance-3 yields
    
    δ_p(α,β)    ≡δ_p(α,α')+δ_p(α',β')+δ_p(β',β)  m,

    where we have used the consequence (p)∈ m of p∈ R_Q. By this congruence, Assertion (<ref>) can be seen by proving that all summands on the right hand side are multiples of m. Hence, that is what we now show.
    Step 1.2: Color distances of α and α' and of β and β'. Because α,α'∈ B_1 are legs of the same block of p and because p∈ R_Q, Lemma <ref> guarantees δ_p(α,α')∈ m. Likewise, β,β'∈ B_2 implies δ_p(β,β')∈ m. 
    Step 1.3: Color distance of α' and β'. Because B_1≠ B_2, necessarily α'≠β' and T={α',β'}. As T is a turn, the neighboring points  α' and β' have opposite normalized colors. We infer δ_p(α',β')=0 or δ_p(α',β')=(p), depending on whether α' is left of β' or not. As (p)∈ m due to p∈ R_Q, it thus follows δ_p(α',β')∈ m in any case. And that is what we needed to see.
    Case 2: Part <ref>.
    In this case, l_Q=m+2m and k_Q=2m mean that Assertion (<ref>) can be expressed equivalently as
    
    δ_p(α,β)!∈σ_p({α,β})/2m+2m.

    If B is a block of p, then p∈ R_Q implies (<ref>) immediately. Hence, we only need to prove  (<ref>) for the case that B is not a block of p.  We again want to use Lemma [lemma:color-distance-3]<ref>lemma:color-distance <ref>lemma:color-distance-3. Hence, we must find a suitable choice of points α' and β' to decompose δ_p(α,β) with.
    Step 2.1: Finding points α' and β'.
    As B is not a block of p, there exist blocks B_1 and B_2 of p with B_1∩ T, B_2∩ T≠∅, with T⊆ B_1∪ B_2 and with B=(B_1∪ B_2)\ T. In particular,  α,β∈ B_1∪ B_2.
    The points α and β belong to different blocks in p: If there existed  i∈{1,2} with α,β∈ B_i, then it would follow B_i={α,β} since α≠β and |B_i|≤ 2 by p∈_≤ 2. As α,β∉ T, it would have to hold B_i=B_i\ T={α,β} and thus 2=|B_i|=|B_i\ T|+|B_i∩ T|=2+|B_i∩ T|. Because we have assumed B_i∩ T≠∅, this would be a contradiction.
    By renaming B_1 and B_2 we can assume α∈ B_1 and β∈ B_2. Since distinct blocks are disjoint, B_1∩ B_2=∅. Because α,β∉ T and  B_1∩ T,B_2∩ T≠∅ and because |B_1|,|B_2|≤ 2, we find α'∈ B_1∩ T and β'∈ B_2∩ T such that B_1={α,α'} and B_2={β,β'}.
    Step 2.2: Relating the normalized colors of α, α', β and β'.  Since  B_1≠ B_2, since B=(B_1∪ B_2)\ T and since σ_p(T)=0,
    
    σ_p({α,β})=σ_p(B)   =σ_p(B_1)+σ_p(B_2)=σ_p({α,α'})+σ_p({β',β}),

    as in the proof of Lemma <ref>.
    Step 2.3: Color distances of α and α' and of β and β'.    The assumption p∈ R_Q furthermore guarantees
    
    δ_p(α,α')∈σ_p({α,α'})/2m+2m  and  δ_p(β',β)∈σ_p({β',β})/2m+ 2m

    because α,α'∈ B_α and α≠α' on the one hand and β',β∈ B_β and β'≠β on the other hand.
    Step 2.4: Color distance of α' and β'.    Because T={α',β'} is a turn, δ_p(α',β')=0 or δ_p(α',β')=(p). From p∈ R_Q follows (p)∈ k_Q=2m and thus
    
    δ_p(α',β')∈ 2m.

    We now have all ingredients to prove  (<ref>).
    Step 2.5: Synthesis.     Lemma [lemma:color-distance-3]<ref>lemma:color-distance <ref>lemma:color-distance-3, yields
    rCl
      δ_p(α,β)    ≡    δ_p(α,α')+δ_p(α',β')+δ_p(β',β) 2m,

   (<ref>)≡   δ_p(α,α')+δ_p(β',β) 2m.

   (<ref>)≡   σ_p({α,α'})/2m+σ_p({β',β})/2m 2m

   (<ref>)≡   σ_p({α,β})/2m 2m.                                                           
    
    With the proof of (<ref>) thus complete, so is the proof overall.




 §.§ Color Distances between Legs of Crossing Blocks

The last family of elements of 𝖰 we treat exhibits non-trivial X-com­ponents. 

  
  If Q=(,,m,m,m,\ E)  for some m∈{0}∪ and some E⊆ with E=-E=E+m (as in  Lemma <ref> below), we can employ Lemma <ref> to understand the set R_Q. It is given by the set of all partitions such that the color distance between any two legs of the same block is a multiple of m and such that two points belong to non-crossing blocks whenever the color distance between them is an element of E.
  While the F- and V-components of Q, of course, effectively induce no conditions at all via Z({ · })≤ Q, the -, L- and K-conditions are non-trivial and they are not implied by the X-constraint. 




  
  For every m∈{0}∪ and  E⊆ with E=-E=E+m the set R_Q is a category if 
    Q=(,,m,m,m,\ E).




  Let Q be of the kind described in the claim. Once more, we use the Alternative Characterization (Lemma <ref>) to show that R_Q is a category. By Lem­ma [lemma:categories-3-1]<ref>lemma:categories-3 <ref>lemma:categories-3-1 it suffices to consider the X-component of Z, i.e. to prove that the constraint X({ · })⊆\ E is invariant under rotation, tensor products, verticolor reflection and erasing turns.
  Rotation: With the help of Lemma <ref> we can infer X({p^r})=X({p})⊆\ E for all p∈ R_Q and all r∈{,,,}. Thus, X({ · })⊆\ E is preserved by rotations.
  Verticolor reflection: For all p∈ R_Q holds X({p̃})=-X({p})⊆ -\ E by Lemma <ref>. Since our assumption E=-E implies -\ E=\ E, this proves the condition X({ · })⊆\ E stable under verticolor reflection.
  Tensor product: Let p,p'∈ R_Q be arbitrary. Then, (p),(p')∈ m implies ((p),(p))⊆ m. Consequently, Lemma [lemma:Z-tensor-product-4]<ref>lemma:Z-tensor-product <ref>lemma:Z-tensor-product-4 yields X({p⊗ p'})⊆ X({p,p'})+((p),(p'))⊆\ E+m. As we have assumed E=E+m, also \ E=\ E+m and thus X({p⊗ p'}) ⊆\ E.  In conclusion, tensor products respect  X({ · })⊆\ E.
  Erasing: Let p∈ R_Q have a turn T. We show X({E(p,T)})⊆\ E. Since X({ · })⊆\ E is preserved by rotations, no generality is lost assuming that p has no upper points and that the two rightmost lower points of p make up T.
  Let α_1 and α_2 be points in E(p,T) whose blocks B_1 and B_2 cross in E(p,T). By Lemma [lemma:erasing-tricks-3]<ref>lemma:erasing-tricks <ref>lemma:erasing-tricks-3 all we have to show is
  
    δ_p(α_1,α_2)∉ E.

  If B_1 and B_2 are both blocks of p as well, then (<ref>) is true by the assumption p∈ R_Q. Thus, we only need to show (<ref>) in the opposite case. If so, then, by nature of the erasing operation, exactly one of the blocks B_1 and B_2 is also a block of p.
  Step 1: Reduction to the case that B_1 is not a block of p. The assumption p∈ R_Q ensures (p)∈ m. Hence,  Lemma [lemma:color-distance-2]<ref>lemma:color-distance <ref>lemma:color-distance-2 implies  δ_p(α_1,α_2)≡-δ_p(α_2,α_1) m.  Because we assume \ E=-(\ E)+m, that means δ_p(α_1,α_2)∈\ E if and only if δ_p(α_2,α_1)∈\ E. Hence, it suffices to prove (<ref>) for the case that B_1 is not a block of p.
  Step 2: Expressing δ_p(α_1,α_2) as the color distance between some blocks B_1,i and B_2 of p.    Since B_1 is not a block of p, there are (not necessarily distinct) blocks B_1,1 and B_1,2 of p with B_1,1∩ T,B_1,2∩ T≠∅ with T⊆ B_1,1∪ B_1,2 and with B_1=(B_1,1∪ B_1,2)\ T. By renaming if necessary, we can always achieve α_1∈ B_1,1.  We show that, for all β_1∈ B_1,1∪ B_1,2 and all β_2∈ B_2,
  
    δ_p(α_1,α_2)≡δ_p(β_1,β_2) m.

  Two cases must be distinguished.
  Case 2.1: β_1 and α_1 belong to the same block. Let β_1∈ B_1,1.
  By Lemma [lemma:color-distance-3]<ref>lemma:color-distance <ref>lemma:color-distance-3,
  
    δ_p(α_1,α_2)≡δ_p(α_1,β_1)+δ_p(β_1,β_2)+δ_p(β_2,α_2) m.

  According to Lemma <ref> the assumption
   p∈ R_Q implies δ_p(α_1,β_1),δ_p(β_2,α_2)∈ m since α_1,β_1∈ B_1,1 and β_2,α_2∈ B_2. Hence, the only term on the right side of (<ref>) possibly surviving  is δ_p(β_1,β_2). That proves (<ref>) in this case.
  Case 2.2: β_1 and α_1 belong to different blocks. Now, suppose β_1∈ B_1,2 instead.  Since we have assumed B_1,1∩ T,B_1,2∩ T≠∅ we can infer the existence of γ_1,1∈ B_1,1 and γ_1,2∈ B_1,2 with γ_1,1≠γ_1,2 and T={γ_1,1,γ_1,2}. Again, we use Lemma [lemma:color-distance-3]<ref>lemma:color-distance <ref>lemma:color-distance-3 to derive
  rCl
    
    δ_p(α_1,α_2)   ≡    δ_p(α_1,γ_1,1)+δ_p(γ_1,1,γ_1,2)+δ_p(γ_1,2,β_1)

          +δ_p(β_1,β_2)+δ_p(β_2,α_2) m.
  
  All summands on the right hand side of (<ref>) except for possibly δ_p(β_1,β_2) are multiples of m:
  Because T is a turn, γ_1,1 and γ_1,2 are neighbors of different normalized colors, meaning δ_p(γ_1,1,γ_1,2)=0 or δ_p(γ_1,1,γ_1,2)=(p)∈ m and thus δ_p(γ_1,1,γ_1,2)∈ m in any case.
  And, thanks to p∈ R_Q, from α_1,γ_1,1∈ B_1,1, from γ_1,2,β_1∈ B_1,2 and from β_2,α_2∈ B_2 follow δ_p(α_1,γ_1,1),δ_p(γ_1,2,β_1),δ_p(β_2,α_2)∈ m according to Lemma <ref>.
  Thus  (<ref>) verifies (<ref>) in this case.
  Step 3: Showing that B_1,i and B_2 cross in p.
  If we can establish that B_1,1 and B_2 or B_1,2 and B_2 cross each other in p, then Equation (<ref>) proves (<ref>) as \ E=(\ E)+m. So, this is all we have to show.
  

  Because B_1 and B_2 cross in E(p,T) we find points ϵ_1,1,ϵ_1,2∈ B_1 and η_2,θ_2∈ B_2 such that (ϵ_1,1,η_2,ϵ_1,2,θ_2) is ordered in E(p,T) (and thus also in p). If ϵ_1,1,ϵ_1,2∈ B_1,1∪ B_1,2 both belong to B_1,1 or both to B_1,2, then we have already found the desired crossing with B_2. Hence, we can assume that neither is the case, i.e., that B_1,1≠ B_1,2 and that ϵ_1,1 and ϵ_1,2 belong to different blocks in p. If ϵ_1,1∉ B_1,1, then we rename ϵ_1,1↔ϵ_1,2 and η_2↔θ_2. Thus, we can achieve that  ϵ_1,1∈ B_1,1 and ϵ_1,2∈ B_1,2 while maintaining  η_2,θ_2 ∈ B_2 and while keeping (ϵ_1,1,η_2,ϵ_1,2,θ_2) ordered.
  Let  γ_1,1 and γ_1,2 be as before, i.e., γ_1,1∈ B_1,1∩ T and γ_1,2∈ B_1,2∩ T. Since they are neighbors, they have the same position in the cyclic order of p with respect to the tuple (ϵ_1,1,η_2,ϵ_1,2,θ_2). Let i∈{1,2} be such that γ_1,i lies left of γ_1, i (in the native ordering). Then, the below table shows that, no matter how the points are arranged, we can find a crossing between B_1,1 and B_2 or between B_1,2 and B_2:
  
    ordered tuple   crossing   between
    
     (ϵ_1,1,η_2,ϵ_1,2,θ_2,γ_1,i,γ_1, i)      (ϵ_1,1,η_2,γ_1,1,θ_2)     B_1,1 and B_2 
    
    (ϵ_1,1,η_2,ϵ_1,2,γ_1,i,γ_1, i,θ_2)      (ϵ_1,1,η_2,γ_1,1,θ_2)     B_1,1 and B_2
    
    (ϵ_1,1,η_2,γ_1,i,γ_1, i,ϵ_1,2,θ_2)     (ϵ_1,1,η_2,γ_1,1,θ_2)    B_1,1 and B_2
    
    (ϵ_1,1,γ_1,i,γ_1, i,η_2,ϵ_1,2,θ_2)     (γ_1,2,η_2,ϵ_1,2,θ_2)    B_1,2 and B_2

  So, in combination with what we showed in Step 2, that completes the proof.




 §.§ Synthesis

As announced we can combine the previous results to prove that all sets defined in Section <ref> are categories. Recall Notation <ref>.

  
  For every Q∈𝖰 the set R_Q is a non-hyperoctahedral category: 
   
    R:  𝖰→.

  

For every family 𝖰'⊆𝖰 holds by Lemma <ref> 
  
    ⋂{ R_Q| Q∈𝖰'}= R_⋂ _×𝖰'.

  We show that for every Q∈𝖰 there exists a set  𝖰'⊆𝖰 such that Q=⋂_×𝖰' and such that R_Q' is a category for every Q'∈𝖰'. As  is a complete lattice, that then proves that R_Q is a category of two-colored partitions for every Q∈𝖰. It is straightforward to check that indeed ∈ R_Q or ∉ R_Q for every Q∈𝖰.
  For this proof only, we use names for specific elements of the set 𝖰: The below table applies for all m∈{0}∪ and E⊆ with E=-E=E+m. 


    [                                                                                                   F                                                 V                                                                                                   L                                                 K                                                 X;                                               F_2                                               {2}                                            ±{0,2}                                                 2                                                                                                                                                       <ref>lemma:categories-2 <ref>lemma:categories-2-2;                                             F_≤ 2                                             {1,2}                                          ±{0,1,2}                                                                                                                                                                                                         <ref>lemma:categories-2 <ref>lemma:categories-2-1;                                               V_0                                               {2}                                               {0}                                               {0}                                                 ∅                                                                                                     <ref>lemma:categories-2 <ref>lemma:categories-2-3;                                              V_01                                             {1,2}                                            ±{0,1}                                                                                                   ∅                                                                                                     <ref>lemma:categories-2 <ref>lemma:categories-2-4;                                               S_m                                                                                                                                                     m                                                                                                                                                                                 <ref>lemma:categories-1;                                               K_m                                                                                                                                                     m                                                 m                                                 m                                                   <ref>lemma:categories-3 <ref>lemma:categories-3-1;                                              K_m^                                             {1,2}                                          ±{0,1,2}                                                2m                                              m+2m                                                2m                                                   <ref>lemma:categories-3 <ref>lemma:categories-3-2;                                             X_m,E                                                                                                                                                     m                                                 m                                                 m                                               \ E                           <ref>lemma:categories-4 ]

The corresponding sets R_Q for all these elements Q∈𝖰 are categories of partitions, as shown by the respective lemma cited in the last column.


The ensuing table lists how every element of 𝖰 can be written as a meet in 𝖫 of the specific elements defined above. Here, u∈{0}∪, m∈, D⊆{0}∪⌊m/2⌋, E⊆{0}∪ and N is a subsemigroup of (,+).
                    
    [                         F                         V                         S                         L                         K                         X                       ⋂_×;                          ;                       {2}                   ±{0, 2}                       2um                         m                         m                                   F_2,  S_2um,  K_m;                       {2}                   ±{0, 2}                       2um                      m+2m                        2m                                  F_2,  S_2um,  K_m^;                       {2}                   ±{0, 2}                       2um                      m+2m                        2m                       \ m F_2,  S_2um,  K_m^, X_m,m;                       {2}                       {0}                       {0}                         ∅                         m                                           V_0,  K_m;                       {2}                   ±{0, 2}                       {0}                       {0}                       {0}                     \ N_0             F_2,  X_0,N_0;                       {2}                       {0}                       {0}                         ∅                       {0}                     \ N_0             V_0,  X_0,N_0;                       {2}                       {0}                       {0}                         ∅                       {0}                    \ N_0'            V_0,  X_0,N_0';                     {1,2}                ±{0, 1, 2}                        um                         m                         m                     \ D_m    F_≤ 2,  S_um,  X_m,D_m;                     {1,2}                ±{0, 1, 2}                       2um                      m+2m                        2m                     \ D_m    S_2um,  K_m^,  X_m,D_m;                     {1,2}                   ±{0, 1}                        um                         ∅                         m                     \ D_m     V_01,  S_um,  X_m,D_m;                     {1,2}                ±{0, 1, 2}                       {0}                       {0}                       {0}                     \ E_0           F_≤ 2,  X_0,E_0;                     {1,2}                  ±{0,  1}                       {0}                         ∅                       {0}                     \ E_0            V_01,  X_0,E_0;                                                                            um                         m                         m                     \ D_m            S_um,  X_m,D_m;                                                                           {0}                       {0}                       {0}                     \ E_0                   X_0,E_0 ]

                  That concludes the proof.




§ FURTHER CATEGORIES

Although we are only interested in non-hyperoctahedral categories in this article, it deserves pointing out that the proof Lemma <ref> also shows the existence of certain hyperoctahedral categories.

  
The set { p∈| V({p})⊆ g} is a category for every g∈{0}∪.


  We use the Alternative Characterization (Lemma <ref>) of categories. Lemmata <ref>, <ref> and [lemma:Z-tensor-product-2]<ref>lemma:Z-tensor-product <ref>lemma:Z-tensor-product-2 imply that V({ · })⊆ g is stable under rotations, verticolor reflection and tensor products. While verifying Lemma <ref> we showed that for every  p∈, every turn T in p and every block B of E(p,T) the following is true: Either B is a block of p and then σ_E(p,T)(B)=σ_p(B) or B is not a block of p and then there are blocks B_1 and B_2 of p such that σ_E(p,T)(B)=σ_p(B_1)+σ_p(B_2) or σ_E(p,T)(B)=σ_p(B_1). Thus,  V({E(p,T)})⊆ V({p})∪ (V({p})+V({p})), which proves the claim.





