# From word-representable graphs to altered Tverberg-type theorems

D. Oliveros and A.J. Torres

###### Abstract

Tverberg’s theorem says that a set with sufficiently many points in
$\mathbb{R}^{d}$ can always be partitioned into $m$ parts so that the
$(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the
parts. In [LHOY18] the authors investigate how other simplicial complexes
arise as nerve complexes once we have a set with sufficiently many points. In
this paper we relate the theory of word-representable graphs [KL15] as a way
of codifying $1$-skeletons of simplicial complexes to generate nerves. In
particular, we show that every $2$-word-representable triangle-free graph,
every circle graph, every outerplanar graph, and every bipartite graph could
be induced as a nerve complex once we have a set with sufficiently many points
in $\mathbb{R}^{d}$ for some $d$.

Keywords: Tverberg’s Theorem, Radon’s Lemma, $k$-Word-Representable Graphs,
Geometric Set Partitions, Cyclic Polytopes, Nerve Complexes, Order Types,
Geometric Ramsey Theory, Erdős-Szekeres Theorems, Combinatorial Convexity,
Convex Set–Representable Complexes.

##  1 Introduction

One of the most celebrated and beautiful theorems in discrete geometry is due
to H. Tverberg and states the following:

######  Theorem 1 (Tverberg Theorem 1966 [Tve66]).

Every set $S$ with at least $(d+1)(m-1)+1$ points in Euclidean $d$-space
$\mathbb{R}^{d}$ can be partitioned into $m$ parts $P=S_{1},\dots,S_{m}$ such
that all the convex hulls of these parts have nonempty intersections. The
special case of a bi-partition $m=2$ is known as Radon’s Lemma.

Given a family $F=\\{F_{1},\dots,F_{m}\\}$ of convex sets in $\mathbb{R}^{d}$,
the nerve $\mathcal{N}(F)$ of $F$ is the simplicial complex with vertex set
$[m]:=\\{1,2,...,m\\}$ whose faces are $I\subset[m]$ such that $\cap_{i\in
I}F_{i}\neq\emptyset$ (see [Mat13],[Tan13] for details).

Given a finite collection of points $S\subset\mathbb{R}^{d}$ and an
$m$-partition into $m$ color classes $P=S_{1},\dots,S_{m}$ of $S$, the nerve
of the partition, $\mathcal{N}(P)$ is the nerve complex
$\mathcal{N}(\\{\operatorname{conv}(S_{1}),\dots,$
$\operatorname{conv}(S_{m})\\})$, where $\operatorname{conv}(S_{i})$ is the
convex hull of the elements in the color class $i$.

We will say that a simplicial complex $\mathcal{K}$ is _partition-induced_ on
a finite set of points $S\subset\mathbb{R}^{d}$ if there exists a partition
$P$ of $S$ into color classes such that the nerve of the partition is
isomorphic to $\mathcal{K}$. Then $\mathcal{K}$ is _$d$ -Tverberg_ if there
exists a constant $\mathit{Tv}(\mathcal{K},d)$ such that for every point set
$S$ with at least $\mathit{Tv}(\mathcal{K},d)$ points, there exists a
partition $P$ such that $\mathcal{N}(P)=\mathcal{K}$. The minimal such
constant $\mathit{Tv}(\mathcal{K},d)$ is called the _Tverberg number_ for
$\mathcal{K}$ in dimension $d$.

We can think of Tverberg’s theorem as a Ramsey-type theorem, where one studies
how every sufficiently large system (set of points) must contain a large well-
organized sub-system. “Sufficiently large” means that for every set of points
$S$ with at least $\mathit{Tv}(\mathcal{K},d)=(d+1)(m-1)+1$ points, there
always exists a partition $P$ into $m$ color classes of $S$ such that
$\mathcal{N}(P)=\mathcal{K}$, where $\mathcal{K}$ is a simplex. This allows us
to rephrase Tverberg’s theorem as follows:

######  Theorem 2 (Tverberg’s theorem rephrased).

The $(m-1)$-simplex is a $d$-Tverberg complex for all $d\geq 1$, with Tverberg
number $\mathit{Tv}(\mathcal{K},d)=(d+1)(m-1)+1$.

In [LHOY18], the authors investigate other possible nerves and point out that
Tverberg’s theorem can be seen as a special case of a more general situation,
showing some new Ramsey-Tverberg-type results. In particular, nerves such as
trees and cycles are $d$-Tverberg for every $d$. In this paper we follow the
same philosophy and find an interesting relationship between this problem and
word-representable graphs.

One of the most recent findings in the area of graph theory is the notion of
word-representable graphs, which is a common generalization of several well-
studied classes of graphs such as circle graphs, comparability graphs,
$3$-colorable graphs and graphs of degree at most $3$ (also known as subcubic
graphs). A graph $G=(V,E)$ is _word-representable_ if there exists a word $W$
over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and
only if $\\{x,y\\}\in E$ for each $x\not=y$. In particular, if the length of
the alternation is $k$, the word is called $k$-_representable_. (See
Definitions 1 and 2 and the book [KL15] by S. Kitaev and V. Lozin, for the
state of art of this topic.) Not all graphs are word-representable, but we
have observed that with a slight generalization and allowing the word $W$ to
contain a $k$-alternating subword (see Definition 3), every graph is “
_general_ $k$-_word-representable_ ” for some $k$; see Definition 2. Now
determining the minimum $k$ seems an interesting problem on its own.

In this paper we observe an engaging connection between general $k$-word-
representable graphs and Ramsey-Tverberg-type results, where nerve structures
are shown to arise once we have sufficiently many points.

The main philosophy for the proofs of our results relies on the following four
steps.

  * Step 1)

Suppose we would like to show that a certain simplicial complex $\mathcal{K}$
with vertex set $[m]$ is $d$-Tverberg for some $d$. Assume that the
$1$-skeleton of $\mathcal{K}$ is a graph that is general $r$-word-
representable by some word $W$ with $m$ letters. Then use $d=r-2$.

  * Step 2)

Apply the famous Ramsey-type Erdős-Szekeres result in dimension $d=2$ [ES87]
or the multi-dimensional version of the Erdős-Szekeres theorem (due to
Grünbaum [GKPS67] and Cordovil and Duchet [CD00]; see also [BLVS+93, Chapter
9], and the survey [MS16]) in dimension $d\geq 2$ with sufficiently many
points to ensure we have at least $m$ points on some curve that is
combinatorially equivalent to moment curve.

  * Step 3)

Label these $m$-vertices on this “moment curve” according to the letters of
the word $W$. This naturally yields a partition. Using the Radon partitions
for cyclic polytopes given by M. Breen in [Bre73] we obtain the desired nerve
$\mathcal{K}$ for points in convex position in $\mathbb{R}^{d}$.

  * Step 4)

Extend the partition to the rest of the points in $\mathbb{R}^{d}$ without
altering the desired nerve $\mathcal{K}$.

These tools are enough to show the existence of a Tverberg number
$\mathit{Tv}(\mathit{Tn},d)$, but the bounds are far from tight.

Although it is probably true, it seems a difficult problem at this point to
show whether all four steps work for all simplicial complexes. In this paper
we are able to prove, among other things, the following theorems.

######  Theorem 3.

Every general $2$-word-representable triangle-free graph $G=(V,E)$ is
$2$-Tverberg.

######  Theorem 4.

Every circle graph is $2$-Tverberg.

These two theorems imply that graphs such as outerplanar graphs are
$2$-Tverberg, and prove that trees and cycles are $2$-Tverberg.

######  Theorem 5.

Every triangle free graph $G=(V,E)$ is partition induced on a sufficiently
large set of points in convex position for every integer $d\geq d_{0}$ for
some $d_{0}\geq 1$.

######  Theorem 6.

Every bipartite graph is $d$-Tverberg for some $d$.

In Section 2 we review the concept of word-representable and $k$-word-
representable graphs, and give a generalization of these concepts which helps
us to codify every given triangle-free graph $G$. We further give examples of
graphs that are planar, bipartite and not general $2$-word-representable
graphs, thus not $2$-Tverberg.

In Section 3 we show that every triangle-free general $d$-word-representable
graph is partition-induced on a set of points in convex position in dimension
$d$. In Section 4 we show that if a simplicial complex is partition-induced on
a set of points in convex position in the plane, then it is $2$-Tverberg
(Theorem 11), which proves in particular Theorems 3 and 4. Finally, in the
last part of this section we will prove Theorem 6 using an specific new
codification in $d$-word-representable graphs and Radon partitions on cyclic
polytopes.

##  2 Word-representable graphs

###  2.1 Word-representable and _k_ -word-representable graphs

A _word_ $W$ in an alphabet $A(W)$ is simply a finite sequence of letters
$a_{1}a_{2}\dots a_{r}$ with $a_{i}\in A(W)$. A _sub-word_ $W^{\prime}$ of $W$
is a subsequence of $W$, and a word $\overline{W}$ is a _factor_ of $W$ if
$W=W_{1}\overline{W}W_{2}$ for possibly empty words $W_{1}$ and $W_{2}$. We
denote the letter of $W$ in position $i$ by $W(i)$ for $1\leq i\leq|W|$.

The concepts of word-representable graphs and $k$-word-representable graphs
were introduced by Kitaev in [KMS05] and are defined as follows.

######  Definition 1.

A graph $G=(V,E)$ is _word-representable_ if there exists a word $W$ over the
alphabet $V$ such that two letters $x$ and $y$ alternate in $W$ if and only if
$\\{x,y\\}\in E$. We say that $W$ represents $G$, and $W$ is called a word-
representant for $G$.

######  Definition 2.

A graph $G$ is _$k$ -word-representable_ if there exists a word $W$ such that
$W$ is a word-representant for $G$ and every letter appears in $W$ exactly $k$
times. The minimal $k$ such that a graph is $k$-word-representable is called
the graph’s representation number, and it is denoted by $\mathcal{R}(G)$.
Also, $\mathcal{R}_{k}=\\{G|\mathcal{R}(G)=k\\}$.

There are many interesting results and contributions in word-representable and
$k$-word-representable graphs (see [KL15] for an excellent treatment of the
theory). Below we will list some of them that we will use in this work.

######  Lemma 1 ([KP08]).

A graph $G$ is word-representable if and only if it is $k$-word-representable
for some $k$.

######  Lemma 2 ([KP08]).

A $k$-word-representable graph $G$ is also $(k+1)$-word-representable.

######  Lemma 3 ([HKP11]).

For a graph $G$ different from a complete graph, $G$ is $2$-word-representable
if and only if $G$ is a circle graph (intersection graph of a set of chords
with vertices on a circle).

###  2.2 General _d_ -word-representable graphs

We now introduce a very similar concept that results in a generalization of
$k$-word-representable graphs.

######  Definition 3.

Given a word $W$ in an alphabet $A(W)$ for $|A(W)|\geq 2$, we say that two
letters $a,b\in A(W)$ are _$d$ -intersecting_ for $d\geq 1$ if there exists an
alternating sub-word of $W$ of length at least $d+2$.

######  Definition 4.

A graph $G=(V,E)$ is general $d$-word-representable if there exists a word $W$
from alphabet $V$ such that for every edge $\\{x,y\\}\in E$, letters $x$ and
$y$ are $d$-intersecting in $W$. We say that $W$ general $d$-word-represents
the graph $G$, and $W$ is called a general $d$-word-representant for $G$. The
minimal $d$ such that a graph is general $d$-word-representable is called the
graph’s general $d$-word-representation number, and it is denoted by
$\mathcal{GR}(G)$.

Clearly, this last definition is more general in the sense that every word-
representable graph $G$ is also general $d$-word-representable for some $d$.
In fact, using the same proof that the authors gave for the lemma 2 in [KP08]
the following result is true.

######  Proposition 1.

Every general $d$-word-representable graph $G$ is also general $(d+1)$-word-
representable.

Now consider, for example, the wheel $W_{5}$ (see Figure 1). It is known
[KP08] that this graph is not $k$-word-representable; however, the word
$W=156216326436546$ general $4$-word-represents this graph. Note for instance
that for $1$ and $6$, the sub-word $1616666$ is not an alternating sequence,
but contains one of length $4$ that is $1616$. The following proposition
observes that every graph is a general $d$-representable graph.

Figure 1: $W_{5}$ and a polygon arrangement that induces it as polygon-circle
graph.

######  Proposition 2.

Every graph $G=(V,E)$ is general $d$-word-representable for $d\geq m-1$ where
$m=|E|$.

###### Proof.

Assume the edges of $G$ are given by $E=\\{e_{1},e_{2},\dots,e_{m}\\}$ and
$W=W_{1}W_{2}\dots W_{m}$ where every factor $W_{i}$ is an alternating
sequence from alphabet $e_{i}=\\{x_{i},y_{i}\\}$, $i\in\\{1,2,\dots,m\\}$ and
length $d+2\geq m+1$. Observe that $W$ $d$-word-represents $G$. Clearly by
construction every edge of $G$ is induced by $W$. On the other hand, suppose
that $W$ induces an edge $e^{*}=\\{x,y\\}\notin E$. Consider an alternating
sub-sequence $S\subset W$ with letters $\\{x,y\\}$ and length at least $d+2$.
Note that each letter in $S$ lies in a unique $W_{i}$; otherwise we have that
$e^{*}\in E$, but this yields a contradiction on the length of $S$. ∎

Finding the minimal $d$, $\mathcal{GR}(G)$ such that a graph $G$ is general
$d$-word-representable seems an interesting problem on its own.

###  2.3 2-word-representable graphs and polygon circle graphs

######  Definition 5.

A graph $G$ is a _polygon-circle_ graph if it is the intersection graph of
convex polygons inscribed in a circle.

The $5$-wheel graph $W_{5}$, for example, belongs to the family of polygon-
circle graphs (see Figure 1). In [EK19] Enright and Kitaev proved that the two
classes of polygon-circle and word-representable graphs do not contain one
another. But in the case of general $d$-word-representable graphs we have that
the following proposition is true.

######  Proposition 3.

A graph $G=(V,E)$ is a polygon-circle graph if and only if $G$ is a general
$2$-word-representable graph.

###### Proof.

Let $G$ be a general $2$-word-representable graph, and suppose that $W$ is a
word, say with length $n$ and alphabet $V$, that general $2$-word-represents
$G$. Consider a set of $n$ points in a circle and color them with the labels
of the letters of $W$ in the clockwise direction. Clearly, the convex hulls of
the chromatic classes are a polygon arrangement that induces $G$ as a polygon-
circle graph. Reciprocally, if $G$ is a polygon-circle graph, and $\tilde{G}$
its realization in the circle, then we may label the vertices of every polygon
with a different color. This clockwise coloring sequence starting at any
vertex generates a word $W$ that general $2$-word-represents the graph $G$. ∎

######  Observation 1.

If a word $W$ general $2$-word-represents a graph $G$, then every word
generated by a cyclic permutation of $W$ also general $2$-word-represents $G$.
This clearly does not hold for $d$ odd. Consider the word $W=12121$; this word
induces $K_{2}$ as a general $3$-word-representable graph but its cyclic
permutation $W^{\prime}=11212$ induces two disjoint vertices.

One natural question to ask is what kind of graphs are not general $2$-word-
representable. Using a data base from [Com] we have computationally checked
all connected planar graphs with at most nine vertices and have found that the
following three planar graphs are not general $2$-word-representable, been the
most right not only planar but bipartite (see Figure 2).

Figure 2:  All graphs up to $9$ vertices that are not general $2$-word-
representable.

##  3 Nerves over the cyclic polytope and its relation with words

The _moment_ curve $M_{d}$ is the curve in $\mathbb{R}^{d}$ defined by
$\\{x(t)=(t,t^{2},\dots,t^{d})\colon t\in\mathbb{R}\\}$. Consider a set
$T=\\{t_{1},\dots,t_{r}\\}\subset\mathbb{R}$ with $r>d+1$ and let
$x(T)=\\{x(t_{i})\colon i=1,2,\dots,r\\}$. Then $\operatorname{conv}(x(T))$ is
known as a cyclic $d$-polytope. As all $r$-point sets in the moment curve are
combinatorially equivalent [GKKZ03], we will refer to any of them as $x(r,d)$
and denote their convex hull by $C(r,d)$. The following classical theorem
gives the combinatorial structure of cyclic polytopes.

######  Theorem 7 (Gale’s evenness condition [Gal63]).

Let $T=\\{t_{1},\dots,t_{r}\\}\subset\mathbb{R}$ with
$t_{1}<t_{2}<\cdots<t_{r}$ and $T_{d}\subset T$ a subset with $d$ points. Then
$C(x(T_{d}))$ is a facet of $C(x(T))$ if and only if $T\setminus T_{d}$ are
separated by an even number of elements of $T_{d}$ in the sequence
$(t_{1},\dots,t_{r})$.

In [Bre73] M. Breen shows that the set $A\cup B$ is a primitive Radon
partition in the cyclic polytope $C(r,d)$ if the following condition holds.

######  Theorem 8 ([Bre73]).

Let $A$ and $B$ be two point sets in $V(|A\cup B|,d)$. Then
$\operatorname{conv}(A)\cap\operatorname{conv}(B)\neq\emptyset$ if and only if
there exists an alternating sequence of length $d+2$ along the moment curve.

As we will see next, Breen’s theorem gives us the key to the relationship
between realization of nerves in convex position and general $k$-word-
representable graphs and $k$-word-representable graphs. In particular, Theorem
5 shows that every triangle-free general $d$-word-representable graph is
partition-induced on a set of points in $M_{d}$. Furthermore we are ready to
prove Theorem 5.

Proof of theorem 5

###### Proof.

By propositions 1 and 2 we have that for every $d\geq d_{0}=\mathcal{GR}(G)$
there exist a word $W$ with alphabet $V$, such that $W$ general $d$-word-
represents $G$, set $m=|W|$. Let $S$ be any set of $m$ points in general
position in dimension $d$. And let set $S_{1}$ be a set of $m$ points;
$S_{1}=\\{s_{1},s_{2},\dots,s_{m}\\}\subset M_{d}$ on the moment curve. Color
them by $\mathcal{C}\colon S_{1}\mapsto V$ defined by
$\mathcal{C}(s_{i})=W(i)$ and let $P=\\{P_{1},P_{2},\dots,P_{m}\\}$ be the set
where $P_{i}$ is the convex hull of the color class $v_{i}$. By Theorem 8, two
sets $P_{i}$ and $P_{j}$ intersect if and only if colors $v_{i}$ and $v_{j}$
are $d$-intersecting in $W$. This implies that $G$ is preserved as the
$1$-skeleton of $\mathcal{N}(P)$. Furthermore, since $G$ does not have
triangles, there are no intersections of three or more sets of $P$; therefore,
there are no faces with a dimension greater than $1$, so we conclude that
$\mathcal{N}(P)=G$. Finally we know, than any two set in convex position have
the same order type thus, there exist a bijection $\sigma$ that preserves
orientation between $S_{1}$ and $S$. By Lemma 2 of [LHOY18], the partition
$\mathcal{P}=(P_{1},P_{2},\dots,P_{m})$ of $S_{1}$ and the corresponding
partition of $S$ via $\sigma$, denoted
$\sigma{\mathcal{P}}=\\{\sigma(P_{1}),\sigma(P_{2}),\dots,\sigma(P_{n})\\}$,
have the same intersection graph $G$. ∎

Observe that since $k$-word-representable graphs are also general $d$-word-
representable graphs, using Theorem 5 and Lemmas 1, 2 and 3 the following
results are true.

######  Theorem 9.

Every bipartite graph $G=(U\cup V,E)$ is partition-induced for a point set
$S\subset M_{d}$ for some positive integer $d$.

######  Theorem 10.

Every circle graph (intersection graph of a set of chords with vertices on a
circle) $G=(V,E)$ is partition-induced for a point set $S\subset M_{d}$ for
$d\geq 2$.

##  4 Extensions and Tverberg-type theorems

In a similar way to [LHOY18], we will extend the colorings from specific point
sets to more general configurations. Let us define the extension concept
formally.

######  Definition 6.

Let $S$ be a set of points in $\mathbb{R}^{d}$ and let $P$ be a color
partition of $S$ into color classes $P=S_{1},S_{2},\dots,S_{m}$. Then there is
a function $\mathcal{C}\colon S\mapsto\\{c_{1},c_{2},\dots,c_{m}\\}$, a
“coloring” that yields a specific nerve $\mathcal{N}(P)$. We say that
$\mathcal{C}$ is extendable if for every point set $\overline{S}$ containing
$S$, there is a color partition
$\overline{P}=\overline{S}_{1},\overline{S}_{2},\dots\overline{S}_{m}$ and a
coloring
$\overline{\mathcal{C}}\colon\overline{S}\mapsto\\{c_{1},c_{2},\dots,c_{m}\\}$
such that $\overline{C}|_{S}=\mathcal{C}$ and $\mathcal{N}(\overline{P})$ is
isomorphic to $\mathcal{N}(P)$.

In this section we will show that every simplicial complex $\mathcal{K}$ that
is partition-induced on a set of points in convex position in the plane is
extendable; therefore, it is $2$-Tverberg. It turns out that in dimension
$d\geq 3$, the extension is not that clear (see Figure 3); however, we will
show that some subfamilies of graphs are extendable.

###  4.1 Extensions in $\mathbb{R}^{2}$

We begin by extending colorings of partitions of point sets in convex position
contained in $\mathbb{R}^{2}$, and obtain the following theorem.

######  Theorem 11.

Every simplicial complex $\mathcal{K}$ that is partition-induced in any set of
points in convex position is $2$-Tverberg.

###### Proof.

Suppose that a simplicial complex $\mathcal{K}$ is partition-induced in a set
of $n$ points in convex position in the plane. In order to show that
$\mathcal{K}$ is $2$-Tverberg, we need to show that it is partition-induced in
an arbitrary set of points with sufficiently many points. By the Erdős-
Szekeres Theorem[ES87], every point set $\bar{S}$ with sufficiently many
points contains a subset ${S}$ of $n$ points in convex position. By
hypothesis, there is a partition ${P}$ of $S$ into color classes
${S_{1}},{S_{2}},\dots S_{m}$ obtained by a coloring function
$\mathcal{C}\colon S\mapsto\\{c_{1},c_{2},\dots,c_{m}\\}$ such that its nerve
$\mathcal{N}({P})=\mathcal{K}$. (Furthermore such a partition yields a polygon
circle graph and by Proposition 3, it is a $2$-word-representable graph.) It
remains to show that we can extend $\mathcal{C}$ to the set $\bar{S}$ with a
new coloring $\bar{\mathcal{C}}$ as in Definition 6.

We proceed by induction on the number $m$ of different chromatic classes of
$\mathcal{C}$. If there are only two different colors, say $c_{1}$ and
$c_{2}$, then two possible scenarios may occur:
$\operatorname{conv}(S_{1})\cap\operatorname{conv}(S_{2})=\emptyset$ or
$\operatorname{conv}(S_{1})\cap\operatorname{conv}(S_{2})\not=\emptyset$. If
the first case happens, it is clear that there exists a support line $h$
through one edge of the polygon $\operatorname{conv}(S_{1})$ that leaves
$\operatorname{conv}(S_{1})$ on one side of $h$, say $H^{+}$, and
$\operatorname{conv}(S_{2})$ on the other, $H^{-}$. Next, we color the points
in $\bar{S}\subset H^{+}$ with color $c_{1}$ and the points in $\bar{S}\subset
H^{-}$ with color $c_{2}$. If
$\operatorname{conv}(S_{1})\cap\operatorname{conv}(S_{2})\not=\emptyset$, then
we may color every point in $\bar{S}\setminus S_{1}$ with color $c_{2}$.

Given a color $c_{i}\in\\{c_{1},c_{2},\dots,c_{m}\\}$ and a supporting line
$h_{ij}$ of $\operatorname{conv}(S_{i})$, denote by $H^{+}_{ij}$ the half-
plane that contains $S_{i}$ (if $S_{i}$ consists of an interval take either
side) and denote by
$\mathcal{C}_{ij}^{c}=\\{c_{ij_{1}},c_{ij_{2}},\dots,c_{ij_{n_{ij}}}\\}$ the
set of colors such that the corresponding sets $S_{ij_{k}}$ are in
$H^{+}_{ij}$ and
$\operatorname{conv}(S_{i})\cap\operatorname{conv}(S_{ij_{k}})=\emptyset$ for
$k=1,\dots,n_{ij}$.

Note that if $|\mathcal{C}_{ij}^{c}|=0$ for some $i$ and some $j$, then we can
ignore the points of $S$ having color $c_{i}$, and by the induction hypothesis
there exists an $m-1$ coloring on the remaining points that induce $G$ as a
nerve. Now we may re-color all the points in $\bar{S}\setminus S$ that lie in
$H_{ij}^{+}$ with color $c_{i}$ preserving $G$ as a nerve.

For every color $c_{i}$, we may suppose that for every supporting line
$h_{ij}$ we have that $|\mathcal{C}_{ij}^{c}|\not=0$. Assume that $c_{1}$ is
the color and $h_{11}$ the supporting line of $conv(S_{1})$ such that
$|\mathcal{C}_{11}^{c}|$ is minimum for all $i=1,\dots,m$. We now observe that
there is a color, say $c_{r}\in\mathcal{C}_{11}^{c}$, and therefore it is in
$H_{11}^{+}$ and a supporting line $h_{rs}$ such that
$|\mathcal{C}_{rs}^{c}|\leq|\mathcal{C}_{11}^{c}|$, contradicting the
minimality of $|\mathcal{C}_{1}^{c}|$. ∎

Proof of Theorem 3 and Theorem 4

Although it is possibly true that all general $2$-word-representable graphs
are $2$-Tverberg, as it was pointed out in Lemma 2 and Fig. 2 of [LHOY18],
dealing with simplicial complexes that contain triangles is hard, simply
because order types do not register the difference between a $2$-simplex and a
triangle.

By Proposition 3, we know that for every general $2$-word-representable graph
$G=(V,E)$, $|V|=m$, there exists an arrangement of polygons
$Q_{1},Q_{2},\dots,Q_{m}$ on a circumference that induces $G$ as an
intersection graph. Furthermore, the $1$-skeleton of every triangle-free graph
satisfies that it is equal to the intersection graph of its polygon
arrangement. Then Theorem 11 yields Theorem 3. This implies in particular that
$m$-cycles, for $m\geq 3$, are $2$-Tverberg.

######  Observation 2.

Given a _circle graph_ $G=(V,E)$; that is, the intersection graph of a set of
chords with vertices on a circle, it is not difficult to see that for a large
enough set of points in convex position we can choose $2|V|$ of them and
construct a chord arrangement without triple chord intersections such that it
induces $G$ as a circular graph.

Then by Theorem 11, it is clear that the family of _circle graphs_ are
$2$-Tverberg, which proves Theorem 4. This implies that families of graphs
such as outerplanar graphs, trees and cycles [Sac85, pp. 207-210] are
$2$-Tverberg.

###  4.2 Extensions in $\mathbb{R}^{d}$

As we observed in the previous subsection, Theorem 11 allows us to extend any
given partition on a sufficiently large set of points $S$ in convex position
in the plane to an arbitrary set of points $\bar{S}$ containing $S$,
preserving the nerve of the partition in $S$. This theorem uses the fact that
a color $c$ and a line $h$ always exist, leaving all the points of color $c$
on one half-plane together with all possible points such that their color
classes intersect with color $c$.

Unfortunately this is not always the case in higher dimensions. Consider, for
example, a set $S=\\{x(t_{i})\colon 1\leq i\leq 9\text{ with
}t_{1}<t_{2}<\cdots<t_{9}\\}$ and $\mathcal{C}\colon S\mapsto\\{r,b,g\\}$ such
that $\mathcal{C}\\{x(t_{1}),x(t_{2}),x(t_{6})\\}=b$,
$\mathcal{C}\\{x(t_{3}),x(t_{5}),x(t_{7})\\}=r$ and
$\mathcal{C}\\{x(t_{4}),x(t_{8}),x(t_{9})\\}=g$ (see Figure 3).

Figure 3: Convex hulls of the chromatic classes of $S$.

###  4.3 Bipartite graphs as general $k$-word-representable graphs

Although we know that bipartite graphs are $k$-word-representable graphs
[KL15] there is still no specific word description. In this section, we will
prove that bipartite graphs are general $d$-word-representable graphs for some
$d$. This implies, from Theorem 5 that bipartite graphs are a partition
induced on a set $S$ with as many points as the length of the corresponding
word in the moment curve $M_{d}$. Having the great advantage that such
partition is extendable, will later on allow us to prove Theorem 6.

######  Theorem 12.

Every bipartite graph $G=(U\cup V,E)$ is general $d$-word-representable for
$d\geq|V|$.

###### Proof.

We start to labeling the vertices of $G$ as $V=\\{v_{1},v_{2},...,v_{d}\\}$
and $U=\\{u_{1},u_{2},...,u_{m}\\}$ with $d\leq m$. For every pair of vertices
$v_{i}\in V$ and $u_{j}\in U$, $1\leq i\leq d$ and $1\leq j\leq m$ we define
the word $F_{i}({u_{j}})$ as an alternating sequence of length $d+2$ with
letters $\\{v_{i},u_{j}\\}$ if $\\{v_{i},u_{j}\\}\in E$, and as the empty word
if $\\{v_{i},u_{j}\\}\notin E$. Next define the word $W_{i}$ as follows.

$W_{i}=\left\\{\begin{aligned}
&F_{i}({u_{1}})F_{i}(u_{2})...F_{i}(u_{m})&\text{ when }i\text{ odd}\\\
&F_{i}(u_{m})F_{i}(u_{m-1})...F_{i}(u_{1})&\text{ when }i\text{
even}\end{aligned}\right.$

Now we claim that $W=W_{1}W_{2}....W_{d}$ general $d$-word-represents $G$.

By construction it is clear that for every edge $e\in E(G)$ there is an
alternating sequence of length $d+2$ in $W$ with alphabet in $e$, then every
edge of $G$ is induced by $W$. Now, we will prove that $W$ does not induce
further edges. Consider two different vertices $v_{p},v_{q}\in V$, since these
vertices are not adjacent they appear in different factors $W_{p}$ and
$W_{q}$. Then, note that every alternating sequence with maximum length in $W$
with letters $\\{v_{p},v_{q}\\}$ has at most length $2$. Therefore $W$ does
not induce the edge $\\{v_{p},v_{q}\\}$.
For $\\{u_{p},u_{q}\\}\notin E$ with $u_{p},u_{q}\in U$, by construction every
$W_{i}$ contributes at most with a copy of $u_{p}$ and at most one of $u_{q}$
to every alternating sequence with letters $\\{u_{p},u_{q}\\}$. Furthermore,
these elements are ordered in the reverse order in any two consecutive factors
$W_{i}$’s, then every alternating sequence has at most length $n+1$ and $W$
does not induce the edge $\\{u_{p},u_{q}\\}$.

Finally, given an edge $\\{v_{p},u_{q}\\}\notin E(G)$ with $v_{p}\in V$ and
$u_{q}\in U$, by construction there exist a factor $W^{*}\supseteq W_{p}$ in
$W$ containing all the copies of $v_{p}$ and only other letters for which
there are alternating subsequences of size $d+2$ with $v_{p}$. Then,
$u_{q}\notin W^{*}$ and the maximum length of an alternating sequence with
these two letters in $W$ has a size of at most $3$. We conclude that $W$
general $d$-word-represents $G$. ∎

###  4.4 General-bipartite graphs are _k_ -Tverberg complexes

We begin with the following useful Lemma that follows directly from Gales’s
evenness condition Theorem 7.

######  Lemma 4.

Every hyperplane $H$ generated by $d$ points on the moment curve $M_{d}$,
$H=\langle\\{p_{i}\\}\rangle:=\langle\\{p_{i}=x(l_{i})\in M_{d}\colon
l_{i}\in\mathbb{R}\text{ such that }l_{1}<l_{2}<\cdots<l_{d}\\}\rangle$
divides $\mathbb{R}^{d}$ into two half-hyperplanes one of them, say $H^{+}$
containing all points lying in $R_{i}$ for even $i$ and the other $H^{-}$
containing all points lying in $R_{i}$ for odd $i$. Where $R_{0}=\\{x(l)\colon
l<l_{1}\\}$, $R_{i}=\\{x(l)\colon l_{i-1}<l<l_{i}\\}$ with $1<i\leq d$ and
$R_{d+1}=\\{x(l)\colon l>l_{d}\\}$.

Let $G=(U\cup V,E)$ be a bipartite graph with $V=\\{v_{1},v_{2},...,v_{d}\\}$
and $U=\\{u_{1},u_{2},...,u_{m}\\}$ with $d\leq m$. Let $W=W_{1}W_{2}...W_{d}$
be the word that general $d$-word-represents $G$ constructed as described in
the Theorem 12 with $d\geq|V|$. Consider a point set
$S=\\{s_{i}=x(t_{i}):t_{1}<t_{2}<...<t_{|W|}\in\mathbb{R}\\}$ of $|W|$ points
on the moment curve $M_{d}$. Then, by Theorem 5 the coloring
$\mathcal{C}_{G}:S\mapsto V$ given by $\mathcal{C}_{G}(s_{i})=W(i)$ induce a
partition $P_{G}$ such that $\mathcal{N}(P_{G})=G$.

######  Lemma 5.

Given any set of points $\bar{S}$ such that
$S\subset\bar{S}\subset\mathbb{R}^{d}$, $\mathcal{C}_{G}$ is extendable in
such a way that $\mathcal{N}(\bar{P}_{G})=G$.

###### Proof.

Let us begin by looking closely to the $W=W_{1}W_{2}...W_{d}$ described in
Theorem 12.

$W=\underbrace{F_{1}({u_{1}})F_{1}(u_{2})\dots
F_{1}(u_{m})}_{W_{1}}\underbrace{F_{2}(u_{m})F_{2}(u_{m-1})\dots
F_{2}(u_{1})}_{W_{2}}F_{3}(u_{1})\cdots
F_{d-1}({u_{1}})\underbrace{F_{d}({u_{1}})F_{d}(u_{2})...F_{d}(u_{m})}_{W_{d}}$
if $d$ is odd, or

$W=\underbrace{F_{1}({u_{1}})F_{1}(u_{2})\dots
F_{1}(u_{m})}_{W_{1}}\underbrace{F_{2}(u_{m})F_{2}(u_{m-1})\dots
F_{2}(u_{1})}_{W_{2}}F_{3}(u_{1})\cdots
F_{d-1}(u_{m})\underbrace{F_{d}(u_{m})\dots F_{d}(u_{2})F_{d}(u_{1})}_{W_{d}}$

if $d$ is even.

Since some of the factors $F_{i}(u_{1})$ may be empty for some $1\leq i\leq
d$. We begin by observing that the factors that contain the letter $u_{1}$ can
be grouped in most $\lfloor\frac{d}{2}\rfloor+1$ subwords of length one or
two. This observation will allow us to insert $d$ “separators”
$\bf{p_{1},p_{2},\dots,p_{d}}$ that isolate these
$\lfloor\frac{d}{2}\rfloor+1$ subwords containing letter $u_{1}$, as follows:

$\tilde{W}=\underbrace{F_{1}({u_{1}}){\bf{p_{1}}}F_{1}(u_{2})\dots
F_{1}(u_{m})}_{W_{1}}\underbrace{F_{2}(u_{m})F_{2}(u_{m-1})\dots\bf{p_{2}}F_{2}(u_{1})}_{W_{2}}F_{3}(u_{1}){\bf{p_{3}}}\dots$

$\dots\hskip
14.22636pt{\bf{p_{d-1}}}F_{d-1}({u_{1}})\underbrace{F_{d}({u_{1}}){\bf{p_{d}}}F_{d}(u_{2})...F_{d}(u_{m})}_{W_{d}}$
if $d$ is odd, or

$\tilde{W}=\underbrace{F_{1}({u_{1}}){\bf{p_{1}}}F_{1}(u_{2})\dots
F_{1}(u_{m})}_{W_{1}}\underbrace{F_{2}(u_{m})F_{2}(u_{m-1})\dots\bf{p_{2}}F_{2}(u_{1})}_{W_{2}}F_{3}(u_{1}){\bf{p_{3}}}\dots$

$\dots\hskip 48.36958pt\underbrace{F_{d}(u_{m})\dots
F_{d}(u_{2}){\bf{p_{d}}}F_{d}(u_{1})}_{W_{d}}$ if $d$ is even.

Now, these separators will not be part of $S$ and therefore will not be
colored by $\mathcal{C}_{G}$. The aim of these separators is only to choose a
point in $M_{d}\setminus S$ in the corresponding interval between the two
original points in $S$. That is, if $W=\dots W(k){\bf{p_{r}}}W(k+1)\dots$ then
we may chose any $t$ such that $t_{k}<t<t_{k+1}$ and $p_{r}:=x(t)$ for $1\leq
r\leq d$. This will allow us to choose $d$ points
$\\{p_{1},p_{2},\dots,p_{d}\\}$ at the moment curve $M_{d}$. By Lemma 4, there
exists a hyperplane $H_{1}=\langle p_{i}\rangle$, containing no points of
$\overline{S}$, that divides $\mathbb{R}^{d}$ in two half-planes $H_{1}^{+}$
and $H_{1}^{-}$. This is done so that the points of $S$, that are related with
a non-empty factor of the form $F_{i}(u_{1})$ colored with colors $u_{1}$ or
$v_{i}$, lie on one of the half-planes, say $H_{1}^{+}$, and every other point
in $S$ lies in $H_{1}^{-}$. Similarly, for color $u_{2}$ we can define a new
hyperplane $H_{2}$ and a convex region $R_{2}=(\mathbb{R}^{d}\setminus
H_{1}^{+})\cap H_{2}^{+}$ such that every point related with the factors of
the form $F_{i}(u_{2})$ lie in the interior of this convex region $R_{2}$.
Recursively, for $3\leq j\leq m-1$ we can define hyperplanes $H_{j}$ and
convex regions $R_{j}=(\mathbb{R}^{d}\setminus\cup_{k=1}^{j}R_{k-1})\cap
H_{j}^{+}$ such that every point related with the factors of the form
$F_{i}(u_{j})$ lies in the interior of these convex regions. Finally we define
the region $R_{m}=\mathbb{R}^{d}\setminus R_{m-1}^{+}$.
Observe that for every point $s\in S$ lying in the interior of the region
$R_{j}$, we have one of two cases: $\mathcal{C}(s)=u_{i}$ or the convex hull
of the points with color $\mathcal{C}(s)$ intersects the convex hull of the
points with color $u_{j}$. By construction of the regions $R_{j}$, every pair
of adjacent points of $S$ with color $v\in V$ lie in the same region or in
adjacent regions. Then, if $v\notin R_{j}$ for some $j$, the convex hull of
the points with color $v$ does not intersect the convex hull of the points
with color $u_{j}$. Coloring the points of $(\bar{S}\setminus S)\cap R_{j}$
with the color $u_{j}$ yields that the resulting coloring extension preserves
$G$ as a nerve. ∎

Proof of Theorem 6

###### Proof.

Consider a bipartite graph $G=(U\cup V,E)$ and let $W$ a word of length $|W|$
that general $d$-word represents $G$ for some integer $d$, constructed as
described in Theorem 12. By the multi-dimensional version of Erdős-Szekeres
theorem (due to Grunbaum[GKPS67] and Cordovil and Duchet [CD00], we know that
every large enough set of points in $\bar{S}\subset\mathbb{R}^{d}$ contains a
set of $|W|$ points $S$ of some curve $\gamma$ that is combinatorially
equivalent to a set $S_{1}$ on the moment curve $M_{d}$. Then by Lemma 2 of
[LHOY18] there is a bijection $\sigma$ from $S_{1}$ to $S$ that preserves the
orientation of any $(d+1)$-tuple in $S$. Then any partition
$\mathcal{P}=(P_{1},P_{2},\dots,P_{n})$ of $S_{1}$ and the corresponding
partition of $S$ via $\sigma$, denoted
$\sigma{\mathcal{P}}=\\{\sigma(P_{1}),\sigma(P_{2}),\dots,\sigma(P_{n})\\}$,
have the same intersection graph $\mathcal{N}^{1}(\mathcal{P})$. Since
bipartite graphs are triangle-free simplicial complexes, their nerve complexes
equal to their $1$-skeleton. This implies that the partition induced by the
word $W$ in Theorem 5 is also partition induced in $S$. Then finally by Lemma
5 $G$ is $d$-Tverberg. ∎

##  5 Conclusions

In this paper we have obtained new Ramsey-type Tverberg theorems by showing
that some infinite families of graphs (such as cycle graphs and bipartite
graphs) are always induced as the nerve of some partition of any sufficiently
large set of points in $\mathbb{R}^{d}$, and we relate $k$-word-representable
and general $k$-word-representable graphs to realizations of nerves in convex
position in $\mathbb{R}^{d}$. We believe that it is possible to generalize
Theorem 3 to every general $2$-word-representable graph without the triangle-
free restriction, but at this time it seems difficult to prove.

We observe that if a graph $G$ is general $d$-word-representable and
extendable for some integer $d$, then $G$ is $d$-Tverberg. Furthermore, if a
graph $G$ is not general $d$-word-representable for a fixed $d$, then $G$ is
not partition-induced in $\mathbb{R}^{d}$. Thus it would be interesting to
study how to determine the minimum $d$ for which a certain given graph $G$ is
general $d$-Tverberg. This would provide us with a tool to see how to decide
if a given graph $G$ is not $(d-1)$-Tverberg. We conjecture that the following
bipartite graph (see Figure 4) is not $3$-Tverberg, but even with the
assumption that each vertex contributes only $3$ copies to the possible word,
there are $\sum_{i=1}^{10}{{3i}\choose{3}}$ words to be checked, which is hard
to compute. Then clearly another approach for future work is needed.

Figure 4: Graph that may not be 3-Tverberg.

## Acknowledgements

The authors will like to thank the support from grants PAPIIT 104915 and
CONACyT 166306.

## References

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# Multi-groups

Tatyana A<EMAIL_ADDRESS>Regional Scientific and
Educational Mathematical Center of Tomsk State University, 36 Lenin Ave.,
634050, Tomsk, Russia.

###### Abstract

In the present paper we define homogeneous algebraic systems. Particular cases
of these systems are: semigroup (monoid, group) system. These algebraic
systems were studied by J. Loday, A. Zhuchok, T. Pirashvili, N. Koreshkov.
Quandle systems were introduced and studied by V. Bardakov, D. Fedoseev, V.
Turaev.

We construct some group systems on the set of square matrices over a field
$\Bbbk$. Also we define rack system on the set $V\times G$, where $V$ is a
vector space of dimension $n$ over $\Bbbk$, $G$ is a subgroup of
$\operatorname{GL}_{n}(\Bbbk)$.Finally, we find connection between skew braces
and dimonoids.

_Key words:_ algebraic system, group, monoid, semigroup system, groupoid,
multi-group, quadle system, multi-quandle, dimonoid, homogeneous algebraic
system, multy-rack, skew brace.

_Mathematics Subject Classification_ : 20M05, 08B20

###### Contents

  1. 1 Introduction
  2. 2 Homogeneous algebraic systems
  3. 3 Group systems and multi-groups
  4. 4 Rack systems
  5. 5 Connection between skew braces and dimonoids

##  1 Introduction

In the theory of algebraic systems there exist algebraic systems with a set of
one type algebraic operations. Let us give some examples of these algebraic
systems.

Brace (skew brace) is a set with two group operations, which satisfy some
axiom (see [1]-[2]). A generalization of skew braces was suggested in the
paper Bardakov-Neshchadim-Yadav [3], where were introduced brace systems as a
set with a family of group operations which connected by some axioms.

Dimonoids were introduced by J.-L. Loday [4] in his construction of a
universal enveloping algebra for the Leibniz algebra. Dimonoid is a set with
two semigroup operations, which are connected by a set of axioms. The
construction of a free dimonoid generated by a given set was presented in [4]
and applied to the study of free dialgebras and a cohomology of dialgebras.
Structural properties of free dimonoids have been investigated by A. V.
Zhuchok in [5]. In [6] was presented a construction of a free product of
arbitrary dimonoids which generalizes a free dimonoid and describe its
structure. Dimonoids are examples of duplexes, which were introduced by T.
Pirashvili in [7]. A duplex is an algebraic system with two associative binary
operations (without added connections between these operations). T. Pirashvili
constructed a free duplex generated by a given set via planar trees and proved
that the set of all permutations forms a free duplex on an explicitly
described set of generators.

In [8], N. Koreshkov introduced $n$-tuple semigroup as an algebraic system

$\mathcal{S}=(S,*_{i},i\in I)$

such that $(S,*_{i})$ is a semigroup for any $i\in I$ and with the following
axiom which connects these operations,

$(a*_{i}b)*_{j}c=a*_{i}(b*_{j}c),\leavevmode\nobreak\ \leavevmode\nobreak\
a,b,c\in S,\leavevmode\nobreak\ \leavevmode\nobreak\ i,j\in I.$

The free $n$-tuple semigroup of an arbitrary rank was first constructed in
[9].

In the present paper we define homogeneous algebraic systems (see Definition
2.1). Particular cases of these systems are: semigroup (monoid, group) system
$\mathcal{G}=(G,*_{i},i\in I)$, where $(G,*_{i})$ is a semigroup (monoid,
group) for any $i\in I$. An example of a semigroup system with two operations
is a duplex. We will call $\mathcal{G}$ by multi-semigroup (multi-monoid,
multi-group) if the operations are connected by the following condition

$(a*_{i}b)*_{j}c=a*_{i}(b*_{j}c),\leavevmode\nobreak\ \leavevmode\nobreak\
a,b,c\in G,\leavevmode\nobreak\ \leavevmode\nobreak\ i,j\in I.$

An example of a multi-semigroup with $n$ operations is an $n$-tuple semigroup
[8].

V. G. Bardakov and D. A. Fedoseev considered quandle system
$\mathcal{Q}=(Q,*_{i},i\in I)$ in [10], where $(Q,*_{i})$ is a quandle for any
$i\in I$, and defined a multiplication $*_{i}*_{j}$ of the operations $*_{i}$
and $*_{j}$ by the rule

$p(*_{i}*_{j})q=(p*_{i}q)*_{j}q,\leavevmode\nobreak\ \leavevmode\nobreak\
p,q\in Q.$

In the general case the algebraic system $(Q,*_{i}*_{j})$ is not a quandle,
but if the operations are satisfied the axioms

$(x*_{i}y)*_{j}z=(x*_{j}z)*_{i}(y*_{j}z),\leavevmode\nobreak\
\leavevmode\nobreak\
(x*_{j}y)*_{i}z=(x*_{i}z)*_{j}(y*_{i}z),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x,y,z\in Q,$ (1.0.1)

then $(Q,*_{i}*_{j})$ and $(Q,*_{j}*_{i})$ are quandles. V. Turaev called
quandle systems, which satisfy axioms (1.0.1) for all $i,j\in I$ by multi-
quandles and gave them a topological interpretation (see [11]).

In 1971, V. M. Buchstaber and S. P. Novikov [12] introduced a notion of
$n$-valued group, in which the product of each pair of elements is an
$n$-multi-set, the set of $n$ elements with multiplicities. An appropriate
survey on $n$-valued groups and its applications can be found in [13].

If we have a group system $\mathcal{G}=(G,*_{i},i\in I)$, where $|I|=n$, we
can can define $n$-valued multiplication

$a*b=[a*_{1}b,a*_{2}b,\ldots,a*_{n}b],\leavevmode\nobreak\
\leavevmode\nobreak\ a,b\in G,$

and study the algebraic system $(G,*)$. In [14] were investigated connections
between group systems and $n$-valued groups and proved that if all groups
$(G,*_{i})$ have the general unit and $(G,*)$ is an $n$-valued group, then
$*_{i}=*_{j}$ for all $1\leq i,j,\leq n$.

In the present paper we are studying connections between skew braces and
dimonoids, define a semigroup systems on the set of square matrices. Also, we
investigate semigroup systems on the set of matrices $M_{n}(\Bbbk)$ and give
an answer on a question from [14]. Also, we construct some rack systems and
multi-racks on the set $V\times G$, where $V$ is a vector space of dimension
$n$ over a field $\Bbbk$, $G$ is a subgroup of $\operatorname{GL}_{n}(\Bbbk)$.

The paper is organized as follows.

In Section 2 we introduce definition of algebraic systems, which include the
algebraic systems from the introduction.

In Section 3 we construct some group systems on the set of square matrices
over a field and give an answer on a question from [14].

In Section 4 will be define rack system on the set $V\times G$,
$G\leq\operatorname{GL}_{n}(\Bbbk)$.

In Section 5 will be established connection between skew braces and dimonoids.

##  2 Homogeneous algebraic systems

In this section we introduce definition of algebraic systems, which include
the algebraic systems from the introduction.

######  Definition 2.1.

Let $\mathcal{A}=(A,f_{i},i\in I)$ be an algebraic system with a set of
algebraic operations $f_{i}$ of arity $n_{i}$. It is said to be
$m$-homogeneous $I$-system if all arities $n_{i}$ are equal to $m$. In
particular, if $|I|=n$, we will say on $n$-system. If $m=2$ we will say
instead $2$-homogeneous $I$-system on groupoid $I$-system or simply on
groupoid system.

The typical example is a ring $(K,+,\cdot)$ that is groupoid 2-system. Other
examples of $2$-homogeneous $I$-system are: semigroup (monoid, group) system
$\mathcal{G}=(G,*_{i},i\in I)$, where $(G,*_{i})$ is a semigroup (monoid,
group) for any $i\in I$. An example of a semigroup system with two operations
is a duplex. We will call a system $\mathcal{G}$ by multi-semigroup (multi-
monoid, multi-group) if the operations are connected by the following
condition

$(a*_{i}b)*_{j}c=a*_{i}(b*_{j}c),\leavevmode\nobreak\ \leavevmode\nobreak\
a,b,c\in S,\leavevmode\nobreak\ \leavevmode\nobreak\ i,j\in I.$

An example of a multi-semigroup with $n$ operations is an $n$-tuple semigroup
(see the introduction).

Let us give other examples of semigroup systems.

Skew braces (see [1], [2]). A triple $(G,\cdot,\circ)$, where $(G,\cdot)$ and
$(G,\circ)$ are groups, is said to be a _skew (left) brace_ if

$g_{1}\circ(g_{2}\cdot g_{3})=(g_{1}\circ g_{2})\cdot
g_{1}^{-1}\cdot(g_{1}\circ g_{3})$ (2.0.1)

for all $g_{1},g_{2},g_{3}\in G$, where $g_{1}^{-1}$ denotes the inverse of
$g_{1}$ in $(G,\cdot)$. We call $(G,\cdot)$ the _additive group_ and
$(G,\circ)$ the _multiplicative group_ of the skew left brace
$(G,\cdot,\circ)$. A skew left brace $(G,\cdot,\circ)$ is said to be a _(left)
brace_ if $(G,\cdot)$ is an abelian group. In this case we will use the
notation $+$ instead $\cdot$ in additive group. We see that a skew left brace
is an example of group system with 2 operations.

Dimonoids (see [4], [15]). A dimonoid is a set X together with two binary
operations $\vdash$ and $\dashv$ satisfying the following axioms:

$\begin{cases}x\dashv(y\dashv z)\overset{1}{=}(x\dashv y)\dashv
z\overset{2}{=}x\dashv(y\vdash z),&\\\ (x\vdash y)\dashv
z\overset{3}{=}x\vdash(y\dashv z),&\\\ (x\dashv y)\vdash
z\overset{4}{=}x\vdash(y\vdash z)\overset{5}{=}(x\vdash y)\vdash
z.&\end{cases}$

for all $x,y$ and $z\in X$. Observe that relations $1$ and $5$ are the
‘‘associativity’’ of the products $\vdash$ and $\dashv$ respectively. The
typical examples of dimonoid are the following.

a) Let $M$ be a monoid. Put $D=M\times M$ and define the products by

$(m,n)\dashv(m^{\prime},n^{\prime}):=(m,nm^{\prime}n^{\prime}),$
$(m,n)\vdash(m^{\prime},n^{\prime}):=(mnm^{\prime},n^{\prime}).$

With these definitions $D=(D,\dashv,\vdash)$ is a dimonoid. Let us check
relation 3 for instance:

$((m,n)\vdash(m^{\prime},n^{\prime}))\dashv(m^{\prime\prime},n^{\prime\prime})=(mnm^{\prime},n^{\prime})\dashv(m^{\prime\prime},n^{\prime\prime})=(mnm^{\prime},n^{\prime}m^{\prime\prime}n^{\prime\prime})$
$(m,n)\vdash((m^{\prime},n^{\prime})\dashv(m^{\prime\prime},n^{\prime\prime}))=(m,n)\vdash(m^{\prime},n^{\prime}m^{\prime\prime}n^{\prime\prime})=(mnm^{\prime},n^{\prime}m^{\prime\prime}n^{\prime\prime}).$

Let $1\in M$ be a unit for $M$. Then $e=(1,1)$ is a bar-unit for $D$, but one
has $e\dashv x\neq x$ and $x\vdash x\neq x$ in $D$ in general. For any
invertible element $m$ the element $(m,m^{-1})\in D$ is a bar-unit.

b) Let $G$ be a group and $X$ be a $G$-set. The following formulas define a
dimonoid structure on $X\times G$:

$(x,g)\dashv(y,h):=(x,gh),$ $(x,g)\vdash(y,h):=(g\cdot x,gh).$

We see that a dimonoid is an example of group system with 2 operations.

##  3 Group systems and multi-groups

Let $M_{n}(\Bbbk)$ be a set of $n\times n$ matrices over a field $\Bbbk$. The
next multiplication was defined in [14]:

$A*_{s,t,M_{1},M_{2}}B=sAM_{1}B+tAM_{2}B,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ s,t\in\Bbbk,\leavevmode\nobreak\
\leavevmode\nobreak\ M_{1},M_{2}\in M_{n}(\Bbbk),$

and was formulated

######  Question 3.1.

What can we say on this product? What algebraic systems one can construct
using these operations? Is there connection of these operations with non-
standard matrix multiplications which were studied in [18]?

Let us find conditions under which $(M_{n}(\Bbbk),*_{s,t,M_{1},M_{2}})$ is a
semigroup. It is need to check axiom of associativity,

$(A*B)*C=A*(B*C),\leavevmode\nobreak\ \leavevmode\nobreak\ A,B,C\in
M_{n}(\Bbbk).$

We have

$(A*B)*C=(sAM_{1}B+tAM_{2}B)*C=s(sAM_{1}B+tAM_{2}B)M_{1}C+t(sAM_{1}B+tAM_{2}B)M_{2}C,$

on the other side,

$A*(B*C)=A*(sBM_{1}C+tBM_{2}C)=sAM_{1}(sBM_{1}C+tBM_{2}C)+tAM_{2}(sBM_{1}C+tBM_{2}C).$

We have a system

$\begin{cases}AM_{1}BM_{1}C=AM_{1}BM_{1}C;&\\\
AM_{2}BM_{1}C+AM_{1}BM_{2}C=AM_{1}BM_{2}C+AM_{2}BM_{1}C;&\\\
AM_{2}BM_{2}C=AM_{2}BM_{2}C.&\end{cases}$

It is easy to see that the associativity axiom holds. Hence, we have proven
the next lemma.

######  Lemma 3.2.

The operation $*_{s,t,M_{1},M_{2}}$ is associative.

######  Corollary 3.3.

The algebraic system
$(M_{n}(\Bbbk),*_{s,t,M_{1},M_{2}},s,t\in\Bbbk,M_{1},M_{2}\in M_{n}(\Bbbk))$
is a semigroup system.

Let us check, is this semigroup system a multi-semigroup? To do it it is need
to take two different operations: $*=*_{s,t,M_{1},M_{2}}$,
$\circ=\circ_{p,q,N_{1},N_{2}}$ and check the axiom

$(A*B)\circ C=A*(B\circ C).$

The left hand side:

$(A*B)\circ C=(sAM_{1}B+tAM_{2}B)\circ
C=p(sAM_{1}B+tAM_{2}B)N_{1}C+q(sAM_{1}B+tAM_{2}B)N_{2}C.$

The right hand side:

$A*(B\circ
C)=A*(pBN_{1}C+qBN_{2}C)=sAM_{1}(pBN_{1}C+qBN_{2}C)+tAM_{2}(pBN_{1}C+qBN_{2}C).$

We get a system

$\begin{cases}AM_{1}BN_{1}C=AM_{1}BN_{1}C;&\\\
AM_{2}BN_{1}C=AM_{2}BN_{1}C;&\\\ AM_{1}BN_{2}C=AM_{1}BN_{2}C;&\\\
AM_{2}BN_{2}C=AM_{2}BN_{2}C.&\end{cases}$

Since, this system true for all matrices, we get

######  Proposition 3.4.

The semigroup system
$(M_{n}(\Bbbk),*_{s,t,M_{1},M_{2}},s,t\in\Bbbk,M_{1},M_{2}\in M_{n}(\Bbbk))$
is a multi-semigroup.

Let us find the unit element:

$A*X=sAM_{1}X+tAM_{2}X=A.$

It means that $t=0,s=1,M_{1}X=E\Rightarrow X=M_{1}^{-1}.$

Hence, $A*X=AM_{1}X,E^{(*)}=M_{1}^{-1}$. On the other side $X*A=XM_{1}A=A$.

######  Lemma 3.5.

We will have the unit element only for multiplication $A*B=AMB$,
${detM}{\neq}0$, $E^{(*)}=M^{-1}.$

Inverse element $A*Y=E^{(*)}\Leftrightarrow AMY=M^{-1}.$ Hence,
$Y=M^{-1}A^{-1}M^{-1}.$

######  Theorem 3.6.

1) Let $M\in M_{n}(\Bbbk)$, ${detM}{\neq}0$. Then $(GL_{n}(\Bbbk),*_{M})$ is a
group with the product $A*_{M}B=AMB$, with the unit element $E^{(*)}=M^{-1}$
and the inverse $\bar{A}^{(*)}=M^{-1}A^{-1}M^{-1}.$

2) The algebraic system $(GL_{n}(\Bbbk),*_{M},M\in GL_{n}(\Bbbk))$ is a multi-
group.

##  4 Rack systems

Some examples of quandle systems and multi-quandles can be found in [10, 11].
In this section we give some other examples. At first, recall basic
definitions.

######  Definition 4.1 ([16], [17]).

A quandle is a non-empty set $Q$ with a binary operation $(x,y)\mapsto x*y$
satisfying the following axioms:

  1. (Q1)

$x*x=x$ for all $x\in Q$,

  2. (Q2)

for any $x,y\in Q$ there exists a unique $z\in Q$ such that $x=z*y$,

  3. (Q3)

$(x*y)*z=(x*z)*(y*z)$ for all $x,y,z\in Q$.

An algebraic system satisfying only (Q2) and (Q3) is called a rack. Many
interesting examples of quandles come from groups.

######  Example 4.2.

  1. 1.

If $G$ is a group, $m$ is an integer, then the binary operation
$a*_{m}b=b^{-m}\,a\,b^{m}$ turns $G$ into a quandle
$\operatorname{Conj}_{m}(G)$ called the $m$-conjugation quandle on $G$. If
$m=1$, this quandle is called conjugation quandle and is denoted
$\operatorname{Conj}(G)$.

  2. 2.

A group $G$ with the binary operation $a*b=ba^{-1}b$ turns the set $G$ into a
quandle $\operatorname{Core}(G)$ called the core quandle of $G$. In
particular, if $G=\mathbb{Z}_{n}$ is the cyclic group of order $n$, then it is
called the dihedral quandle and denoted by $\operatorname{R}_{n}$.

  3. 3.

Let $G$ be a group and $\varphi\in\operatorname{Aut}(G)$. Then the set $G$
with binary operation $a*_{\varphi}b=\varphi(ab^{-1})b$ forms a quandle
$\operatorname{Alex}(G,\ \varphi)$ referred as the generalized Alexander
quandle of $G$ with respect to $\varphi$.

From the last example follows that if $Q=\operatorname{GL}_{n}(\Bbbk)$,
$\varphi\in\operatorname{Aut}(\operatorname{GL}_{n}(\Bbbk))$, then we can
define a quandle system
$(Q,*_{\varphi},\varphi\in\operatorname{Aut}(GL_{n}(\Bbbk)))$.

In the present section we are studying the next question: What rack (quandle)
systems one can defines on $V\times G$, where $V$ is a vector space of
dimension $n$ over a field $\Bbbk$, $G$ is a subgroup of
$\operatorname{GL}_{n}(\Bbbk)$. On the set $Q=V\times G$ we can define the
operation

$(a,A)\circ(b,B)=(Ab,\varphi(AB^{-1})B),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ a,b\in V,\leavevmode\nobreak\
\leavevmode\nobreak\ A,B\in G,\leavevmode\nobreak\ \leavevmode\nobreak\
\varphi\in\operatorname{Aut}(G).$

In this case

$(a,A)\circ(a,A)=(Aa,\varphi(E)A)=(Aa,A).$

It means that $A=E$, hence, $G=\\{E\\}$ is the trivial group.

The second quandle axiom:

$(u,X)\circ(a,A)=(b,B)\Leftrightarrow(u,X)\circ(a,A)=(Xa,\varphi(XA^{-1})A).$

Hence,

$Xa=b,\leavevmode\nobreak\ \leavevmode\nobreak\ X=\varphi^{-1}(BA^{-1})A.$

It means that such element $(u,X)$ exists, but it is not unique.

Let us check the third axiom:

$((a,A)\circ(b,B))\circ(c,C)=((a,A)\circ(c,C))\circ((b,B)\circ(c,C)).$

The left hand side,

$((a,A)\circ(b,B))\circ(c,C)=(Ab,\varphi(AB^{-1})B)\circ(c,C)=(\varphi(AB^{-1})Bc,\varphi(\varphi(AB^{-1})BC^{-1})C).$

The right hand side

$((a,A)\circ(c,C))\circ((b,B)\circ(c,C))=(Ac,\varphi(AC^{-1})C)\circ(Bc,\varphi(BC^{-1})C)=$
$=(\varphi(AC^{-1})CBc,\varphi(\varphi(AC^{-1})C).$

We have the system

$\begin{cases}\varphi(AB^{-1})Bc=\varphi(AC^{-1})CBc;&\\\
\varphi(\varphi(AB^{-1})BC^{-1})C=\varphi(\varphi(AC^{-1})C).&\end{cases}$

Since, $A*B=\varphi(AB^{-1})B$ defines a quandle operation, the second
equation is true. Consider the first equation of the system. It equivalents to
the equality

$\varphi(AB^{-1}CA^{-1})=C,$

which must be true for arbitrary $A,B,C\in G$. Evidently, that for trivial
group it is true.

Let us define the operation (see [19]):

$(a,A)\circ(b,B)=(Ab,ABA^{-1}),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ a,b\in V,\leavevmode\nobreak\ \leavevmode\nobreak\ A,B\in
G,$

and check the left self-distributivity,

$(a,A)\circ((b,B)\circ(c,C))=((a,A)\circ(b,B))\circ((a,A)\circ(c,C)).$

Since

$(a,A)\circ((b,B)\circ(c,C))=(a,A)\circ(Bc,BCB^{-1})=(ABc,ABCB^{-1}A^{-1})$

and

$((a,A)\circ(b,B))\circ((a,A)\circ(c,C))=(Ab,ABA^{-1})\circ(Ac,ACA^{-1})=(ABc,ABCB^{-1}A^{-1}),$

then the left self-distributivity holds.

Let us take $n\in\mathbb{Z}$ and define more general operation,

$(a,A)\circ_{n}(b,B)=(A^{n}b,A^{n}BA^{-n}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ a,b\in V,\leavevmode\nobreak\
\leavevmode\nobreak\ A,B\in G.$

Check the left self-distributivity,

$(a,A)\circ_{n}((b,B))\circ_{n}((c,C))=(a,A)\circ_{n}(B^{n}c,B^{n}CB^{-n})=(A^{n}B^{n}c,A^{n}B^{n}CB^{-n}A^{-n}),$
$((a,A)\circ_{n}(b,B))\circ_{n}((a,A)\circ_{n}(c,C))=(A^{n}b,A^{n}BA^{-n})\circ_{n}(A^{n}c,A^{n}CA^{-n})=(A^{n}B^{n}c,A^{n}B^{n}CB^{-n}A^{-n}).$

Hence, the operation $(a,A)\circ_{n}(b,B)=(A^{n}b,A^{n}BA^{-n})$ is left self-
distributive.

Let us check the left divisibility axiom: $(a,A)\circ_{n}(u,X)=(b,B)$ We have

$(a,A)\circ_{n}(u,X)=(A^{n}u,A^{n}XA^{-n}).$

Hence,

$\begin{cases}u=A^{-n}b;&\\\ X=A^{-n}BA^{n}.&\end{cases}$

Since this system has unique solution, the left divisibility holds.

Summarize the previous calculations, we get

######  Theorem 4.3.

Let $Q=(V,G)$, where $V$ is a vector space of dimension $n$ over a field
$\Bbbk$, $G$ is a subgroup of $\operatorname{GL}_{n}(\Bbbk)$. Then the
algebraic system $(Q,*_{n},n\in\mathbb{Z})$, where

$(a,A)\circ_{n}(b,B)=(A^{n}b,A^{n}BA^{-n}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ a,b\in V,\leavevmode\nobreak\
\leavevmode\nobreak\ A,B\in G$

satisfies the axioms:

1) Left self-distributivity,

$(a,A)\circ_{n}((b,B)\circ_{n}(c,C))=((a,A)\circ_{n}(b,B))\circ_{n}((a,A)\circ_{n}(c,C)),\leavevmode\nobreak\
\leavevmode\nobreak\ a,b,c\in V,\leavevmode\nobreak\ \leavevmode\nobreak\
A,B,C\in G.$

2) Left divisibility,

for any $(a,A)$, $(b,B)\in Q$ there is unique $(u,X)\in Q$ such that

$(a,A)\circ_{n}(u,X)=(b,B).$

From this theorem follows

######  Corollary 4.4.

The algebraic system $(Q,*^{op}_{n},n\in\mathbb{Z})$, where the opposite
operations are defined by the rules

$(a,A)*^{op}_{n}(b,B)=(b,B)*_{n}(a,A)$

is a rack system.

##  5 Connection between skew braces and dimonoids

In this section we find some connections between skew braces and dimonoids.

######  Proposition 5.1.

Let $(G,\cdot)$ be a group.

1) If $a\circ b=a\cdot b$ then we get a skew brace. If $a\vdash b=a\dashv
b=ab$ then we get a dimonoid.

2) If $a\circ b=ba$, then $(G,\cdot,\circ)$ is skew brace. If $a\vdash b=ab$
and $a\dashv b=ba$ then $(G,\dashv,\vdash)$ is not a dimonoid.

###### Proof.

The binary operation $\vdash$ is associative since it corresponds to the
product in the group $G$.

Let us check the following axiom:

$(a\dashv b)\dashv c=a\dashv(b\dashv c).$

To do it, we have to compute both sides of the equation:

$(ba)\dashv c=c(ba),$ $a\dashv(cb)=(cb)a.$

Since they are the same, the operation is associative.

Let us check the following axiom: $a\dashv(b\dashv c)=a\dashv(b\vdash c)$. We
have $cba=bca$. Therefore, it must satisfy $bc=cb$ and this group is an
abelian group. It means that if $a\dashv b=a\vdash b=ab$, then
$(G,\vdash,\dashv)$ is a dimonoid.

∎

If we a have a skew brace $(G,\cdot,\circ)$, then we can define operations
$a\vdash b=ab,a\dashv b=a\circ b$ and formulate the question: is
$(G,\vdash,\dashv)$ a dimonoid?

The next example shows that in the general case the answer is negative.

######  Example 5.2.

Let us take the brace $(\mathbb{Z},+,\circ)$, where $(\mathbb{Z},+)$ is the
infinite cyclic group and $a\circ b=a+(-1)^{a}b$, $a,b\in\mathbb{Z}$. Note
that

$a\circ b=a+(-1)^{a}b=\begin{cases}a+b,&if\ a\ is\ even;\\\ a-b,&if\ a\ is\
odd.\end{cases}$

Put

$a\vdash b=a+b,a\dashv b=a\circ b.$

It is evident that the associativity holds for the binary operations $\vdash$
and $\dashv$. Let us check that

$a\dashv(b\dashv c)=a\dashv(b\vdash c).$

We have that

$a\circ(b\circ c)=a\circ\begin{cases}b+c,&if\ b\ is\ even;\\\ b-c,&if\ b\ is\
odd.\end{cases}=\begin{cases}a+b+c&if\ a,b\ are\ even;\\\ a-b-c&if\ a\ is\
odd\ and\ b\ is\ even;\\\ a+b-c&if\ a\ is\ even\ and\ b\ is\ odd;\\\ a-b+c&if\
a,b\ are\ odd.\end{cases}$

On the other side we get

$a\circ(b+c)=\begin{cases}a+b+c,&if\ a\ is\ even;\\\ a-b-c,&if\ a\ is\
odd.\end{cases}$

If we take $a=2$, $b=3$, $c=4$, then $a\dashv(b\dashv c)=a+b-c=1.$ On the
other side $a\dashv(b\vdash c)=a+b+c=9.$

Therefore, the skew brace $(\mathbb{Z},+,\circ)$ is not a dimonoid.

At the end we formulate the following question.

######  Question 5.3.

Under which conditions a skew brace $(G,\cdot,\circ)$ is a dimonoid with
respect to the operations $a\vdash b=ab,a\dashv b=a\circ b?$

######  Question 5.4.

Let $\mathcal{G}=(G,*_{i},i\in I)$ be a semigroup system. Define a product of
semigroup operations,

$g(*_{i}*_{j})h=(g*_{i}h)*_{j}h,\leavevmode\nobreak\ \leavevmode\nobreak\
g,h\in G.$

Find necessary and sufficient conditions under which $(Q,*_{i}*_{j})$ is a
semigroup.

######  Acknowledgement.

This work was supported by the Ministry of Science and Higher Education of
Russia (agreement No. 075-02-2023-943).

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