# Ordered Ehresmann semigroups and categories

T. Stokes

###### Abstract

Ehresmann semigroups may be viewed as biunary semigroups equipped with domain
and range operations satisfying some equational laws. Motivated by some of the
main examples, we here define ordered Ehresmann semigroups, and consider their
basic properties as well as special cases in which the order is algebraically
definable. In particular, one and two-sided restriction semigroups equipped
with their natural orders are characterised within the class of ordered
Ehresmann semigroups. The main result is an ESN-style theorem for ordered
Ehresmann semigroups with particular reference to the special cases.

Keywords. Ehresmann semigroup; ordered semigroup; ESN Theorem.

2020 Mathematics Subject Classification. 20M50; 06F05

##  1 Introduction

The Ehresmann-Nambooripad-Schein Theorem (“ESN Theorem”) for inverse
semigroups gives an isomorphism between the category of inverse semigroups and
a suitable category of partial algebras. The main significance of this result
is that one can capture the entire semigroup by limited knowledge of the
product: just the compatible products (defined in some way), certain products
of idempotents and arbitary elements (“restriction” and “corestriction”), and
the product of idempotents with one-another (a semilattice operation); one
also retains the natural order determined by the inverse semigroup. The
associated partial algebra is then a small category with additional order-
theoretic structure, making it a so-called “inductive groupoid”, from which
the original inverse semigroup may be reconstructed. Moreover every inductive
groupoid arises from an inverse semigroup in this way, the correspondence is
one-to-one, and there is a suitable category isomorphism between the two
classes.

Onbe of the most successful generalisations of this result to possibly non-
regular semigroups is given by Lawson in [4], applying to what the author
calls Ehresmann semigroups. These may be defined in various ways, the one
adopted here being as biunary semigroups equipped with unary operations here
denoted by $D,R$ that model domain and range in a certain sense. Examples
include (in increasing generality): inverse semigroups in which one defines
$D(s)=ss^{\prime}$ and $R(s)=s^{\prime}s$ for all $s$ (where $s^{\prime}$ is
the inverse of $s$), (two-sided) restriction semigroups, and semigroups
embeddable in the semigroup $\operatorname{\sf Rel}(X)$ of binary relations on
some set $X$ under relational composition, in which for all
$\rho\in\operatorname{\sf Rel}(X)$,

$D(\rho)=\\{(x,x)\mid x\in\mbox{dom}(\rho)\\},\ R(\rho)=\\{(x,x)\mid
x\in\mbox{ran}(\rho)\\}.$

In an Ehresmann semigroup $S$, there is a distinguished semilattice of
idempotents $D(S)$, consisting of all elements of the form $D(s)$
(equivalently, $R(s)$), and this replaces the semilattice of all idempotents
used in the original ESN theorem.

In [4], Lawson showed how to obtain a suitable type of small category from an
Ehresmann semigroup $S$ that retains only those products $st$ where
$R(s)=D(t)$, retains $D$ and $R$, and retains not one but two partial orders
determined by $S$. Defining $s\leq_{l}t$ when $s=D(s)t$, and $s\leq_{r}t$ when
$s=tR(s)$, both $\leq_{l}$ and $\leq_{r}$ are partial orders, each agreeing
with the semilattice order on $D(S)$ when restricted to it. One can use these
two partial orders to characterise products such as $es,sf$ where $e,f\in
D(S)$ are such that $e\leq D(s)$, $f\leq R(s)$ respectively: $es$ is the
unique $t\in S$ such that $t\leq_{l}s$ and $D(t)=e$, and dually for $sf$. One
can then characterise precisely those enriched categories arising in this way:
they have two partial orders satisfying certain conditions, and notions of
restriction and corestriction defined via these partial orders, which allow
products of the form $es,sf$ to be formed, and thence products of arbitrary
category elements so that multiplication is everywere-defined. Lawson
characterised the class of small categories arising in this way and called
them Ehresmann categories in [4]; he then proceeded to obtain a category
isomorphism between Ehresmann semigroups (with morphisms being homomorphisms
preserving $D$ and $R$) and Ehresmann categories (equipped with suitable
morphisms).

Much more recently, in [5] Lawson took a different and rather simpler
approach. Instead of defining partial orders to augment the category structure
determined by an Ehresmann semigroup $S$, he supplemented the category with a
so-called biaction of the identities of the category on the arrows in general,
thereby allowing semigroup products of the form $es,sf$ ($e,f\in D(S),s\in S$)
to be captured; in particular, products of the form $ef$ (where $e,f\in D(S)$)
are captured, giving a semilattice structure on the projections. Lawson
characterised the categories with biaction arising in this way, calling them
categories with Ehresmann biaction. This approach requires less structure and
complexity of axioms than that presented in [4], dispensing with partial
orders entirely, but comes at the expense of requiring all products of the
form $es,se$ to be retained rather than only some (although it is not hard to
slightly reformulate things so that one need only consider products of the
form $es$ where $e\leq D(s)$ and dually). A very similar approach was used by
Fitzgerald and Kinyon in [3], where slightly greater generality was used: the
distinguished semilattice is replaced by a band.

Lawson’s approach to obtaining an ESN-style theorem for Ehresmann semigroups
used in [5] may be far simpler than that used in [4], but we believe the
general ideas in [4] have much value, particularly if the Ehresmann semigroups
one is interested in are themselves ordered. In fact many natural examples of
Ehresmann semigroups are ordered in a way compatible with the operations:
inverse and restriction semigroups by their natural order, and
$\operatorname{\sf Rel}(X)$ by the order of set inclusion (which is not
algebraically definable).

When the Ehresmann semigroups under consideration are ordered, this has
immediate ramifications for the the associated categories in any ESN-style
theorem one seeks: such categories will necessarily be ordered. One could
proceed by attempting to “bolt on” the additional order structure to either an
Ehresmann category (which already has two partial orders) or a category with
Ehresmann biaction (which has none). However, we shall see that one can work
directly with the order without requiring the additional structure of either
an Ehresmann category or a category with Ehresmann biaction – such additional
structure may be derived but need not be assumed. Special cases may be
obtained corresponding to restriction semigroups, Ehresmann semigroups that
are one-sided restriction semigroups, and their common generalisation, the so-
called de Barros semigroups (defined in [4]).

In this work, we first review the details of the earlier work on Ehresmann
semigroups and ESN-style theorems for them, especially as it appears in [4]
and [5]. We then define ordered Ehresmann semigroups to be those admitting a
so-called Ehresmann order, giving examples and obtaining their basic
properties. It is shown that a given Ehresmann semigroup may admit more than
one Ehresmann order or possibly none at all. We consider special cases of
Ehresmann semigroups in which the Ehresmann order may be algebraically
definable, giving alternative characteristions of these classes as ordered
Ehresmann semigroups.

The final section concerns our “ESN Theorem”. We use a combination of some of
the laws considered in [4], together with a new law, to define the appropriate
types of small categories with order, then obtain the hoped-for category
isomorphism. We then obtain special cases of the correspondence applying to
the particular classes of Ehresmann semigroups with algebraically definable
Ehresmann order considered in the previous section.

Throughout, all function and relation compositions will be written using the
left-to-right convention, and functions will be written on the right of their
arguments, aside from unary operations which will be written on the left
($D(s)$ rather than $sD$, and so on).

##  2 Preliminaries

###  2.1 Localisable and Ehresmann semigroups

We start with slightly greater generality than Ehresmann semigroups. As in
[3], a localisable semigroup is a biunary semigroup $(S,\cdot,D,R)$ satisfying
the following laws:

  1. (L1)

$D(s)s=s,sR(s)=s$;

  2. (L2)

$D(R(s))=R(s),R(D(s))=D(s)$;

  3. (L3)

$D(st)=D(sD(t)),R(st)=R(R(s)t)$;

  4. (L4)

$D(D(s)D(t))=D(s)D(t)$.

These are not exactly the laws given in [3]: the final law is different but
follows from those given in [3] since $D(S)=\\{D(s)\mid s\in S\\}=R(S)$ is a
band in any localisable semigroup. Conversely, the law $D(D(s)t)=D(s)D(t)$,
appearing in [3], follows easily from the above laws since
$D(D(s)t)=D(D(s)D(t))=D(s)D(t)$ using (L2) and (L4); dually for the law
$R(sR(t))=R(s)R(t)$.

A localisable semigroup is Ehresmann if it also satisfies

  * •

$D(s)D(t)=D(t)D(s)$.

If $(S,\cdot,D,R)$ is localisable, we call elements of $D(S)=R(S)$
projections; they form a band under multiplication, which is a semilattice if
(and only if) $(S,\cdot,D,R)$ is Ehresmann. (Ehresmann semigroups were first
defined in [4] simply as semigroups, with the unary operations secondary, but
here we assume the unary operations are part of the signature throughout.)

Examples of Ehresmann semigroups include any two-sided restriction semigroup,
hence any inverse semigroup $(S,\cdot,^{\prime})$ in which one defines
$D(s)=ss^{\prime}$ and $R(s)=s^{\prime}s$, where $s^{\prime}$ is the inverse
of $s\in S$. A more generic example is $\operatorname{\sf Rel}(X)$, the
semigroup of binary relations on $X$ equipped with relational composition
(written left-to-right), and unary operations $D$ and $R$, where

$D(\rho)=\\{(x,x)\mid x\in\mbox{dom}(\rho)\\},\ R(\rho)=\\{(x,x)\mid
x\in\mbox{ran}(\rho)\\}.$

Hence, the projections are precisely the restrictions of the identity map on
$X$. Some examples of localisable semigroups that are not Ehresmann are given
in [3].

###### Definition 2.1

Let $(S,\cdot,D,R)$ and $(T,\cdot,D,R)$ be Ehresmann semigroups. We say
$F:S\rightarrow T$ is an Ehresmann semigroup homomorphism if it is a semigroup
homomorphism which respects $D$ and $R$: for all $s\in S$, $D(s)F=D(sF)$ and
$R(s)F=R(sF)$.

This notion of Ehresmann semigroup homomorphism coincides with that of an
admissible homomorphism considered in [4], as shown there (in the discussion
prior to Lemma 4.22). It also coincides with the notion of morphism for
localisable semigroups given in [3].

###  2.2 ESN-style theorems for Ehresmann semigroups

A (small) category here refers to a partial algebra $(C,\circ,D,R)$, in which
$\circ$ is a partial binary operation and $D,R$ are unary domain and range
operations such that the following laws are satisfied for all $x,y,z\in C$:

  * •

$D(x)\circ x=x=x\circ R(x)$;

  * •

$D(R(x))=R(x)$, $R(D(x))=D(x)$;

  * •

$x\circ y$ is defined if and only if $R(x)=D(y)$;

  * •

if $x\circ y$ is defined then $D(x\circ y)=D(x)$, $R(x\circ y)=R(y)$;

  * •

if $(x\circ y)\circ z$ and $x\circ(y\circ z)$ are defined then they are equal.

This is the approach taken in [3] and is a common one in algebra, in which
objects are not needed; one can introduce an object for each element of the
set of identities $D(C)=R(C)=\\{D(x)\mid x\in C\\}$ of $C$, and then the above
definition is easily seen to be equivalent to the usual one for small
categories.

###### Definition 2.2

Given an Ehresmann semigroup $(S,\cdot,D,R)$, define the partial product
$s\circ t$ to equal $st$ but to only be defined when $R(s)=D(t)$.

This is precisely what is done in [4] and [5], and for the larger class of
localisable semigroups in [3]. The resulting partial algebra $(S,\circ,D,R)$
is a category which is used in various ways in these works to obtain ESN-style
theorems for Ehresmann and localisable semigroups.

The following definition is from [4], but follows earlier work due to
Ehresmann (as do most of the subsequent properties for categories considered
below – see [2]).

###### Definition 2.3

Let $C$ be a set. We say $(C,\circ,D,R,\leq)$ is an $\Omega$-structured
category if it satisfies the following axioms on $C$:

  1. (OC1)

$(C,\circ,D,R)$ is a category and $(C,\leq)$ is a partially ordered set;

  2. (OC2)

if $a\leq b$ then $D(a)\leq D(b)$ and $R(a)\leq R(b)$;

  3. (OC3)

if $a\leq b$ and $c\leq d$ with both $a\circ c$ and $b\circ d$ existing, then
$a\circ c\leq b\circ d$.

The definition of ordered category given in [4] also requires the following
law:

  1. (OC4)

if $a\leq b$, $D(a)=D(b)$, and $R(a)=R(b)$ then $a=b$.

Note that (OC4) can be expressed as saying that $\leq$ is trivial on hom-sets.
It can be usefully strengthened in two different ways:

  1. (OC4A)

for all $s,t\in C$ if $s\leq t$ and $D(s)=D(t)$, then $s=t$;

and

  1. (OC4B)

for all $s,t\in C$ if $s\leq t$ and $R(s)=R(t)$, then $s=t$.

As in [4], the $\Omega$-structured category $(C,\circ,D,R,\leq)$ is said to be
sufficiently regular if it satisfies the following conditions (together called
(OC6) in [4]):

  * (OC6a)

for all $x\in C$ and $e\in D(C)$ for which $e\leq D(x)$, there is
$e|x=max\\{y\leq x\mid D(y)\leq e\\}$, and $D(e|x)=e$;

  * (OC6b)

for all $x\in C$ and $e\in D(C)$ for which $e\leq R(x)$, there is
$x|e=max\\{y\leq x\mid R(y)\leq e\\}$, and $R(x|e)=e$.

In an $\Omega$-structured category $(C,\circ,D,R,\leq)$, if (OC6a)
(respectively (OC6b)) is satisfied, $x\in C$ and $e\in D(C)$ with $e\leq D(x)$
(resp. $e\leq R(x)$), we call $e|x$ (resp. $x|e$) the restriction (resp.
corestriction) of $x$ by $e$.

A further possible law for $\Omega$-structured categories which is considered
in [4] is the following:

  1. (OC7)

if $a\leq b\circ c$, then there are $b^{\prime}\leq b,c^{\prime}\leq c$ such
that $a=b^{\prime}\circ c^{\prime}$.

A regular category is a sufficiently regular one satisfying (OC7).

A further possible law holding in an $\Omega$-structured category $C$
considered in [4] is the following, which asserts that $D(C)$ is an order-
ideal:

  1. (OCI)

if $a\in C$ and $a\leq e\in D(C)$, then $a\in D(C)$.

As Lawson notes in [4], de Barros makes the following definition in [1].

###### Definition 2.4

An $\Omega$-structured category satisfying (OC4), (OC6), (OC7) and (OCI) is a
totally, regularly ordered category.

A stronger property (strictly, pair of properties) than (OC6) for
$\Omega$-structured categories was considered in [4] and called (OC8) there:

  1. (OC8a)

for all $x\in C$ and $e\in D(C)$ such that $e\leq D(x)$, $e|x$ is the unique
$y\leq x$ such that $D(y)=e$;

  2. (OC8b)

for all $x\in C$ and $e\in D(C)$ such that $e\leq R(x)$, $x|e$ is the unique
$y\leq x$ such that $R(y)=e$.

In [4], an $\Omega$-structured category satisfying (OC8a) (resp, (OC8b)) was
said to have restrictions (resp. corestrictions). (Strictly speaking, Lawson
in [4] uses terminology dual to that used here, since compositions are read
right-to-left there.) Hence, the notions of restriction and corestriction
defined here based on (OC6) are more general than Lawson’s.

It was noted in Proposition 2.8 of [4] that if (OC8) holds in an
$\Omega$-structured category, then so do (OC4), (OC7) and (OCI), and of course
(OC6) obviously holds also, so the category is totally, regularly ordered.
Examining the short proof of this result in [4] shows that (OC8a) is
sufficient for not just (OC4), (OC7) and (OCI), but also the stronger
condition (OC4A); likewise (OC8b) is sufficient for (OC4B). It is also easy to
see that (OC4A) and (OC6) together imply (OC8a); indeed (OC4A) and (OC8a) are
equivalent in the presence of (OC6). We have therefore shown the following.

###### Proposition 2.5

For an $\Omega$-structured category, (OC8a) is equivalent to (OC4A) and (OC6),
and (OC8) is equivalent to (OC4A), (OC4B) and (OC6).

The Ehresmann categories of [4], shown there to be equivalent to (unordered)
Ehresmann semigroups, can now be defined. (Again note that Lawson used dual
definitions in [4].)

###### Definition 2.6

A structure $(C,\circ,D,R,\leq_{l},\leq_{r})$ is said to be an Ehresmann
category if $(C,\circ,D,R)$ is a category, and $\leq_{l}$ and $\leq_{r}$ are
partial orders on $C$ such that:

  * •

$(C,\circ,D,R,\leq_{l})$ is $\Omega$-structured and satisfies (OC8a);

  * •

$(C,\circ,D,R,\leq_{r})$ is $\Omega$-structured and satisfies (OC8b);

  * •

$\leq_{l}$ and $\leq_{r}$ agree on $D(C)$ (call it $\leq$ there);

  * •

$(D(C),\leq)$ is a meet-semilattice;

  * •

$\leq_{l}\circ\leq_{r}=\leq_{r}\circ\leq_{l}$ (so each of these equals the
join of $\leq_{l},\leq_{r}$);

  * •

if $x\leq_{r}y$ and $e\in D(C)$ then $(D(x)\wedge e)|x\leq_{r}(D(y)\wedge
e)|y$;

  * •

if $x\leq_{l}y$ and $e\in D(C)$ then $x|(R(x)\wedge e)\leq_{l}y|(R(y)\wedge
e)$.

This is how the defining laws appeared in [4], except that the first law above
was stated in terms of $(C,\circ,D,R,\leq_{l})$ being an ordered category
satisfying (OC8a). But we have seen that amongst $\Omega$-structured
categories, (OC8a) implies (OC4A) and hence (OC4). So it is sufficient to
require that $(C,\circ,D,R,\leq_{l})$ be $\Omega$-structured satisfying
(OC8a); similarly for the second law. Note that assuming (OC8a) holds, the
restriction of $\leq_{l}$ to $D(C)$ serves to define $\leq_{l}$ on all of $C$,
since $s\leq_{l}t$ if and only if $D(s)\leq D(t)$ and $s=D(s)|t$.

Lawson showed in [4] that an Ehresmann semigroup $(S,\cdot,D,R)$ gives rise to
an Ehresmann category $(S,\circ,D,R,\leq_{l},\leq_{r})$ where $\circ$ is as in
Definition 2.2 and $\leq_{l}$, $\leq_{r}$ are defined via $x\leq_{l}y$
whenever $x=D(x)y$ and $x=yR(x)$ respectively, and then $e|s=es$ (if $e\leq
D(s)$) and $s|f=sf$ (if $f\leq R(s)$). Conversely, he showed that every
Ehresmann category $(C,\circ,D,R,\leq_{l},\leq_{r})$ gives rise to an
Ehresmann semigroup $(C,\otimes,D,R)$ by setting $s\otimes t=(s|(R(s)\wedge
D(t))\circ(R(s)\wedge D(t))|t)$. He showed that these constructions are
mutually inverse, and indeed that the categories are isomorphic (where one
equips the category of Ehresmann semigroups with Ehresmann semigroup
homomorphisms and the category of Ehresmann categories with morphisms that are
so-called strongly ordered functors – functors preserving both the given
partial orders as well as meets of elements of $D(C)$).

We now turn to the rather simpler approach used by Lawson in [5]. Thus, let
$(C,\circ,D,R)$ be a (small) category with $D(C)$ its set of identities.
Lawson defined an Ehresmann biaction on $C$ as follows:

  1. (E1)

$D(C)$ is equipped with the structure of a commutative, idempotent semigroup;
denote the operation by $\wedge$ and interpret it as meet (so define $e\leq f$
whenever $e=e\wedge f$).

  2. (E2)

There is a left action $D(C)\times C\rightarrow C$ denoted by
$(e,a)\rightarrow e\cdot a$ (hence satisfying $(e\wedge f)\cdot
a=e\cdot(f\cdot a)$) such that $D(a)\cdot a=a$ and $D(e\cdot a)=e\wedge D(a)$;
there is a right action $C\times D(C)\rightarrow C$ denoted by
$(a,e)\rightarrow a\cdot e$ such that $a\cdot R(a)=a$ and $R(a\cdot
e)=R(a)\wedge e$.

  3. (E3)

The biaction property $(e\cdot a)\cdot f=e\cdot(a\cdot f)$ holds.

  4. (E4)

For all $e\in D(C)$ and $a\in S$, $e\cdot a=e\wedge a$ and $a\cdot e=a\wedge
e$ if $a\in D(C)$.

  5. (E5)

$R(e\cdot a)\leq R(a)$ and $D(a\cdot e)\leq D(a)$ for all $a\in C$ and $e\in
D(C)$.

  6. (E6)

When $a\circ b$ exists for some $a,b\in C$, then for all $e\in D(C)$,
$e\cdot(a\circ b)=(e\cdot a)\circ(R(e\cdot a)\cdot b)$ and $(a\circ b)\cdot
e=(a\cdot D(b\cdot e))\circ(b\cdot e)$.

In [5], Lawson showed that Ehresmann semigroups give rise to categories with
Ehresmann biactions and conversely. If $(S,\cdot,D,R)$ is an Ehresmann
semigroup, turn it into a category by defining the partial products $a\circ b$
as in Definition 2.2 above, by defining the left action via $e\cdot a=ea$ and
similarly for the right action. In the other direction, given a category $C$
with Ehresmann biaction, define the “pseudoproduct” $a\bullet b=(a\cdot
D(b))\circ(R(a)\cdot b)$, and then $(C,\bullet,D,R)$ is an Ehresmann
semigroup; moreover these constructions are mutually inverse (and there is a
suitable category isomorphism, as follows from results in [3]). We use these
facts in what follows.

A more general approach in which the identities $D(C)$ have band structure
rather than semilattice structure, is considered in [3], and an equivalence
between such so-called transcription categories and localisable semigroups is
established there. Indeed the results in the first main section of [5] can be
viewed as a special case of those obtained in [3]. For reasons that will
become clear shortly, we concentrate on Lawson’s approach here.

##  3 Ordered Ehresmann semigroups

###  3.1 Basic properties and examples

In the current work, we consider Ehresmann semigroups endowed with a
compatible but not algebraically definable partial order. In fact we do this
in what initially seems to be greater generality.

###### Definition 3.1

Let $S$ be a set. We say $(S,\cdot,D,R,\leq)$ is an ordered localisable
semigroup if:

  1. (OS1)

$(S,\cdot,D,R)$ is a localisable semigroup and $(S,\leq)$ is a partially
ordered set;

  2. (OS2)

if $a,b\in S$ and $a\leq b$ then $D(a)\leq D(b)$ and $R(a)\leq R(b)$;

  3. (OS3)

if $a,b\in S$ and $a\leq b$ and $c\leq d$, then $ac\leq bd$ (that is,
$(S,\cdot,\leq)$ is an ordered semigroup);

  4. (OS6)

if $a\in S$ and $e\in D(S)$, then $ae\leq a$ and $ea\leq a$; and

  5. (OSI)

if $a\in S$ and $e\in D(S)$, then $a\leq e$ implies $a\in D(S)$.

(The seemingly odd labelling of the defining laws above is to help make clear
how the various properties for the semigroup case correspond to those defined
in the category case as in Subsection 2.2.)

We might as well have restricted attention to Ehresmann semigroups all along
because of the following.

###### Proposition 3.2

Every ordered localisable semigroup is Ehresmann, and then the given order
agrees with the semilattice order on $D(S)$: for all $e,f\in D(S)$, $e\leq f$
if and only if $e=ef$.

Proof. Let $(S,\cdot,D,R,\leq)$ be an ordered localisable semigroup. Then for
all $e,f\in D(S)$, $ef=D(ef)ef=efef\leq fef\leq fe$ by (OS6), so $ef\leq fe$;
dually, $fe\leq ef$ and so $ef=fe$. Hence $S$ is Ehresmann.

If $e=ef$ then $e\leq f$ by (OS6). Conversely, if $e\leq f$, then by (OS3) and
(OS6), $e=ee\leq ef\leq e$, so $e=ef$. $\Box$

Hence, from now on we refer only to ordered Ehresmann semigroups.

###### Definition 3.3

Let $\leq$ be a partial order on the Ehresmann semigroup $(S,\cdot,D,R)$. We
say it is an Ehresmann order if $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann
semigroup.

Ordered Ehresmann semigroups are easily seen to generalise Boolean Ehresmann
monoids as defined in [5]. Boolean Ehresmann monoids include as examples the
Ehresmann semigroup $\operatorname{\sf Rel}(X)$ (for any non-empty set $X$)
equipped with the partial order of inclusion, and hence any subsemigroup of
$\operatorname{\sf Rel}(X)$ closed under $D$ and $R$ is an ordered Ehresmann
semigroup. Generalising this example is the Boolean Ehresmann monoid $P(C)$
for any category $C$ (see [5] for its definition); hence any subsemigroup
closed under $D$ and $R$ will be ordered Ehresmann.

A given Ehresmann semigroup can be Ehresmann ordered in more than one way.

###### Example 3.4

A two-element Ehresmann semigroup with two Ehresmann orders.

Let $S=\\{0,1\\}$ be the two-element monoid with zero $0$, and define
$D(x)=1=R(x)$ for all $x\in S$. Then $(S,\cdot,D,R)$ is easily seen to be an
Ehresmann semigroup, and it has two non-isomorphic Ehresmann orders,
$\leq_{1}$ and $\leq_{2}$. In the first, $1\leq_{1}0$. (This example may be
realised as binary relations on $X$ equipped with composition, $D,R$ and
inclusion, with $1$ the diagonal relation and $0$ the full relation.) In the
second, $0,1$ are not related, so that $\leq_{2}$ is equality. (Verification
of the defining laws for Ehresmann orders is routine.)

On the other hand, we have the following.

###### Example 3.5

An Ehresmann semigroup with no Ehresmann order.

Consider the subsemigroup $S=\\{c,d,P_{x},P_{y},P_{z},1\\}$ of the full
tranformation semigroup on $X=\\{x,y,z\\}$, in which
$c=\\{(x,x),(y,y),(z,x)\\},d=\\{(x,x),(y,y),(z,y)\\}$, $P_{x}$ is the
projection onto $\\{x\\}$, and similarly for $P_{y},P_{z}$, and $1$ is the
identity map on $X$. It is easily checked that $S$ is closed under
composition, and has the following multiplication table.

$\circ$ | $c$ | $d$ | $P_{x}$ | $P_{y}$ | $P_{z}$ | $1$
---|---|---|---|---|---|---
$c$ | $c$ | $c$ | $P_{x}$ | $P_{y}$ | $P_{z}$ | $c$
$d$ | $d$ | $d$ | $P_{x}$ | $P_{y}$ | $P_{z}$ | $d$
$P_{x}$ | $P_{x}$ | $P_{x}$ | $P_{x}$ | $P_{y}$ | $P_{z}$ | $P_{x}$
$P_{y}$ | $P_{y}$ | $P_{y}$ | $P_{x}$ | $P_{y}$ | $P_{z}$ | $P_{y}$
$P_{z}$ | $P_{x}$ | $P_{y}$ | $P_{x}$ | $P_{y}$ | $P_{z}$ | $P_{z}$
$1$ | $c$ | $d$ | $P_{x}$ | $P_{y}$ | $P_{z}$ | $1$

Notice that $(S,\cdot)$ is a band. Define $D(P_{x})=D(P_{y})=D(P_{z})=P_{z}$
and $D(c)=D(d)=D(1)=1$, with $R(P_{x})=R(P_{y})=R(c)=R(d)=R(1)=1$ and
$R(P_{z})=P_{z}$. Then $(S,\cdot,D,R)$ is an Ehresmann semigroup with
$D(S)=\\{P_{z},1\\}$. To verify this, note that $D(S)$ is a semilattice, and
that all the Ehresmann semigroup laws follow immediately except (L3). For
this, note that for $s,t\in S$, $D(st)=D(sD(t))$ and $R(ts)=R(R(t)s)$ are
trivially true if $t\in D(S)$. Otherwise, we consider cases.

So consider $D(ct)$ for $t\in\\{c,d,P_{x},P_{y}\\}$. Then $D(ct)=D(c)$ if
$t=c,d$, but for such $t$, $D(t)=1$, and so $D(ct)=D(cD(t))$; if $t=P_{x}$ or
$t=P_{y}$, then $D(ct)=D(t)=P_{z}$, whereas
$D(cD(t))=D(cP_{z})=D(P_{z})=P_{z}=D(ct)$. So in all cases, $D(ct)=D(cD(t))$.
The argument for $D(dt)=D(dD(t))$ is very similar. From the table we see that
$D(P_{x}t)=P_{z}$ for all $t$, so $D(P_{x}t)=D(P_{x}D(t))=P_{z}$. Similarly
for $P_{y},P_{z}$. And $D(1t)=D(t)=D(1D(t))$. So $D(st)=D(sD(t))$ holds for
all $s,t\in S$.

Next, consider $R(tc)$, $t\in\\{c,d,P_{x},P_{y}\\}$. If $t=c,d$ then
$R(tc)=1=R(c)=R(1c)=R(R(t)c)$; if $t=P_{x}$ or $t=P_{y}$ then
$R(tc)=R(t)=P_{z}=R(P_{z}c)=R(R(t)c)$, so in all cases $R(tc)=R(R(t)c)$.
Again, the argument that $R(td)=R(R(t)d)$ is very similar. Finally,
$R(tP_{x})=R(P_{x})$ for all $t$, so in particular, $R(tP_{x})=R(R(t)P_{x})$
for all $t$, and similarly for $R(tP_{y})$ and $R(tP_{z})$. Finally,
$R(t1)=R(R(t)1)$. So in all cases, $R(ts)=R(R(t)s)$ for all $t,s\in S$.

This completes the proof that $(S,\cdot,D,R)$ is an Ehresmann semigroup.
Suppose the partial order $\leq$ on $S$ is an Ehresmann order. Now $P_{z}c\leq
c$ and so $P_{y}(P_{z}c)\leq P_{y}c$, and similarly $P_{x}P_{z}d\leq P_{x}d$.
Direct computation then gives that $P_{x}=P_{y}P_{z}c\leq P_{y}c=P_{y}$, but
also $P_{y}=P_{x}P_{z}d\leq P_{x}d=P_{x}$, and so $P_{x}=P_{y}$, a
contradiction.

###### Definition 3.6

Let $(S,\cdot,D,R,\leq)$ and $(T,\cdot,D,R,\leq)$ be ordered Ehresmann
semigroups. Then $F:S\rightarrow T$ is an ordered Ehresmann semigroup
homomorphism if it is an Ehresmann semigroup homomorphism that respects the
Ehresmann order: for all $s,t\in S$, $s\leq t$ implies $sF\leq tF$.

It is easily seen that the class of ordered Ehresmann semigroups forms a
category in which the morphisms are ordered Ehresmann semigroup homomorphisms.
(One need only verify closure under composition). Moreover, using the
terminology of [3], these are nothing but order-preserving localisable
semigroup morphisms.

An ordered Ehresmann semigroup $(S,\cdot,D,R,\leq)$ can satisfy semigroup
versions of the other properties previously considered for categories:

  1. (OS4)

for all $s,t\in S$, if $s\leq t,D(s)=D(t)$ and $R(s)=R(t)$, then $s=t$;

  2. (OS4A)

for all $s,t\in S$ if $s\leq t$ and $D(s)=D(t)$, then $s=t$;

  3. (OS4B)

for all $s,t\in S$ if $s\leq t$ and $R(s)=R(t)$, then $s=t$;

  4. (OS7)

for all $s,t\in S$, if $u\leq st$ then there are $s^{\prime}\leq
s,t^{\prime}\leq t$ for which $u=s^{\prime}t^{\prime}$.

###  3.2 When the order is algebraically definable

Let $(S,\cdot,D,R)$ be an Ehresmann semigroup. Following [4], denote by
$\leq_{l}$ the partial order given by $s\leq_{l}t$ if and only if $s=D(s)t$,
and define $\leq_{r}$ dually, via $s\leq_{r}t$ if and only if $s=tR(s)$. It is
noted in [4] and easy to see that these two partial orders permute, and hence
their join is simply their composite as binary relations; define $\leq_{e}$ to
be this join. As noted in [4], $\leq_{e}$ is a partial order, agreeing with
the natural partial order on the (meet-)semilattice $D(S)$ (Proposition 3.11
applied to an Ehresmann semigroup), and $s\leq_{e}t$ if and only if $s=gth$
for some $g,h\in D(S)$ (Proposition 3.13 in [4]).

###### Lemma 3.7

Suppose $(S,\cdot,D,R)$ is an Ehresmann semigroup. Then for all $s,t\in S$,
$s\leq_{e}t$ if and only if $s=D(s)tR(s)$. Hence any Ehresmann order $\leq$ on
$S$ contains $\leq_{e}$ (in the sense that any pair related by $\leq_{e}$ is
also related by $\leq$).

Proof. Suppose $s,t\in S$ satisfy $s\leq_{e}t$, so that $s=gth$ for some
$g,h\in D(S)$. Then $D(s)=D(gth)=D(gD(th))=gD(th)$, so $D(s)\leq g$, and
similarly, $R(s)\leq h$. Hence $s=D(s)sR(s)=D(s)gthR(s)=D(s)tR(s)$. The
converse direction is obvious.

Hence, if $\leq$ is an Ehresmann order on $S$, and $s\leq_{e}t$ for some
$s,t\in S$, then $s=D(s)tR(s)\leq tR(s)\leq t$, so $s\leq t$. $\Box$

The partial order $\leq_{e}$ is important partly because in many important
types of Ehresmann semigroup, it is an Ehresmann order. In fact it is always
close to being an Ehresmann order, as the following shows.

###### Proposition 3.8

Suppose $(S,\cdot,D,R)$ is an Ehresmann semigroup. Then $\leq_{e}$ satisfies
the laws (OS1), (OS2), (OS6) and (OSI) of Ehresmann orders, but may not
satisfy (OS3).

Proof. That $\leq_{e}$ is a partial order has already been noted, so (OS1) is
satisfied. Suppose $a,b\in S$ and $e\in D(S)$. If $a\leq_{e}b$ then
$a=D(a)bR(a)$, so $D(a)=D(D(a)bR(a))=D(a)D(bR(a))\leq D(bR(a))\leq D(b)$ since
$D(b)bR(a)=bR(a)$, and similarly $R(a)\leq R(b)$; hence (OS2) holds. Now
$ae=D(ae)aeR(ae)=D(ae)aeR(R(a)e)=D(ae)aeR(a)$, so $ae\leq_{e}a$; similarly,
$ea\leq_{e}a$; hence (OS6) holds. Finally, if $a\leq e$ then $a=D(a)eR(a)\in
D(S)$; hence (OSI) holds.

It follows that if $\leq_{e}$ satisfies (OS3), then $(S,\cdot,D,R,\leq_{e})$
is an ordered Ehresmann semigroup. But Example 3.5 shows that not every
Ehresmann semigroup can be given an Ehresmann order. Hence (OS3) need not be
satisfied by $\leq_{e}$ in general. $\Box$

In [4], an Ehresmann semigroup $(S,\cdot,D,R)$ is said to be de Barros if it
is such that $\leq_{e}$ satisfies (OS3); from Proposition 3.8, this is
equivalent to requiring that $\leq_{e}$ be an Ehresmann order on $S$.

We have the following easy consequence of Lemma 3.7.

###### Corollary 3.9

In a de Barros Ehresmann semigroup, $\leq_{e}$ is the smallest Ehresmann order
on $S$.

A left restriction semigroup with range $(S,\cdot,D,R)$ will here mean an
Ehresmann semigroup additionally satisfying the law $sD(t)=D(st)s$; this is
nothing but a left restriction semigroup equipped with a range operation
satisfying the Ehresmann semigroup laws for range. In [4], these are referred
to as Ehresmann semigroups satisfying the condition $IC_{l}$. The main example
is perhaps $\operatorname{\sf PT}(X)$, the subsemigroup of $\operatorname{\sf
Rel}(X)$ consisting of partial transformations. This example also satisfies
the law $st=su\Rightarrow R(s)t=R(s)u$, and conversely, it was shown in [6]
that any left restriction semigroup with range satisfying this further law
admits a functional representation; accordingly, any left restriction
semigroup with range satisfying this further law will be called functional.

There is an obvious dual definition of right restriction semigroup with
domain; in [4], these are the Ehresmann semigroups satisfying $IC_{r}$. A
restriction semigroup is an Ehresmann semigroup $(S,\cdot,D,R)$ such that
$(S,\cdot,D)$ is a left restriction semigroup and $(S,\cdot,R)$ is a right
restriction semigroup; equivalently, $(S,\cdot,D,R)$ is both a left
restriction semigroup with range and a right restriction semigroup with
domain. As is well-known, every inverse semigroup is a restriction semigroup
if one defines $D(s)=ss^{\prime}$ and $R(s)=s^{\prime}s$ for all $s$ (where
$s^{\prime}$ is the inverse of $s$).

It is well-known that in a left restriction semigroup $(S,\cdot,D)$ (possibly
without range), $\leq_{l}$ satisfies (OS3) and moreover that $\leq_{e}$ (which
still makes sense in this one-sided case) equals $\leq_{l}$. Hence a left
restriction semigroup with range is de Barros; hence so is a right restriction
semigroup with domain. In particular, every restriction semigroup
$(S,\cdot,D,R)$ is de Barros, a fact noted in [4].

It follows easily from Proposition 3.7 that the class of de Barros semigroups
may be viewed as a quasivariety of Ehresmann semigroups, but in fact it is a
variety.

###### Proposition 3.10

An Ehresmann semigroup $(S,\cdot,D,R)$ is a de Barros semigroup if and only if
for all $s,t\in S$ and $e\in D(S)$, $set=D(set)stR(set)$.

Proof. If $(S,\cdot,D,R)$ is de Barros, then $(S,\cdot,D,R,\leq_{e})$ is an
ordered Ehresmann semigroup, so for all $s,t\in S$ and $e\in D(S)$,
$se\leq_{e}s$ and so $(se)t\leq_{e}st$, so by Proposition 3.7,
$set=D(set)stR(set)$. Conversely, if $(S,\cdot,D,R)$ satisfies
$set=D(set)stR(set)$ for all $s,t\in S$ and $e\in D(S)$, then $set\leq_{e}st$
for all such $s,t,e$. Suppose $x_{1}\leq_{e}y_{1},x_{2}\leq_{e}y_{2}$ for some
$x_{1},x_{2},y_{1},y_{2}\in S$. Then $x_{1}=f_{1}y_{1}g_{1}$ and
$x_{2}=f_{2}y_{2}g_{2}$ for some $f_{1},f_{2},g_{1},g_{2}\in D(S)$. But
$g_{1}f_{2}\in D(S)$, so $y_{1}g_{1}f_{2}y_{2}\leq_{e}y_{1}y_{2}$ and hence
$y_{1}g_{1}f_{2}y_{2}=h_{1}y_{1}y_{2}h_{2}$ for some $h_{1}h_{2}\in D(S)$.
Hence
$x_{1}x_{2}=f_{1}y_{1}g_{1}f_{2}y_{2}g_{2}=f_{1}h_{1}y_{1}y_{2}h_{2}g_{2}$ so
because $f_{1}h_{1},h_{2}g_{2}\in D(S)$, $x_{1}x_{2}\leq_{e}y_{1}y_{2}$. So
$\leq_{e}$ satisfies (OS3) and so $(S,\cdot,D,R)$ is de Barros. $\Box$

The de Barros property can be related to property (OS4) given earlier.

###### Proposition 3.11

Let $(S,\cdot,D,R)$ be an Ehresmann semigroup. If it is de Barros then the
ordered Ehresmann semigroup $(S,\cdot,D,R,\leq_{e})$ satisfies (OS4).
Conversely, if $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann semigroup
satisfying (OS4), then $(S,\cdot,D,R)$ is de Barros and $\leq$ is $\leq_{e}$.

Proof. Suppose $(S,\cdot,D,R)$ is de Barros. Suppose $a,b\in S$ are such that
$a\leq_{e}b$, $D(a)=D(b)$ and $R(a)=R(b)$. Then $a=D(a)bR(a)=D(b)bR(b)=b$. So
the ordered Ehresmann semigroup $(S,\cdot,D,R,\leq_{e})$ satisfies (OS4).

Conversely, suppose $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann semigroup
satisfying (OS4). Suppose $a,b\in S$. If $a\leq b$ then $a=D(a)aR(a)\leq
D(a)bR(a)\leq b$, so

$D(D(a)bR(a))=D(D(a)D(bR(a)))=D(a)D(bR(a))\leq D(a)\leq D(D(a)bR(a)),$

whence $D(a)=D(D(a)bR(a))$; similarly $R(a)=R(D(a)bR(a))$, so by (OS4),
$a=D(a)bR(a)$, and so $a\leq_{e}b$. If $a\leq_{e}b$ then $a=D(a)bR(a)\leq b$.
So $\leq$ and $\leq_{e}$ agree. Since $\leq$ satisfies (OS3), so must
$\leq_{e}$, so $(S,\cdot,D,R)$ is de Barros. $\Box$

###### Proposition 3.12

Let $(S,\cdot,D,R)$ and $(T,\cdot,D,R)$ be de Barros semigroups, with
$F:S\rightarrow T$ an Ehresmann semigroup homomorphism. Then $F$ is an ordered
Ehresmann semigroup homomorphism from $(S,\cdot,D,R,\leq_{e})$ to
$(T,\cdot,D,R,\leq_{e})$.

Proof. If $s\leq_{e}t$ then $s=D(s)tR(s)$, so

$sF=(D(s)tR(s))F=(D(s)F)(tF)(R(s)F)=D(sF)(tF)R(sF),$

and so $sF\leq_{e}tF$. $\Box$

By the above result and Proposition 3.11, the categories of de Barros
semigroups and ordered Ehresmann semigroups satisfying (OS4) are isomorphic.

###### Proposition 3.13

For ordered Ehresmann semigroups, (OS4) implies (OS7), but the converse is
false.

Proof. Suppose $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann semigroup
satisfying (OS4); then $\leq$ is $\leq_{e}$ by Proposition 3.11. Suppose
$u\leq st$ for some $u,s,t\in S$. Then $u\leq_{e}st$, so $u=D(u)stR(u)$ by
Proposition 3.7, so letting $s^{\prime}=D(u)s\leq s$ and $t^{\prime}=tR(u)\leq
t$, clearly $u=s^{\prime}t^{\prime}$. Hence (OS7) holds.

Let $S=\\{0,1\\}$ be the two-element monoid as in Example 3.4, equipped with
the Ehresmann order $\leq_{1}$ as in that example. Then $(S,\cdot,D,R,\leq)$
satisfies (OS7), as an easy case analysis shows. However, $D(1)0R(1)=0\neq 1$,
so $1\not\leq_{e}0$ by Proposition 3.7, and so $\leq$ and $\leq_{e}$ are
distinct, so $(S,\cdot,D,R,\leq)$ does not satisfy (OS4) by Proposition 3.11.
$\Box$

Recall that one and two-sided restriction semigroups are de Barros semigroups.
There are specialisations of Proposition 3.11 applying to them.

###### Proposition 3.14

Let $(S,\cdot,D,R)$ be an Ehresmann semigroup.

If $(S,\cdot,D,R)$ is a left restriction semigroup with range (resp. right
restriction semigroup with domain), then the Ehresmann order $\leq_{e}$ on
$(S,\cdot,D,R)$ satisfies (OS4A) (resp. (OS4B)). Conversely, if
$(S,\cdot,D,R,\leq)$ is an ordered Ehresmann semigroup satisfying (OS4A)
(resp. (OS4B)), then it is a left restriction semigroup with range (resp.
right restriction semigroup with domain) in which $\leq$ equals $\leq_{e}$.

Hence if $(S,\cdot,D,R)$ is a restriction semigroup, then
$(S,\cdot,D,R,\leq_{e})$ is an ordered Ehresmann semigroup satisfying (OS4A)
and (OS4B). Conversely, if $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann
semigroup satisfying (OS4A) and (OS4B), then it is a restriction semigroup in
which $\leq$ equals $\leq_{e}$.

Proof. Suppose $(S,\cdot,D,R)$ is a left restriction semigroup with range. As
noted prior to Proposition 3.10, $S$ is de Barros and so $\leq_{e}$ is an
Ehresmann order on $(S,\cdot,D,R)$. It remains to check (OS4A). But if
$s\leq_{e}t$ and $D(s)=D(t)$, then $s=D(s)t=D(t)t=t$.

Conversely, suppose $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann semigroup
satisfying (OS4A). Then it satisfies (OS4), and so by Proposition 3.11, it is
de Barros, and $\leq$ is $\leq_{e}$. Moreover, for all $x,y\in S$,
$xD(y)=D(xD(y))xD(y)=D(xy)xD(y)\leq D(xy)x$, and

$D(xD(y))=D(xy)=D(D(x)xy)=D(D(x)D(xy))=D(D(xy)D(x))=D(D(xy)x),$

so by (OS4A), $xD(y)=D(xy)x$. Hence, $(S,\cdot,D,R)$ is a left restriction
semigroup with range. The dual case now follows also, and the second claim
follows by combining the two one-sided cases. $\Box$

##  4 An ESN-style theorem for ordered Ehresmann semigroups

###  4.1 Motivation

One approach to obtaining an ESN-style theorem for ordered Ehresmann
semigroups would be to be guided by [5], simply building in order information
to a category with Ehresmann biaction. However, we are mainly interested in a
second more elegant approach, which although necessarily equivalent to the one
just outlined, is more along the lines of [4].

Let $(S,\cdot,D,R)$ be an Ehresmann semigroup with $e,f\in D(S)$, and $s\in
S$. The approach used in [4] rested on the observation that if $e\leq D(s)$
and $f\leq R(s)$ then $es$ is the unique $t\in S$ for which $t\leq_{l}s$ and
$D(t)=e$, and dually for $sf$. However, if $\leq$ is an Ehresmann order on
$S$, then $es$ is the largest $t\in S$ (with respect to $\leq$) for which
$t\leq s$ and $D(t)\leq e$, and then $D(t)=e$; dually for $sf$. Therefore,
part of our approach is to replace (OC8a) and (OC8b) applying to $\leq_{l}$
and $\leq_{r}$ respectively by (OC6) applying to $\leq$. A relatively small
number of order-theoretic properties of the category with partial order
$(S,\cdot,D,R,\leq)$ then suffice to characterise the partially ordered
categories that arise in this way from ordered Ehresmann semigroups.
Remarkably, this approach gives simpler laws than those of Ehresmann
categories which model unordered Ehresmann semigroups, but it is also simpler
than an approach in which one enriches the structure of a category with
Ehresmann biaction.

###  4.2 Ehresmann-ordered categories

We now turn to the category-theoretic analogs of ordered Ehresmann semigroups
we shall work with here. To do this, we must consider a further possible
property of an $\Omega$-structured category $(C,\circ,D,R,\leq)$, which is a
weakening of (OC7):

  1. (OC7’)

if $a\leq b\circ c$, then there are $b^{\prime}\leq b,c^{\prime}\leq c$ such
that $D(a)=D(b^{\prime}),R(a)=R(c^{\prime})$, $b^{\prime}\circ c^{\prime}$
exists, and $a\leq b^{\prime}\circ c^{\prime}$.

If an $\Omega$-structured category satisfies (OC4) and (OC7’) then it is easy
to see that it satisfies (OC7) as well. In the general case, we are interested
in cases satisfying (OC7’) but not necessarily (OC7), hence perhaps not (OC4)
either.

###### Definition 4.1

We say an $\Omega$-structured category $(C,\circ,D,R,\leq)$ is an Ehresmann-
ordered category if:

  * •

it satisfies (OC6), (OC7’) and (OCI);

  * •

$(D(C),\leq)$ is a meet-semilattice.

This definition parallels fairly closely that of ordered Ehresmann semigroups,
with only (OC7’) and the meet-semilattice assumption having no parallels
(although the latter parallels one of the properties of Ehresmann semigroups).
We contrast this relatively simple definition with that of Ehresmann
categories as in Definition 2.6. Here, there is only one partial order and the
first two laws for Ehresmann categories are replaced by the first law above,
and only the meet-semilattice property of $(D(C),\leq)$ is retained from all
the other laws for Ehresmann categories.

In fact, every ordered Ehresmann semigroup yields an Ehresmann-ordered
category.

###### Proposition 4.2

If $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann semigroup, then defining
$\circ$ as in Definition 2.2 makes
$\operatorname{\mathcal{C}}(S)=(S,\circ,D,R,\leq)$ an Ehresmann-ordered
category in which for all $a\in S$, if $e\leq D(a)$ then $e|a=ea$, and if
$e\leq R(a)$ then $a|e=ae$.

Proof. Suppose $(S,\cdot,D,R)$ is an Ehresmann semigroup. That $(S,\circ,D,R)$
is a category has been discussed already. Suppose $\leq$ is an Ehresmann order
on $(S,\cdot,D,R)$. Only (OC6) and (OC7’) are not obvious in showing that
$(S,\circ,D,R,\leq)$ is an Ehresmann-ordered category.

If $e\leq D(a)$, then $e|a=ea\leq a$ and $D(e|a)=D(ea)=D(eD(a))=D(e)=e$, and
if $y\leq a$ with $D(y)\leq e\leq D(a)$, then $y=D(y)y\leq ea=e|a$. Dually for
$a|e$ if $e\leq R(a)$. So (OC6) holds.

For (OC7’), suppose $z\leq x\circ y$, so $D(z)\leq D(x\circ y)=D(x)$. Let
$x^{\prime}=D(z)x\leq x$ and $y^{\prime}=R(D(z)x)y\leq y$. Then because
$R(D(z)x)\leq R(x)=D(y)$, we obtain

$D(y^{\prime})=D(R(D(z)x)y)=R(D(z)x)D(y)=R(D(z)x)=R(x^{\prime}),$

so $x^{\prime}\circ y^{\prime}$ exists and $x^{\prime}\circ y^{\prime}\leq
x\circ y$. Moreover

$z=D(z)z\leq D(z)xy=D(z)xR(D(z)x)y=x^{\prime}\circ y^{\prime},$

with

$D(x^{\prime}\circ
y^{\prime})=D(x^{\prime})=D(D(z)x)=D(D(z)D(x))=D(D(z))=D(z).$

Applying the dual argument to $z\leq x^{\prime}\circ y^{\prime}$ gives
$x^{\prime\prime}<x^{\prime}<x,y^{\prime\prime}<y^{\prime}<y$ such that $z\leq
x^{\prime\prime}\circ y^{\prime\prime}$ and $R(y^{\prime\prime})=R(z)$, whilst
$D(z)\leq D(x^{\prime\prime})\leq D(x^{\prime})=D(z)$, so
$D(z)=D(x^{\prime\prime})$, as required. $\Box$

Recall that a morphism between ordered Ehresmann semigroups is an order-
preserving semigroup homomorphim that respects $D$ and $R$.

###### Definition 4.3

A morphism between Ehresmann-ordered categories $(C_{1},\circ,D,R,\leq)$ and
$(C_{2},\circ,D,R,\leq)$ is a functor
$F:(C_{1},\circ,D,R)\rightarrow(C_{2},\circ,D,R)$ satisfying the following:

  * •

if $s,t\in C_{1}$ and $s\leq t$ then $sF\leq tF$;

  * •

if $e,f\in D(C_{1})$ then $(e\wedge f)F=eF\wedge fF$;

  * •

if $s\in C_{1}$ and $e\leq D(s)$ then $(e|s)F=(eF)|(sF)$, and dually if $f\leq
R(s)$ then $(s|f)F=(sF)|(fF)$.

The definition used by Lawson in [4] omitted the final requirement given
above, but as shown there in the proof of Lemma 4.23, it follows from the
first two requirements, at least in that setting, and the argument carries
over easily to the current setting if (OC8) is assumed. In fact (OC8) is
necessary for this simplification, as the following example shows.

Let $X$ be a set with at least two elements, with
$S=\\{0,1,\nabla\\}\subseteq\operatorname{\sf Rel}(X)$, where $0$ is the empty
relation, $1$ is the diagonal relation (identity function on $X$) and $\nabla$
is the full relation on $X$. Evidently, $S$ is an ordered Ehresmann
subsemigroup of $(\operatorname{\sf Rel}(X),\cdot,D,R,\subseteq)$. Define
$F:S\rightarrow S$ as follows: $0F=1F=1,\nabla F=\nabla$. This is routinely
checked to be order-preserving, to preserve all category products on
$\operatorname{\mathcal{C}}(S)$, to respect $D$ and $R$, and to preserve all
meets of members of $D(S)$. So, viewed as a functor
$F:\operatorname{\mathcal{C}}(S)\rightarrow\operatorname{\mathcal{C}}(S)$ on
the Ehresmann-ordered category $\operatorname{\mathcal{C}}(S)$ (upon using
Proposition 4.2), it satisfies the first two conditions of a morphism. But
note that $0\leq 1=D(\nabla)$, and $0|\nabla=0$, so $(0|\nabla)F=0F=1$,
whereas $(0F)|(\nabla F)=1|\nabla=\nabla$, so the third is not satisfied. (It
follows that (OC8) is not satisfied, which can be seen directly because both
$1$ and $\nabla$ are below $\nabla$ even though both have domain $1$.)

Consider the Ehresmann semigroup $S$ as in Example 3.4, along with the two
Ehresmann orders on it, $\leq_{1}$ (in which $1\leq_{1}0$) and $\leq_{2}$ (the
equality relation). Then we know that $(S,\circ,D,R,\leq_{1})$ and
$(S,\circ,D,R,\leq_{2})$ are Ehresmann-ordered categories by Proposition 4.2.
The identity mapping $I:S\rightarrow S$ evidently satisfies the second and
third conditions in Definition 4.3, but fails to satisfy the first, since
$1\leq_{1}0$ yet $1\not\leq_{2}0$. So the first condition in Definition 4.3
cannot in general be inferred from the second and third either.

Clearly morphisms between Ehresmann-ordered categories are closed under
composition, and all identity mappings on Ehresmann-ordered categories are
morphisms, and we define the category of Ehresmann-ordered categories to have
as its arrows the morphisms between Ehresmann-ordered categories.

###### Theorem 4.4

Suppose $(C,\circ,D,R,\leq)$ is an Ehresmann-ordered category. Then it has
Ehresmann biaction in which $D(C)$ is ordered by $\leq$, and for all $e\in
D(C)$ and $x\in S$,

$e\cdot x=(e\wedge D(x))|x,x\cdot e=x|(R(x)\wedge e).$

Proof. First, note that $e\cdot x$ exists, and $e\cdot x\leq x$ and $D(e\cdot
x)\leq e$. Moreover if $y$ is such that $y\leq x,D(y)\leq e$, then because
$D(y)\leq D(x)$, we have that $D(y)\leq e\wedge D(x)$, so $y\leq(e\wedge
D(x))|x=e\cdot x$. Trivially $D(e\cdot x)=e\wedge D(x)$, so $e\cdot x$ is the
largest $y\leq x$ for which $D(y)\leq e$, and then $D(e\cdot x)=e\wedge D(x)$.
Dually for $x\cdot e$.

Now if $e,f\in D(C)$ and $a\in C$, then we must show that $(e\wedge f)\cdot
a=e\cdot(f\cdot a)$. Suppose $y\leq(e\wedge f)\cdot a$. Then $y\leq a$,
$D(y)\leq D(a)\wedge e\wedge f\leq D(a)\wedge f$ and so $y\leq f\cdot a$, and
since $D(y)\leq D(a)\wedge e\wedge f=D(f\cdot a)\wedge e\leq e$, we have that
$y\leq e\cdot(f\cdot a)$. Conversely, if $y\leq e\cdot(f\cdot a)$ then $y\leq
f\cdot a$ and $D(y)\leq D(f\cdot a)\wedge f=D(a)\wedge e\wedge f$, so
$y\leq(e\wedge f)\cdot a$. Hence $e\cdot(f\cdot a)=(e\wedge f)\cdot a$.
Dualise to get that $D(C)$ acts on the right using corestriction. The rest of
(E1) is satisfied by Propostion 3.14.

For $e\in D(C)$ and $a\in C$, $D(e\cdot a)=D(e)\wedge a$ by Propostion 3.14,
and clearly $D(a)\cdot a=a$. Dualising gives (E2).

For (E3), we must show that $e\cdot(a\cdot f)=(e\cdot a)\cdot f$ for all $a\in
C$ and $e,f\in D(C)$. But for $a,y\in C$ and $e,f\in D(C)$, the following are
equivalent: $y\leq(e\cdot a)\cdot f$; $y\leq e\cdot a$ and $R(y)\leq f$;
$y\leq a$, $D(y)\leq e$ and $R(y)\leq f$; $y\leq a\cdot f$ and $D(y)\leq e$;
$y\leq e\cdot(a\cdot f)$.

For (E4), Proposition 3.14 (and its dual) again gives the desired results, and
(E5) follows from monotonicity of $D$ and $R$.

For (E6), we must show that $e\cdot(x\circ y)=(e\cdot x)\circ(R(e\cdot x)\cdot
y)$,and dually. Suppose $z\leq e\cdot(x\circ y)$. Then $z\leq x\circ y$ and
$D(z)\leq e$. By assumption, there are $x^{\prime}\leq x,y^{\prime}\leq y$ for
which $z\leq x^{\prime}\circ y^{\prime}$ and $D(x^{\prime})=D(z)\leq e$, so
$x^{\prime}\leq e\cdot x$, and so $D(y^{\prime})=R(x^{\prime})\leq R(e\cdot
x)$, so $y^{\prime}\leq R(e\cdot x)\cdot y$. So $z\leq x^{\prime}\circ
y^{\prime}\leq(e\cdot x)\circ(R(e\cdot x)\cdot y)$ (the latter existing since
$D(R(e\cdot x)\cdot y)=R(e\cdot x)\wedge D(y)=R(e\cdot x)$ since $R(e\cdot
x)\leq R(x)=D(y)$). So $e\cdot(x\circ y)\leq(e\cdot x)\circ(R(e\cdot x)\cdot
y)$. Conversely, it is evident that $(e\cdot x)\circ R(e\cdot x)\cdot y\leq
x\circ y$, and because $D((e\cdot x)\circ R(e\cdot x)\cdot y)=D(e\cdot
x)=e\wedge D(x)\leq e$, we have that $(e\cdot x)\circ R(e\cdot x)\cdot y\leq
e\cdot(x\circ y)$. The dual case follows similarly. $\Box$

A notion of functor between transcription categories was defined in [3], and
therefore applies to categories with Ehresmann biaction. Thus,
$F:C_{1}\rightarrow C_{2}$ is such a functor providing it is a category
functor additionally respecting the biaction, so that $(e\cdot s)F=eF\cdot sF$
and dually, for any $e\in D(C_{1})$ and $s\in C_{1}$.

###### Proposition 4.5

Let $(C_{1},\circ,D,R,\leq)$ and $(C_{2},\circ,D,R,\leq)$ be Ehresmann-ordered
categories. If $F$ is an Ehresmann-ordered category morphism from the first to
the second, then it satisfies $(e\cdot s)F=(eF)\cdot(sF)$ for all $e\in
D(C_{1})$ and $s\in C_{1}$.

Proof. For all $e\in D(C_{1})$ and $s\in C_{1}$,

$\displaystyle(e\cdot s)F$ $\displaystyle=$ $\displaystyle((e\wedge D(s))|s)F$
$\displaystyle=$ $\displaystyle((e\wedge D(s))F)|(sF)$ $\displaystyle=$
$\displaystyle(eF\wedge D(s)F)|(sF)$ $\displaystyle=$ $\displaystyle(eF\wedge
D(sF))|(sF)$ $\displaystyle=$ $\displaystyle(eF)\cdot(sF),$

as required. $\Box$

Hence an Ehresmann-ordered category morphism is a functor between the induced
categories with Ehresmann biaction (given by Theorem 4.4).

###  4.3 The ESN-style Theorem

First, we observe that there is a converse to Proposition 4.2.

###### Proposition 4.6

If $(C,\circ,D,R,\leq)$ is an Ehresmann-ordered category, then setting

$s\otimes t=s|(R(s)\wedge D(t))\circ(R(s)\wedge D(t))|t,$

$(C,\otimes,D,R,\leq)$ is an ordered Ehresmann semigroup.

Proof. Suppose $C$ is an Ehresmann-ordered category. Then it induces an
Ehresmann biaction by Theorem 4.4, and hence by Theorem 2.7 of [5] can be made
into an Ehresmamm semigroup $\operatorname{\mathcal{S}}(C)$ under $\otimes$ as
given. Equip $\operatorname{\mathcal{S}}(C)$ with the partial order of $C$.
Most of the properties of an Ehresmann order are easily seen, the only non-
obvious one being $s_{1}\leq s_{2},t_{1}\leq t_{2}\Rightarrow s_{1}s_{2}\leq
t_{1}t_{2}$. But $t_{1}t_{2}=t_{1}\otimes
t_{2}=(t_{1}|e)\circ(e|t_{2})\leq(s_{1}|e)\circ(e|s_{2})$ (where
$e=R(t_{1})\wedge D(t_{2})$, noting that $e\leq R(s_{1}),D(s_{2})$), which in
turn is below $s_{1}\otimes s_{2}=(s_{1}\cdot f)\circ(f\cdot s_{2})$ where
$e\leq f=R(s_{1})\wedge D(s_{2})$. $\Box$

###### Theorem 4.7

The two constructions given in Theorem 4.2 and 4.6 are mutually inverse, and
indeed there is a category isomorphism between the category of ordered
Ehresmann semigroups and the category of Ehresmann-ordered categories.

Proof. Viewed as Ehresmann semigroups and categories with Ehresmann biaction,
these constructions are precisely those shown to be mutually inverse in [5]
(and indeed in [3]); since the order does not change under the construction,
this carries over to when they are viewed as Ehresmann-ordered categories and
ordered Ehresmann semigroups respectively. To prove the category isomorphism,
it therefore suffices to prove that for any two ordered Ehresmann semigroups
$S,T$, a function $F:S\rightarrow T$ is an ordered Ehresmann semigroup
homomorphism if and only if it is an Ehresmann-ordered category morphism
$\operatorname{\mathcal{C}}(S)\rightarrow\operatorname{\mathcal{C}}(T)$. If
$F:S\rightarrow T$ is an ordered Ehresmann semigroup homomorphism, then it is
certainly a functor
$\operatorname{\mathcal{C}}(S)\rightarrow\operatorname{\mathcal{C}}(T)$,
moreover one that is order-preserving (since the order has not changed) and
meet-preserving in $D(\operatorname{\mathcal{C}}(S))=D(S)$ (since meet in
$D(\operatorname{\mathcal{C}}(S))$ is just multiplication in $D(S)$), and so
it is an Ehresmann-ordered category functor. Conversely, if
$F:\operatorname{\mathcal{C}}(S)\rightarrow\operatorname{\mathcal{C}}(T)$ is
an Ehresmann-ordered functor, then by Proposition 4.5, it is a transcription
category morphism in the sense of [3], and so by Theorem 4.8 there, it is an
Ehresmann semigroup homomorphism, moreover one preserving the order, and hence
is an ordered Ehresmann sermigroup homomorphism $S\rightarrow T$. $\Box$

If $(S,\cdot,D,R,\leq)$ is an ordered Ehresmann semigroup, then for all
$s,t\in S$, $s\leq t$ if and only if $D(s)\leq D(t)$, $R(s)\leq R(t)$, and
$s\leq D(s)tR(s)$, as is easily seen. In the corresponding Ehresmann-ordered
category $(S,\circ,D,R,\leq)$ (using Theorem 4.7), we have
$D(s)tR(s)=D(s)\cdot t\cdot R(s)$, using the biaction defined in Theorem 4.4.
Note further that $D(s)=D(D(s)sR(s))\leq D(D(s)tR(s))=D(s)D(tR(s))\leq D(s)$,
so that $D(D(s)tR(s))=D(s)$, and similarly $R(D(s)tR(s))=R(s)$. All of this
shows that $\leq$ on all of $S$ may be recovered from knowledge of the
biaction plus the restriction of $\leq$ to the hom-sets of $S$ viewed as a
category. There will therefore be a characterisation of Ehresmann-ordered
categories as categories with Ehresmann biaction such that the hom-sets are
all partially ordered and satisfy a handful of further properties that enforce
precisely the Ehresmann-ordered category laws. We do not pursue the precise
details further here.

###  4.4 Some special cases

We have the following specialisations of Theorem 4.7.

###### Theorem 4.8

There is a category isomorphism between the category of ordered Ehresmann
semigroups satisfying (OS4) and the category of Ehresmann-ordered categories
satisfying (OC4).

Similarly, there is a category isomorphism between the category of ordered
Ehresmann semigroups satisfying (OS7) and the category of Ehresmann-ordered
categories satisfying (OC7).

Proof. The laws (OS4) and (OC4) have identical forms and do not depend on the
product (semigroup or category-theoretic), eatablishing the first claim.

Suppose the ordered Ehresmann semigroup $(S,\cdot,D,R,\leq)$ satisfies (OS7).
Then suppose that in the Ehresmann-ordered category $(S,\circ,\leq)$, $u\leq
s\circ t$. Then $u\leq st$, so there are $s^{\prime}\leq s,t^{\prime}\leq t$
such that $u=s^{\prime}t^{\prime}$. Let
$s^{\prime\prime}=s^{\prime}D(t^{\prime}),t^{\prime\prime}=R(s^{\prime})t^{\prime}$;
then $s^{\prime\prime}t^{\prime\prime}=s^{\prime}t^{\prime}=u$ and

$R(s^{\prime\prime})=R(s^{\prime}D(t^{\prime}))=R(R(s^{\prime})D(t^{\prime}))=R(s^{\prime})D(t^{\prime})=D(R(s^{\prime})D(t^{\prime}))=D(R(s^{\prime})t^{\prime})=D(t^{\prime\prime}),$

so $u=s^{\prime\prime}\circ t^{\prime\prime}$. So the Ehresmann-ordered
category $(S,\circ,\leq)$ satisfies (OC7).

Conversely, suppose the Ehresmann-ordered category $(C,\circ,D,R,\leq)$
satisfies (OC7). Further suppose that in the ordered Ehresmann semigroup
$(C,\otimes,D,R,\leq)$, $u\leq s\otimes t$. Then letting $e=R(s)\wedge D(t)$,
$(s|e)\circ(e|t)$ exists and we have that $u\leq(s|e)\circ(e|t)$, so there are
$s^{\prime},t^{\prime}\in C$ such that $s^{\prime}\leq s|e\leq s$ and
$t^{\prime}\leq e|t\leq t$, with $u=s^{\prime}\circ
t^{\prime}=s^{\prime}\otimes t^{\prime}$. Hence the ordered Ehresmann
semigroup $(C,\otimes,D,R,\leq)$ satisfies (OS7). $\Box$

From the above together with Propositions 3.11 and 3.12, we have the
following.

###### Corollary 4.9

There is a category isomorphism between the category of de Barros semigroups
and the category of Ehresmann-ordered categories satisfying (OC4).

In [4], Lawson considers those Ehresmann categories in which
$\leq_{e}=\leq_{l}\circ\leq_{r}$ satisfies (OC3): he calls them de Barros
categories, and shows that they correspond to de Barros semigroups, as a
special case of his main theorem describing the correspondence of Ehresmann
semigroups and categories. Lawson characterises de Barros categories as
Ehresmann categories $(C,\circ,D,R,\leq_{l},\leq_{r})$ such that
$(C,\circ,D,R,\leq_{e})$ is a totally, regularly ordered category, that is, it
is an Ehresmann-ordered category satisfying (OC4), and hence also (OC7).

From the first part of Theorem 4.8, we immediately have the following.

###### Corollary 4.10

Every Ehresmann-ordered category $(C,\circ,D,R,\leq)$ satisfying (OC4) (that
is, every totally, regularly ordered category) is such that
$(C,\circ,D,R,\leq_{l},\leq_{r})$ is de Barros in the sense of [4], if we
define $s\leq_{l}t$ providing $s=D(s)|t$ and $s\leq_{r}t$ providing
$s=t|R(s)$, and moreover $\leq$ equals $\leq_{e}$.

So de Barros categories in the sense of [4] are nothing but arbitrary totally,
regularly ordered categories $(C,\circ,D,R,\leq)$ in which $\leq_{l},\leq_{r}$
are defined as in the above corollary, and $\leq$ equals
$\leq_{l}\circ\leq_{r}$.

Corollary 4.9 has further specialisations. In [4], the author specialised his
result relating de Barros semigroups and categories to one involving (two-
sided) restriction semigroups equipped with their natural order $\leq_{e}$ and
what he called inductive1 categories: these are $\Omega$-structured categories
satisfying (OC8) (hence also (OC4), (OC7) and (OCI) by Proposition 2.8 in [4],
or equivalently by Lemma 2.5, (OC4A), (OC4B) and (OC6)), and such that
$(D(C),\leq)$ is a meet-semilattice; hence they are nothing but Ehresmann-
ordered categories satisfying (OC8). This result can also be obtained from our
results, using the fact that (OC4A) and/or (OC4B) hold in an Ehresmann-ordered
category if and only if (OS4A) and/or (OS4B) hold in the corresponding ordered
Ehresmann semigroup (that is, it is a one or two-sided restriction semigroup).
Together with Proposition 3.14, this gives the following.

###### Corollary 4.11

There is a category isomorphism between the category of restriction semigroups
and the category of inductive1 categories.

Also a consequence is the following result for left restriction semigroups
with range; there is an easy dual.

###### Corollary 4.12

There is a category isomorphism between the category of left restriction
semigroups with range and the category of Ehresmann-ordered categories
satisfying (OC4A).

We may further specialise this case.

###### Corollary 4.13

There is a category isomorphism between the category of functional left
restriction semigroups with range and the category of Ehresmann-ordered
categories satisfying (OC4A) in which every element is an epimorphism.

Proof. Suppose $(S,\cdot,D,R)$ is a functional left restriction semigroup with
range. Then in the category $(S,\circ,D,R)$, if $s\circ t=s\circ u$ then
$st=su$ so $R(s)t=R(s)u$ and so $t=u$ since $R(s)=D(t)=D(u)$.

Conversely, suppose $C$ is an Ehresmann-ordered category satisfying (OC4A) in
which every element is an epimorphism. Then for $s,t,u\in C$, if $s\otimes
t=s\otimes u$ then $(s|e)\circ(e|t)=(s|e)\circ(e|u)$ where $e=R(s)\wedge
D(t)=R(s)\wedge D(u)$ since $D(t)=D(u)$, so $e|t=e|u$, and so $e\otimes
t=e\otimes u$, or $R(s)\otimes t=(R(s)\otimes D(t))\otimes t=(R(s)\otimes
D(u))\otimes u=R(s)\otimes u$. So $(C,\otimes,D,R)$ is a functional left
restriction semigroup with range. $\Box$

##  5 Open Problems

Two subvarieties of Ehresmann semigroups are those of left restriction
semigroups with range and right restriction semigroups with domain, the meet
being the variety of restriction semigroups. Each of these two varieties is
contained in the variety of de Barros semigroups, whence so is their join, but
it is not known if this join equals the class of de Barros semigroups.

## References

  * [1] de Barros, C. M. (1969). Sur les catégories ordonnées régulières. Cahiers Topologie Géom. Différentielle 11:23–55.
  * [2] Ehresmann, C. 1980–1984). Oeuvres complètes et commentées (A. C. Ehresmann, Ed.). Suppl. Cahiers Top Géom. Diff., Amiens.
  * [3] FitzGerald, D.G., Kinyon, M.K. (2021). Trace- and pseudo-products: restriction-like semigroups with a band of projections. Semigroup Forum 103:848–866.
  * [4] Lawson, M.V. (1991). Semigroups and ordered categories I: the reduced case. J. Algebra 141:422–462.
  * [5] Lawson, M.V. (1991). On Ehresmann semigroups. Semigroup Forum 103:953–965.
  * [6] Schein, B.M. (1970). Restrictively multiplicative algebras of transformations. Izv. Vysš. Učebn. Zaved. 1970 no. 4 (95):91–102. [Russian].

T. Stokes
Department of Mathematics
The University of Waikato
Hamilton, New Zealand
phone: +64 7 8384131
email<EMAIL_ADDRESS>


B397-G336 | 0:47:27.23 | 41:12:10.4 | 16.53 | old | HS |  | B | S
BA10 | 0:47:56.28 | 42:28:43.5 | 19.10$\pm 0.03$ |  |  | 7.7 | L | L
B398-G341 | 0:47:57.78 | 41:48:45.6 | 17.46 | old | HS |  | B | S
B399-G342 | 0:47:59.55 | 41:35:28.3 | 17.28 | old | HS |  | B | S
B400-G343 | 0:48:01.45 | 42:25:33.2 | 16.50$\pm 0.08$ | old | B | 7.7 | L | L
B401-G344 | 0:48:08.50 | 41:40:41.9 | 16.83 | old | HS |  | B | S
B329D | 0:48:19.40 | 42:08:26.7 | 16.61$\pm 0.29$ |  |  | 11.6 | L | L
B332D | 0:48:29.33 | 41:38:10.9 | 17.21 |  |  |  | G |
B402-G346 | 0:48:36.11 | 42:01:34.8 | 17.42$\pm 0.13$ | old | P | 7.7 | L | L
B503 | 0:48:37.41 | 39:31:03.4 | 16.30 |  |  |  | G |
BA11 | 0:48:45.59 | 42:23:37.7 | 17.66 | old | P |  | B |
DAO99 | 0:48:48.30 | 42:32:43.9 | 19.03 |  |  |  | B |
B334D | 0:48:54.84 | 39:35:56.0 | 17.56 |  |  |  | G |
B335D-D100 | 0:49:01.25 | 42:15:39.0 | 17.29 |  |  |  | G |
B337D | 0:49:11.20 | 41:07:21.0 | 18.23 | old | HS |  | B | S
B338D | 0:49:15.76 | 40:46:23.4 | 17.91 |  |  |  | G |
B403-G348 | 0:49:17.62 | 41:35:08.1 | 16.22 | old | HS |  | B | S
B506 | 0:49:34.90 | 40:00:28.9 | 17.04 |  |  |  | G |
B405-G351 | 0:49:39.80 | 41:35:29.7 | 15.20 | old | HS |  | B | S
B345D | 0:49:52.55 | 40:53:10.1 | 17.25 |  |  |  | G |
B509-D108 | 0:49:52.84 | 42:09:43.3 | 17.52 |  |  |  | G |
B406-D109 | 0:49:59.32 | 42:15:55.9 | 17.19 |  |  |  | G |
B407-G352 | 0:50:09.95 | 41:41:00.9 | 16.09 | old | B |  | B |
B347D | 0:50:13.80 | 40:56:24.2 | 16.84 |  |  |  | G |
G353-BA13 | 0:50:18.21 | 42:35:44.1 | 17.15 | old | P |  | B |
B348D | 0:50:19.21 | 40:58:02.7 | 18.14 |  |  |  | G |
B349D | 0:50:32.00 | 41:13:20.3 | 16.68 |  |  |  | G |
B350D | 0:50:48.85 | 42:21:43.9 | 17.77 |  |  |  | G |
EXT8 | 0:53:14.53 | 41:33:24.4 |  |  | B |  |  |
VDB00 | 01:16:04.1 | 49:36:36.2 | 16.07 |  |  |  | G |

Table 2Young Clusters, Ordered by Mass  aafootnotetext: Not a cluster in Cohen
et al. (2005)

bbfootnotetext: Star in field

ccfootnotetext: Noted as young in van den Bergh (1969)

ddfootnotetext: Noted as young in Hodge (1979)

eefootnotetext: Noted as young in Elson & Walterbos (1988)

fffootnotetext: Noted as young in Barmby et al. (2000)

ggfootnotetext: Noted as young in Fusi Pecci et al. (2005)

hhfootnotetext: Noted as young in Burstein et al. (2004)

iifootnotetext: Noted as young in Beasley et al. (2004)

jjfootnotetext: Noted as young in Williams & Hodge (2001a)

Table 3Stars

aafootnotetext: The suffix “x” indicates that coords correspond to a real
object, but which was wrongly identified in the initial input catalog.

bbfootnotetext: All velocities from Hectospec

ccfootnotetext: Source of photometry: L=this paper; B=Barmby et al. (2000);
G=Galleti et al. (2007)

ddfootnotetext: Source of classification as a star: S=spectrum from this paper
indicates a star; L=LGS image indicates stellar FWHM; H=HST image indicates a
star; B=Barmby et al. (2000) indicated a star

eefootnotetext: NB61 is an M star, the velocity of -646 listed Galleti et al.
(2007) was likely the result of a velocity template mismatch.

fffootnotetext: Multiple stars or asterisms in M31.

ggfootnotetext: The velocity of this F supergiant star indicates is it
probably a member of the M31 giant stream. Details to be presented in a
subsequent paper.

Table 4Possible Stars

aafootnotetext: The suffix “x” indicates that coords correspond to a real
object, but which was wrongly identified in the initial input catalog.

bbfootnotetext: Source of velocity: HS=this paper; B=Barmby et al. (2000);
K=Kim et al. (2007); P=Perrett et al. (2002)

ccfootnotetext: Source of photometry: L=this paper; B=Barmby et al. (2000);
G=Galleti et al. (2007)

ddfootnotetext: Source of classification as a star: S=spectrum from this paper
exists but is inconclusive; L=LGS image indicates stellar FWHM; B=Barmby et
al. (2000) indicated a star

eefootnotetext: The colors of SK106A indicate it is an M star, the velocity of
-890 reported in Kim et al. (2007) was likely the result of a velocity
template mismatch (see also Lee et al., 2008)

Table 5Galaxies

aafootnotetext: All redshifts from Hectospec. Median velocity error is 17 km
s-1 . Entries without redshifts were classified background based on images.

bbfootnotetext: Perrett et al. (2002) had velocity indicating M31 membership

Table 6Missing objects or bad coordinates

Table 7Comparison of ages


# Locomotion and Control of a Friction-Driven Tripedal Robot

Mark Hermes1, Taylor McLaughlin2, Mitul Luhar1 and Quan Nguyen1 This work was
supported by the Office of Naval Research grant number N00014-17-1-2062.1These
authors are with the Department of Aerospace and Mechanical Engineering,
University of Southern California, Los Angeles, CA 90089
<EMAIL_ADDRESS>author is recent graduate from the
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI
48109<EMAIL_ADDRESS>
###### Abstract

This paper presents a novel omnidirectional gait design and feedback control
of a radially symmetric tripedal friction-driven robot. The robot features 3
servo motors mounted on a 3-D printed chassis 7 cm from the center of mass and
separated 120 degrees. These motors drive limbs, which impart frictional
reactive forces on the body. We first introduce a mathematical model for the
robot motion, then show experimental observations performed on a uniform
friction surface, which validated the accuracy of the model. This model was
then used to create an omnidirectional gait that allows the robot to translate
in any direction. Based on this gait, we also introduce a Proportional-
Integral (PI) feedback control framework that enables the robot to closely
follow a desired path. Contrasting with feedforward motion generation, the
proposed feedback controller reduced the tracking error by approximately
$46\%$. We have successfully demonstrated the approach in our robot hardware
for the problem of line following using live feedback from an overhead
tracking camera. Our controller is also able to correct for aerodynamic
disturbances generated by a high-volume industrial fan with a mean flow speed
of $5.5ms^{-1}$, reducing path error by $65\%$ relative to the basic position
update procedure.

##  I Introduction

The vast majority of studies on biological and biomimetic locomotion —
aquatic, terrestrial, or aerial — have focused on body types exhibiting
bilateral symmetry. This has provided substantial insight into the underlying
dynamics, effective gait patterns, and motion control for bilateral actuation.
However, few studies have considered locomotion and control for radially
symmetric bodies. Radial symmetries are particularly common in aquatic
locomotion. Examples include many members of the phyla Cnidaria (e.g.,
swimming medusae, or jellyfish; [1]) and Echinodermata (e.g., crawling
organisms such as sea stars, brittle stars, and sea urchins; [2, 3]). The goal
of this paper is to provide physical insight into the crawling locomotion of
organisms with radially-symmetric body plans and to demonstrate controlled
locomotion using a simple friction-driven tripedal robot. In particular, we
focus on crawling driven by large-scale limb movement. Further, we limit our
consideration to locomotion driven by frictional contact with the substrate
rather than locomotion enabled by reversible attachment. In this paper, we
introduce a novel omnidirectional gait for path following with vision-based
feedback control. We demonstrate the effectiveness of our model with a
physical robot placed in a turbulent wind field. Our novel normal force
estimation procedure, cycle-based gait strategy, and disturbance rejection
feedback control method can be useful to other radially symmetric robots
seeking translation control in the presence of unmodeled forces.

Figure 1: Minimally-actuated tripedal robot demonstrating curve following
capabilities. Experimental video: https://youtu.be/F9UxznYtJGM.

Some advantages that radially-symmetric bodies have over bilaterally-symmetric
bodies include the ability to change direction quickly, a potential redundancy
in the number of appendages driving locomotion, and, generally, an increase in
contact points for resistance to exogenous forcing [4]. For example, it is
well-known that the lead limb can be arbitrarily assigned in brittle stars [5,
3]. This means that the motion strategy can be modified more easily in case of
limb detachment, which is often an escape strategy [6, 7, 8, 9]. Previous
observations also suggest that changes in crawling direction are achieved by
simply defining a new lead limb, and altering the motion of the remaining
limbs to follow [3]. No rotation of the body is necessary. Thus, radial
symmetry, in conjunction with an even distribution of sensory and actuation
systems across all limbs [10], ensures that there is no preferred locomotion
direction relative to body orientation. Finally, having multiple radially-
distributed limbs also creates more contact points for adhesion, which becomes
important in the case of resisting exogenous forces like fluid forcing.

Some previous studies have also developed radially symmetric bioinspired
robots to investigate these body plans. A pentaradial five degree-of-freedom
(DOF) robot was recently shown to adapt to limb amputation and retain the
ability to translate in a specified direction [6]. In an earlier study, a
pentaradial robot with six actuators in each limb was developed [11]. Genetic
algorithms were then used to identify optimal gait sequences for this robot
from block simulations of a brittle-star model in a physics engine. This
effort showed that neither bilateral nor radial symmetries were locally
optimal for translation. Instead, a complex writhing motion was shown to be
most effective. Finally, tripedal robots with wheels attached, otherwise known
as trident snake robots, have been studied for path planning, geometric
control, and optimal control [12, 13, 14, 15]. Other researchers have focused
on the development of soft radially-symmetric robots employing pneumatic
actuation [16], shape-memory-alloy actuation [17], and electric field
manipulation [18] to yield greater geometric adaptability in complex,
unstructured environments.

The main contributions of the paper are listed below.

  * •

We develop a minimally-actuated friction driven tripedal robot (Fig. 1).

  * •

We present a mathematical model for the robot motion and verify the model in
experiment.

  * •

We introduce a novel omnidirectional gait that allows the robot to transition
in any direction.

  * •

We propose a feedback control framework based on PI control to improve
tracking performance and compensate for external disturbances.

  * •

We pursue laboratory experiments to validate the mathematical model and show
the efficacy of our proposed omnidirectional gait, which has an average
deviation of $5\%$ for all directions.

  * •

We show successfully path-following capability in the physical robot: the
proposed feedback control reduces tracking error for a rectangular trajectory
by $46\%$ compared to experiments without PI control.

  * •

We show that our approach is robust to external disturbances: the robot
follows a rectangular trajectory in a turbulent wind flow environment (noisy
force disturbances) using PI feedback control.

##  II Robot and Dynamical Model

In this section, we describe the robot design including measured physical
parameters (e.g. mass, friction coefficient) (II-A), our mathematical model
that we use to describe the motion of the robot (II-B), and compare the
results of our simulations with experiments (II-C).

###  II-A Robot Design and Characterization

The tripedal robot, shown in Fig. 1, and Fig 2, was designed to approximate
the size and geometry of natural sea stars [19]. The robot has limbs of length
$l=7.5$cm evenly distributed around a central structure, attached at an
effective radius of $R=5$cm. Three Hitec HS-5646 submersible servo motors are
mounted onto a photopolymer UV-cured support frame to actuate the limbs. The
specific actuation patterns tested are discussed below. The limbs were 3D
printed and designed to minimize material so that the inertial contribution of
the limbs ($88$g) could be neglected when compared to the more massive central
structure ($800$g). The total robot mass is $M=888$g. The power source and
Arduino-based control system were stored externally and connected to the robot
via a central cable to reduce mass and ease waterproofing.

Disposable tape strips were attached to the contact points to distribute the
normal forces and tune friction. For the model validation experiments shown in
Fig. 3, the contact points were covered with 600-grit sandpaper. Additional
validation tests (not shown here) were also carried out with a lower friction
polymer tape contacts. The experiments described in this letter were conducted
with the robot moving on a sheet of white drafting paper (to ease motion
tracking) layered onto a $6.35$mm-thick leveling mat that was stretched across
an optical table. The kinetic friction coefficients, $\mu$, for the polymer-
paper and sandpaper-paper contacts were measured by releasing a known mass,
$m$, from rest on a slope with angle $\beta$, and measuring the velocity
achieved by the mass, $V$, after traveling a distance $\Delta x$ down the
ramp. Frictional energy loss was estimated from these measurements using the
relation

$\Delta E_{f}=mg\Delta x\sin{\beta}-\frac{1}{2}mV^{2},$ (1)

in which the first term on the right-hand side is the change in gravitational
potential energy for the sliding mass, and the second term is the gain in
kinetic energy. Assuming that the frictional loss can be modeled as $E_{f}=\mu
N\Delta x$ where $N=mg\cos\beta$ is the normal reaction force acting on the
mass, the kinetic friction coefficient can be estimated via the relation

$\mu=\frac{\Delta E_{f}}{\Delta xmg\cos\beta}.$ (2)

For the sandpaper-paper contact, the friction coefficient was estimated to be
$\mu=0.85\pm 0.043$. For the polymer-paper contact, the coefficient of
friction was estimated to be $\mu=0.33\pm 0.012$.

To ensure that the observed robot motion was purely due to limb actuation, we
made sure that the dynamic effects due to imperfections in table leveling and
tension in the cable tether were negligible. An electronic leveling tool
showed that the optical support table where the robot motion was recorded had
a tilt no greater than $0.2^{\circ}$. For a maximum tilt of $0.2^{\circ}$, the
in-plane force due to gravity is

$F_{tilt}=Mg\sin(0.2\pi/180)\approx 0.0035Mg$ (3)

Assuming that the normal force at each of the three contact points is on
average $Mg/3$, the friction force at each contact point for the sandpaper-
paper case is expected to be

$F_{f}\approx\mu\frac{Mg}{3}\approx 0.28Mg.$ (4)

(For clarity, this equation is not used for modeling purposes, but rather to
show the effect of small angle tilts are negligible to the dynamics.) In other
words, for this high-friction case, the in-plane friction force at each
contact is expected to be roughly two orders of magnitude larger than the
total force due to tilting. Similarly, cable tension was measured using a
hanging scale and found to be negligible compared to the total weight of the
system. Thus, cable tension is not expected to play a dynamic role either.

###  II-B Physics-Based Model

Figure 2: Simplified representation of tripedal system. $\mathbf{x}$ is the
position coordinate of the center of mass with respect to a fixed reference
frame, $\xi$ is the rotation of the body based on the hinge point of a
specified limb, and $\phi_{i}$ is the local limb rotation.

The mathematical model is based on conservation laws for linear and angular
momentum with a nonlinear frictional forcing. The model assumes that the
translational inertia and rotational inertia for the limbs are negligible
relative to those for the more massive central structure. The friction forces
are assumed to act at the point of contact at the end of each robot limb (see
Fig. 2).

To model the friction force, we use a simplified version of the Coulomb
friction law in which no distinction is made between static and kinetic
friction [20, 21]. Specifically, the friction force,
$\mathbf{F}_{i}=[F_{i,x},F_{i,y}]$, acting at the contact point at the end of
each limb in the $x-y$ plane, $\mathbf{r}_{i}=[r_{i,x},r_{i,y}]$, is modeled
as

$\mathbf{F}_{i}=-\mu
N_{i}\frac{\dot{\mathbf{r}}_{i}}{|\dot{\mathbf{r}}_{i}|}.$ (5)

Here, $N_{i}$ is the normal force at contact point $i$, $\mu$ is the measured
kinetic friction coefficient, and
$\dot{\mathbf{r}}_{i}=[\dot{r}_{i,x},\dot{r}_{i,y}]$ is the velocity of the
contact point at the end of each limb. Note that $\mathbf{r}_{i}$ and
$\dot{\mathbf{r}}_{i}$ can be expressed in terms of the robot state vector,
$\mathbf{z}=[\mathbf{x},\dot{\mathbf{x}},\xi,\dot{\phi}]$, the actuation
angles and rotation rates, $\phi_{i}$ and $\dot{\phi_{i}}$, and the geometric
constants, $R$ and $l$, using simple trigonometric relations (see Fig. 2).
Normal forces are computed by combining a vertical force balance
($N_{1}+N_{2}+N_{3}=Mg$) with torque balances about the robot center of mass:

$\left[\begin{array}[]{c c c}1&1&1\\\ r_{1,x}-x&r_{2,x}-x&r_{3,x}-x\\\
r_{1,y}-y&r_{2,y}-y&r_{3,y}-y\end{array}\right]\left[\begin{array}[]{c
}N_{1}\\\ N_{2}\\\ N_{3}\end{array}\right]=\left[\begin{array}[]{c }Mg\\\ 0\\\
0\end{array}\right].$ (6)

The second and third lines in the equation above ensure that there are no net
torques about the robot center-of-mass due to the normal forces.

Under the assumptions stated above, conservation laws for linear momentum in
the horizontal plane can be expressed as

$M\ddot{x}=\sum\limits_{i=1}^{3}F_{i,x}$ (7)

and

$M\ddot{y}=\sum\limits_{i=1}^{3}F_{i,y},$ (8)

respectively. The conservation law for angular momentum can be expressed
compactly as

$J\ddot{\xi}=\sum\limits_{i=1}^{3}(\mathbf{r_{i}}-\mathbf{x})\times\mathbf{F}_{i},$
(9)

where $J$ represents the rotational inertia, which is estimated from the mass
distribution of the central support structure.

The system of ordinary differential equations shown in (7-9) is solved
numerically to yield predictions for robot translation ($\mathbf{x}$) and
rotation ($\xi$) using the MATLAB ode45 algorithm for an adaptive time-step
4th-order Runge-Kutta solver. Recall that the location of the contact points
relative to the robot center, $\mathbf{r}_{i}-\mathbf{x}$, depends on the limb
angles, $\phi_{i}(t)$ (Fig. 2). Therefore, the prescribed actuation angles
$\phi_{i}(t)$ appear directly in all three conservation laws via the friction
terms dependent on $\mathbf{r}_{i}$ and $\dot{\mathbf{r}}_{i}$. Also, keep in
mind that these model simulations involve no tuning parameters. All geometric
and dynamic variables appearing in the governing equations (e.g.,
$M,J,l,R,\mu$) are obtained from independent measurements.

Figure 3: Predicted and measured paths $\mathbf{x}(t)$ for the sandpaper-paper
contact for (a) translation gait and (c) rotational gait. For translation, we
show the state $x$ vs $t$ in (b), and for rotation, we show state $\xi$ vs $t$
in (d) as these are the states of interest. Experiment paths are traced with
blue markers, and simulation paths are traced with red lines.

###  II-C Model Validation

Model validation tests demonstrated that the robot can rotate and translate
independently. These motions can be achieved with sinusoidal actuation at the
limbs. If two limbs are in anti-phase, and the third limb is motionless, the
robot translates in the direction of the inactive limb, as shown in Fig.
3(a),(b). Here, the active limbs were prescribed to move sinusoidally with
amplitude 30∘ and frequency $f=1$ Hz. If all three limbs are actuated with the
same amplitude and frequency, but with a phase shift of $120^{\circ}$ relative
to one another, the robot rotates in place as shown in Fig. 3(c),(d). Note
that these experiments also confirm that the mathematical model adequately
reproduces the dynamics of the physical system for the high-friction contact.
Though experimental results agree well with simulation predictions, integrated
errors from unmodeled disturbances do become significant without feedback to
correct trajectories. Some sources of uncertainty include model
simplifications and physical system imperfections (uneven mass distribution,
surface irregularities). We also observed close agreement between model
predictions and experimental results for the low-friction polymer-paper
contract surface with $\mu=0.33\pm 0.012$ (not shown here). This modeling
framework is used for gait and control design in the following section.

##  III Gait and Control Design

In this section, we discuss the development and parametrization of the
omnidirectional gait (III-A and III-B) before discussing controller design
(III-C).

###  III-A Omnidirectional Gait

The predictive model showed that an omnidirectional gait would allow for
immediate translation in any direction. This new gait was a variation of the
previously introduced translational gait. As shown in Fig. 3(a,b), when two
limbs operate in anti-phase to one another, and the third limb is inactive,
the robot translates in the direction of the inactive limb. However,
simulations show that sinusoidal actuation of the inactive limb produces
translation at a non-zero angle relative to the previously inactive limb. This
angle of translation, $\theta_{D}$ varied depending on the amplitude, $\alpha$
of the previously inactive limb’s sinusoidal motion. In this new gait, the
previously inactive limb is defined as the $\alpha$-limb because the amplitude
of its sinusoidal motion is $\alpha$. The other two limbs operate in anti-
phase to one another, with constant amplitude sinusoidal motion. Simulations
generated (details in Section III-B) a map of sinusoid amplitude to angle of
translation, $\mathbf{M}$ (Fig. 4).

Importantly, because of the radial symmetry of the model, any limb can be made
the $\alpha$-limb, allowing the model to translate in any direction from any
initial configuration. The desired direction of motion
($\theta_{D}\in[0,360]^{\circ}$) is divided into six zones spanning
$60^{\circ}$ of translation angle, as shown in Table I. For
$\theta_{D}\in[0,60]^{\circ}$, lead limb amplitude $\alpha$ is calculated
using a mapping of $\theta_{D}$ to
$\mathbf{a}=[a_{1},a_{2},a_{3}]^{\circ}=[\alpha,30,-30]^{\circ}$, in which
$a_{i}$ represents the oscillation amplitude for limb $i$ and $a_{1}$ is the
lead (or $\alpha$) limb. The mapping between $\theta_{D}\in[0,60]^{\circ}$ and
$\alpha$ is plotted in Fig. 4. This function is combined with different
permutations of limb amplitudes to span $\theta_{D}\in[0,360]^{\circ}$.

The outputs of the MATLAB block in Fig. 5 are the gait parameters
$\mathbf{a}=[a_{1},a_{2},a_{3}]$ defining the sinusoidal motion of each limb:

$\phi_{i}=a_{i}\sin(2\pi ft).$ (10)

In the expression above, $f$ is the frequency of limb oscillation, which was
set to 1 Hz in both the simulations and the experiments. The limb angle
$\phi_{i}$ can be visualized in Fig. 2. One of the gait parameters,
$[a_{1},a_{2},a_{3}]$, is set to equal $\alpha$. The value of $\theta_{D}$ is
used to determine what index of $\mathbf{a}$ is set to the $\alpha$-value,
following the structure shown in Table I.

###  III-B Gait Map

Open loop simulations with $\alpha\in[0,30]^{\circ}$ and with non-$\alpha$
limb amplitudes $30^{\circ}$ generated $\mathbf{M}$ (Fig. 4). A maximum value
of $\alpha=30^{\circ}$ was selected due to limitations of the physical model.
Over the course of a trial, $\alpha$ was held constant, and the resulting
translation angle $\theta$ was measured relative to the model reference frame
for each cycle of motion. The $\theta$ values of each cycle were then averaged
over the entire trial. This averaging produced the average angle of
translation, $\theta_{avg}$. There was minimal variation in $\theta$ after the
first two cycles of motion.

Figure 4: A comparison of the simulated average map, $\mathbf{M}_{avg}$ to experimentally collected data points. This map is used to prescribe an $\alpha$-limb gait parameter, $\alpha$, given a desired angle of translation, $\theta$. TABLE I: Relationship between desired translation direction and the gait parameters for each limb. All angles are prescribed in degrees. Zone | Desired Translation Angle ($\theta_{D}$) | Gait Parameters ($\mathbf{a}$)
---|---|---
1 | $0<\theta\leq 60$ | [$\alpha$, 30, $-30$]
2 | $60<\theta\leq 120$ | [$-30$, $-\alpha$, 30]
3 | $120<\theta\leq 180$ | [$-30$, $\alpha$, 30]
4 | $180<\theta\leq 240$ | [30, $-30$, $-\alpha$]
5 | $240<\theta\leq 300$ | [30, $-30$, $\alpha$]
6 | $300<\theta\leq 360$ | [$-\alpha$, 30, $-30$]

This procedure was repeated in three different different friction
environments: $\mu=[0.33,0.59,0.87]$. These values were selected based upon
the friction coefficient values tested with the physical robot. The
differences in the mappings generated from these three friction coefficients
was small. As a result, we assumed a universal mapping, $\mathbf{M}_{avg}$, to
provide sufficient accuracy independent of friction magnitude. Fig. 4 shows
this averaged map obtained using the predictive model (red curve), together
with $[\theta,\alpha]$ measurements made in experiments using the tripedal
robot (blue symbol).

###  III-C Controller Design

Figure 5: Flow chart (top) and diagram (bottom) showcasing feedback processing
and gait selection.

Fig. 5 provides an overview of the feedback tracking procedure implemented in
the experiments. The position vector $\mathbf{x}=[x,y,\xi]$ and trajectory
$\mathbf{s}$ are inputs to the gait mapping implemented in MATLAB. The inputs
to MATLAB, $\mathbf{x}$ and $\mathbf{s}$, are used to determine the desired
angle of translation, $\theta_{D}=G(\mathbf{x},\mathbf{s})$, using the
arctangent function. A simple PI controller, shown in the blue MATLAB block of
Fig. 5, was implemented to compensate for steady-state error. The PI
controller outputs an adjusted desired angle of translation, $\theta_{PI}$
according the equation,

$\theta_{PI}[k]=(K_{p}+\frac{K_{I}T_{s}}{z-1})e[k]+\theta_{D}[k]$ (11)

where $k$ is the discrete time variable, $e$ is the error signal, $K_{P}$ is
the proportional gain, $K_{I}$ is the integral gain, $T_{s}$ is the duration
in seconds, and $z$ is the z-transform operator. An error metric,
$H(\mathbf{s},\mathbf{x})$, estimates $e$ by calculating the minimum distance
from a line connecting target points and the position measurement. The Gait
Selection block uses $\theta_{PI}$, along with the average gait map,
$\mathbf{M}_{avg}$ to select $\alpha$ and send amplitudes, $\mathbf{a}$, to
the motors. Depending on $\theta_{PI}$, one of the six zones listed in Table I
is selected and the mapping $\mathbf{M}_{avg}$ is used to calculate the value
of $\alpha$ and the gait parameter vector $\mathbf{a}$.

Figure 6: Four points are used for the robot to trace a square path in the
presence of a variable wind flow field where (a)-(c) are without error
compensation and (d)-(f) are with PI error compensation. (a),(d) Plots are
without wind flow, (b),(e) plots are with 0∘ wind flow, and (c),(f) plots are
wind flow at 25∘ with respect to the x-direction. Blue markers show
experimental measurements while red curves show simulation results.

##  IV Path Following Demonstration

To illustrate path following capabilities in simulation and experiments, a
rectangular path specified by 4 target points was used (see Fig. 6). In the
physical experiments, we used a surface with an unknown friction coefficient
between the low-friction polymer-paper case ($\mu=0.33$) and the high-friction
sandpaper-paper case ($\mu=0.88$) to demonstrate the friction-independence of
$\mathbf{M_{avg}}$. In addition, an industrial floor fan was used to generate
wind flow across the test surface to study the effect of aerodynamic
disturbances on path following capabilities. A Protmex 6252A handheld
anemometer measured wind speed at both ends of the testing platform. The edge
located closest to the fan had a flow speed of $6.4\pm 0.2$ms-1 and the edge
furthest from the fan had a flow speed of $4.5\pm 0.2$ms-1. Thus, there were
significant velocity gradients across the surface, making for a complex flow
field.

In the numerical simulations, aerodynamic drag was modeled as $F_{d}=(1/2)\rho
C_{d}A_{d}v^{2}$ where $\rho$ is the density of air, $C_{d}$ is a drag
coefficient, $A_{d}$ is the frontal area of the robot, and $v$ is wind speed.
The drag coefficient was assumed to be $C_{d}=1$, corresponding roughly to the
drag coefficient of a cubic bluff body. The frontal area for the robot was
estimated to be $A_{d}=0.02$m2. The wind speed was set at $v=5.5$ms-1, an
average of the velocities measured at each end of the test section. We note
that these values were for simulation purposes only to provide a reference for
the experiment results. The controller does not use the fluid dynamics of the
system for gait selection. The flow is considered to be a general disturbance.
This model may differ from the physical system where we assume the flow field
is assumed uniform and constant, there are no ground effects, the drag
coefficient does not change with position, and there are no lift effects.

Proportional gain was $K_{P}=15$ deg cm-1 for rapid correction to path
deviations, and integral gain was $K_{I}=1$ deg cm-1 to reduce steady-state
error. We selected these values based on experimental tuning with an update
rate of $f=1$ Hz (i.e., every actuation cycle for the limbs). From Fig. 6(a)
it can be seen that, without proportional-integral (PI) error compensation,
the robot does not follow a straight line. This is likely a result of error
from unmodeled disturbances in $\mathbf{M_{avg}}$, imperfections in robot
manufacturing, error in gait execution, imbalance from cable tether, and
experiment surface inhomogeneity. Fig. 6(d) shows that the PI controller
corrects for this unmodeled drift. Fig. 6(b),(c) show the effect of wind in
the environment on path following without PI error compensation. The robot is
particularly sensitive to wind speed angled at 25∘ with respect to the
x-direction, where deviations from the nominal trajectory are amplified. The
PI controller successfully compensates for this in Fig. 6(f). Note that the
simulation results shown Fig. 6 are in reasonable qualitative agreement with
the experimental tracks for the cases with background flow.

Table II shows two performance metrics for the tracking experiments:
cumulative path following error $\Delta=\sum|e|$ (m), where $e$ is the
perpendicular distance to the nominal path, and the completion time $T_{c}$
(s). In all flow environments, $\Delta$ is reduced by implementation of the PI
controller. For example, $\Delta$ is reduced by over 65% for the cases with
flow in the $x$-direction. However, $T_{c}$ increases with added control for
both the no flow and angled flow cases. This is because the robot spends time
correcting its trajectory rather than going directly to the way points. A
time-optimal path in the presence of flow is not straightforward, and is a
topic of further investigation.

TABLE II: Experimental performance in terms of cumulative path following error ($\Delta$, m) and completion time ($T_{c}$, s). Rows and columns correspond to Fig. 6. |  | No flow |  $0^{\circ}$ flow |  $25^{\circ}$ flow
---|---|---|---|---
|  | $\Delta$ | $T_{c}$ | $\Delta$ | $T_{c}$ | $\Delta$ | $T_{c}$
No PI | 3.39 | 16 | 8.12 | 39 | 5.34 | 21
With PI | 1.81 | 18 | 2.81 | 26 | 2.73 | 36

Another demonstration of robot path following capability is provided in Fig.
1, which shows that complex curved paths can be tracked by placing
intermediate target points. For this figure, we show tracking with no error
compensation using the standard $\mathbf{M}_{avg}$ map.

##  V Conclusion

We have shown that a momentum-conservation model can accurately reproduce
motion for a friction driven tripedal robot. Normal forces are accurately
predicted by solving an inverse problem from a force/torque balance system of
equations. This principle can be extended to other three-point systems where
normal forces are of interest. This model facilitated generation of a gait-map
which predicted that the robot can translate in any direction independent of
the current state. Open loop experiments confirmed this, and, with this
mapping, the physical system demonstrated effective path following with an
unknown friction coefficient and in a high-speed wind flow environment. The
empirical gait-map control strategy illustrated in this work can be applied to
other nonlinear, non-holonomic systems where analytical approaches may not be
possible. This study shows how a minimally actuated, radially symmetric robot
can achieve path following by exploiting asymmetrical gait patterns.

Future work will target time optimal path following in the presence of
background flows, fluid dynamic characterization of the physical robot, and
locomotion on heterogeneous surfaces.

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