# Ehresmann Semigroups from a Range Restriction Viewpoint

Authors: Wadii Hajji
Journal: Journal of Mathematics
 (2021)

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## Abstract

The first theorem in this article provides the connection between Ehresmann semigroups and range prerestriction semigroups defined by the author. By this connection, we can redefine any Ehresmann semigroups by two unary operations and eight axioms. This connection leads us to a generalization of Ehresmann’s theorem for a range prerestriction categories; as special cases, we obtain Ehresmann’s theorems for range restriction categories and for inverse categories.


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## Body

## 1. Introduction

This paper establishes the link between several classes of semigroups defined in semigroup theory and certain kinds of categories, namely, the category of partial maps and the category of partial relations. Our first theorem is the equivalence between Ehresmann semigroups defined by Lawson in [1] and range prerestriction semigroups defined by the author. A range prerestriction semigroup is defined to capture the notion of partiality on the domain and on the range of a partial relation on a set (RelX) which will denote the set of all relations between the set (X). As a corollary of this theorem, we obtain the equivalence between range restriction semigroups and Ehresmann semigroups with an extra condition. Victoria Gould gave a more precise and general overview about these kind of semigroups [2, 3]. The equivalence of inverse semigroups and inductive ordered groupoids was known to Ehresmann. In order to construct an inverse semigroup from an inductive ordered groupoid, we need to define the “pseudoproduct” of elements of an ordered groupoid. Ehresmann was interested in ordered categories not just ordered groupoids. Lawson in [1] attempts to answer the following question: what partial order does one need on a category in order to define an associative pseudoproduct? This generalization of Ehresmann theorem led us to our generalization at the categorical level. Therefore, we are going to add new theorems to the category of partial maps’ range restriction category defined by Cockett and Guo [4] in 2004 and the category of partial relations range prerestriction category defined by the author.Independently, in 1967, Sklar and Schweizer [5] found an axiomatic characterization of (the set of all partial functions on a set (X). They found a class of semigroups with seven axioms, called function semigroups. We remark that there is a connection between range restriction semigroups and function semigroups. The range restriction categories with one object are exactly function semigroups; for the proof of this remark, see [6].


## 2. Preliminaries

We begin these preliminaries with an equivalent definition of a category where we omit the set of objects.Definition 1.
A category is a setC, equipped with a partial multiplication satisfying the following axioms:(i)
C1xyz is defined if and only if xyz is defined in this case and they are equal(ii)
C2xy and yz are defined and then xyz is defined(iii)
C3 For each x∈C, there exists an identity e and an identity f such that xe=x and fx=x
One can prove that the identitye (resp. f) such that xe=x (resp. fx=x) is unique and call the domain of x, denoted by dx (resp. range or codomain of x, denoted by rx).Definition 2.
A categoryC is ordered if it satisfies the following axioms: OC1C,⩽ is a posetOC2x⩽y implies rx⩽ry and dx⩽dyOC3 If x′⩽x and y′⩽y and both x′y′ and xy exist, then x′y′⩽xyOC4 If rx=ry and dx=dy and x⩽y, then x=y (that is, ⩽ is trivial on hom-sets)
We can view an ordered category as an internal category in poset (the category which has posets for objects and ordered-preserving maps as arrows) with additional condition (see Proposition 4.3.3 in [6]).Definition 3.
LetC be an ordered category and C0 the set of identities:(1)
C has restrictions if every x∈C and e⩽dx, where e∈C0, then there exists a unique element called the restriction of x to e, which will be denoted by x|e, such that x|e⩽x and dx|e=e(2)
C has corestrictions if every x∈C and e⩽rx, where e∈C0, then there exists a unique element called the corestriction of x to e, which will be denoted by e|x, such that e|x⩽x and re|x=e
To a category theorist, a restriction condition in an ordered category is equivalent to saying that the mapd:C⟶C0 is a discrete fibration (we consider C as a poset category) [6].


## 3. Semigroups

In this section, we give the definition and properties of each of the classes of semigroups that we shall consider, namely, Ehresmann semigroups and range prerestriction semigroups and the link between them in Theorem2. As a corollary, we prove that the range restriction semigroups are precisely Ehresmann semigroups with an additional condition.


### 3.1. Ehresmann Semigroups

This section follows Lawson [1] and Gomes and Gould [7]. Let S be a semigroup and let U⊆ES be a fixed, nonempty subset of the idempotents of S. Before starting our definitions, we define two binary relations:(1)x,y∈ℒ˜,if for alle∈U,we havexe=x⟺ye=y,x,y∈ℛ˜,if for alle∈U,we haveex=x⟺ey=y.Remark 1.
The binary relationsℒ˜ and ℛ˜ are equivalence relations on S.Notation 1.
Letx be an element in S. We denote the classes ℒ˜− and ℛ˜−classes containing the element x by Lx˜ and Rx˜, respectively.Definition 4.
LetS be a semigroup. An equivalence relation ρ on S is a right (resp. left) congruence if and only if every a,b∈ρ and c∈S imply ac,bc∈ρ (resp. ca,cb∈ρ).Definition 5.
LetS be a semigroup. Then, S,U is an Ehresmann semigroup if U is a commutative subsemigroup of S and if each ℛ˜-class and each ℒ˜-class contains at least one element of U. If ℒ˜ is a right congruence and ℛ˜ is a left congruence, in this case, we say that S has the congruence condition.
Now, we provide a useful proposition to establish the equivalence between Ehresmann semigroups and range prerestriction semigroups.Proposition 1.
LetS,U be an Ehresmann semigroup. Then,(1)
Eachℒ˜ and ℛ˜ class contains a unique element in U. Let x be in S. We denote by x¯ (resp.x^) the unique element of U∩Lx˜ (resp. U∩Rx˜).(2)
For eache∈U, both e¯=e and e^=e.(3)
The congruence condition inS is equivalent to these two axioms:(2)xy¯=x¯y¯,xy^=xy^^.Proof.
(1)
We will prove the statement forℒ˜. The proof for ℛ˜ is similar. Let e,f∈U such that eℒ˜f. Then, ef=e and fe=f. As U is a commutative sub-semigroup, e=ef=fe=f.(2)
Too trivial because each class contains one idempotent element inU.(3)
Suppose thatS satisfies the congruence condition. Let x be in S; then, xℒ˜x¯. Then, for any y∈S, we have xyℒ˜x¯y; therefore, xy¯=x¯y¯.
Now, suppose thatxy¯=x¯y¯ holds for all x,y∈S. Suppose that xℒ˜y, and let z be in S. Then, x¯=y¯ so x¯z=y¯z; hence, xz¯=x¯z¯=y¯z¯=yz¯. Therefore, xzℒ˜yz. The second axiom is dual.


### 3.2. Range Prerestriction Semigroups

Manes, in [8], introduced guarded semigroups as a restriction category with one object and without necessarily an identity. Restriction categories were defined by Cockett and Lack in [9] to capture the notion of partiality on the domain of partial functions. Following similar ideas, we define range restriction semigroup as a range restriction category (defined by Cockett and Guo [4] to capture the notion of partiality not only on the domain but also on the range) with one object and without necessarily the identity.In 2005, Manes and Cockett defined prerestriction categories to capture the notion of partiality on the domain of partial relations in the category of sets and partial relations. Following similar ideas to Manes and Cockett, we now define range prerestriction semigroups, which have the almost the same axioms as range restriction semigroups, except for one conditionR4, which we weaken.Recall from [4] that a range restriction semigroup is a semigroup with two unary operations ¯ and ^ which satisfy the following axioms:(i)
R1xx¯=x(ii)
R2x¯y¯=y¯x¯(iii)
R3xy¯¯=x¯y¯(iv)
R4x¯y=yxy¯(v)
RR1x^¯=x^(vi)
RR2x^x=x(vii)
RR3y¯x^=y¯x^(viii)
RR4yx^^=yx^Example 1.
(i)
LetX be a set and PfnX the semigroup of all partial functions on X. It is a range restriction semigroup. Indeed, for any element f in PfnXf:X⟶X, the restriction structure is given by f¯=1domf and the range structure f^=1ranf.(ii)
Any monoid with identitye has trivial restriction structure and trivial range structure given by(3)f¯=f^=e.
However, the problem is thatR4 is not valid in RelX. So, instead of R4, we introduce pR4xy¯=x¯y¯. This axiom is true in RelX.
Now, we can define officially a range prerestriction semigroup as follows.Definition 6.
A semigroupS is a range prerestriction semigroup if and only if there are two unary operations ¯ and ^ satisfying the following axioms:(i)
R1xx¯=x(ii)
R2x¯y¯=y¯x¯(iii)
R3xy¯¯=x¯y¯(iv)
pR4xy¯=x¯y¯(v)
RR1x^¯=x^(vi)
RR2x^x=x(vii)
RR3y¯x^=y¯x^(viii)
RR4yx^^=yx^Proposition 2.
A range restriction semigroup is a range prerestriction semigroup.Proof.
x¯y¯=yxy¯¯=y¯xy¯=xy¯y¯=xyy¯¯=xy¯ using R4, R2, R3, and R1.
From this proposition, we can deduce that all the proprieties for range prerestriction semigroups remain true for range restriction semigroups. Some basic properties of range prerestriction semigroups are recorded in the following proposition.Proposition 3.
LetS be a range prerestriction semigroup. Then, for every element x and y in S, we have the following:(1)
x¯ is an idempotent element(2)
x¯yx¯=yx¯(3)
x¯¯=x¯(4)
y¯x¯¯=y¯x¯(5)
xy¯=x⇔x¯=x¯y¯(6)
x^y^=y^x^(7)
x^y¯=y¯x^(8)
x^y^=x^y^(9)
x^ is an idempotent element(10)
x^^=x^(11)
x¯^=x¯(12)
xy^x^=xy^(13)
x^y^^=x^y^(14)
x^y=y⇔y^=x^y^Proof.
(1)
ByR3 and R1, we have x¯x¯=xx¯¯=x¯.(2)
ByR2, R3, and R1, we have x¯yx¯=yx¯x¯=yxx¯¯=yx¯.(3)
By (1),R3, R2, and R1,we havex¯¯=x¯x¯¯=x¯¯x¯=x¯x¯¯=x¯.(4)
ByR3,y¯x¯¯ = y¯¯x¯ = y¯x¯.(5)
Suppose thatxy¯=x. By R3, we get x¯=xy¯¯  =  x¯y¯. Suppose that x¯=x¯y¯. And, by R1, we have xx¯=x. Therefore, xx¯y¯=x; thus, xy¯=x.(6)
ByRR1 and R2, x^y^=x^¯y^¯=y^¯x^¯=y^x^.(7)
ByRR1 and R2, x^y¯=x^¯y¯=y¯x^¯=y¯x^.(8)
ByRR1 and RR3, x^y^=x^¯y^=x^¯y^=x^y^.(9)
ByRR1 and (1), x^x^=x^¯x^¯=x^¯=x^.(10)
ByRR2, RR4, and (9), x^^=x^x^^=x^x^=x^.(11)
FromRR2, we have x^x=x for each x∈S. In particular, we get x¯^x¯=x¯. Now, we can use (7) x¯x¯^=x¯^x¯=x¯. Thus, using RR3, x¯x¯^=x¯x¯^=x¯. An application of (1) gives x¯^=x¯.(12)
By (6), (8), andRR2, xy^x^=x^xy^=x^xy^=xy^.(13)
ByRR1, RR2, and (10), x^y^^=x^¯y^^=x^¯y^^=x^y^.(14)
Suppose thatx^y=y. By (8), we get y^=x^y^=x^y^. Suppose now that y^=x^y^. By RR2, we have y^y=y. Therefore, x^y^y=y; thus, x^y=y.
We shall use the properties established in Proposition3 without further comment.


### 3.3. Ehresmann Semigroups Are Equivalent to Range Prerestriction Semigroups

LetS be a range prerestriction semigroup. To distinguish between idempotent elements and the elements of the form x¯ and x^, where x∈S, we will introduce the following definition.Definition 7.
LetS be range prerestriction (range restriction) semigroup. An element s in S is a restriction idempotent if there is an element x in S such that s=x¯.Theorem 1.
In a given range prerestriction (range restriction) semigroupS, the following statements are equivalent:(1)
s is a restriction idempotent(2)
s=s^(3)
There is an elementx in S such that s=x^(4)
s=s¯Proof.
Ifs is a restriction idempotent, then s=x¯ for some x in S. Hence,(4)s^=x¯^=x¯=s.
Thus, (1) implies (2). The remaining implications are trivial.
An idempotent and commutative semigroup is called a semilattice.Lemma 1.
In range prerestriction (range restriction) semigroups, the set of all restriction idempotent forms a semilattice.Proof.
The proof of this theorem is trivial because, for every two restriction idempotents and t in S, we have that s=s¯ and t=t¯, and these two elements commute using R2. The fact that they form a semigroup is again obvious using R3.Theorem 2.
LetS be a semigroup. Then, S is an Ehremann semigroup if and only if S is a range prerestriction semigroup.Proof.
LetS be a range prerestriction semigroup and let U be the set of all restriction idempotents. By Lemma 1, U is a commutative subsemigroup of S. Each ℒ˜-class contains at least one element; indeed, x,x¯∈ℒ˜ because, for all e∈U, we have xe=x⟺x¯e=x¯. Dually, we can prove x,x^∈ℛ˜ too, i.e., we can prove that, for all e∈U, we have ex=x⟺ex^=x^. By Proposition 1 (3), we conclude that any range prerestriction semigroup is Ehresmann semigroup by taking the set U to be all the restriction idempotents.
Conversely, suppose now thatS,U is an Ehresmann semigroup. Therefore, for every element x in S, there exist unique elements (using Proposition 1) x¯ and x^ in U such that xx¯=x and x^x=x; thus, R1 and RR2 are verified. Now, from Proposition 1 (2), for every element e in U, e¯=e and e^=e. In particular, when e=x^, we get x^¯=x^; thus, RR1 is satisfied. From Proposition 1 (3), S satisfies the congruence condition, so xy¯=x¯y¯ and xy^=xy^^. Therefore, pR4 and RR4 hold.
Now, for all elementsx and y in S, x¯ and y¯ are in U, so by definition, they commute; thus, R2 is verified.
To proveR3, we have to use a particular case of pR4xy¯=x¯y¯, namely, xy¯¯=x¯y¯¯. Then, x¯y¯¯=x¯y¯ because x¯y¯ belongs to U (U is a semigroup). So, by Proposition 1 (2), for every element e in U, e¯=e thus R3. It remains to show RR3y¯x^=y¯x^; for that, we are going to use RR4. By RR4, we have y¯x^=y¯x^^ which is equal to y¯x^ (U is a semigroup, and by Proposition 1 (2), for every element e in U, e^=e).
Now, we have obtained this connection between Ehresmann semigroups and range prerestriction semigroups, and we can redefine Ehresmann semigroups by eight axioms. In particular, Ehresmann semigroups form a variety of semigroups with two unary operations, so free Ehresmann semigroups exist.
We can easily pass to the categorical level and generalize several theorems. In this paper, we will give the generalizations for the following theorem:Theorem 3.
(see Theorem 4.24 in [1]). The category of Ehresmann semigroups and admissible homomorphisms in the sense of Lawson [1] is isomorphic to the category of Ehresmann categories and strongly ordered functors.Remark 2.
From the fact that Ehresmann semigroups are range prerestriction semigroups, we can easily check the following result: admissible homomorphisms are equivalent to semigroups’ homomorphisms preserving the two unary operations¯ and ^.Definition 8.
We define an Ehresmann category to be a categoryC equipped with two relations ⩽l and ⩽r, satisfying the following axioms:E1C,⩽r is an ordered category with corestrictionsE2C,⩽l is an ordered category with restrictionsE3 If e,f∈C0, then e⩽rf⟺e⩽lf, and put ⩽=⩽r=⩽l on C0E4C0,⩽ is a meet semilatticeE5⩽r∘⩽l=⩽l∘⩽r, andwe define the relation ⩽e equal to ⩽r∘⩽lE6 If f⩽ry and f∈C0, then f|df∧f⩽ry|dy∧fE7 If f⩽ly and f∈C0, then rf∧f|f⩽lry∧f|y
As mentioned above, a range restriction semigroup is exactly an Ehresmann semigroup (range prerestriction semigroup) with an extra condition which isR4. Therefore, in the next section, we will give Ehresmann’s theorem for range restriction semigroups using this later fact.


### 3.4. Range Restriction Semigroups

We will state, as a corollary of Theorem3, the isomorphism between range restriction semigroups and R4-Ehresmann categories which are Ehresmann categories with a condition, which we will define later.Now, we can define three different orders on any range prerestriction (range restriction) semigroup following Lawson (Proposition 3.13 in [1]):(i)x⩽ey⟺, and there exist e,f∈U such that x=eyf(ii)x⩽ry⟺, and there exists e∈U such that x=ey(iii)x⩽ly⟺, and there exists f∈U such that x=yfLetS be an Ehresmann (range Prerestriction) semigroup. Then, R4 (which is equivalent to Lawson’s ICr condition [1]) holds if and only if ⩽r⊆⩽l. Indeed, let x⩽ry. Then, there exists e∈U such that x=ey. By applying the R4 condition, we get that x=ey=yey¯; thus, x⩽ly. The converse is clear; see also Lemma 3.15 in [1]. Therefore, we have to add this condition to our definition of Ehresmann category to get an R4-Ehresmann category C. And, we have more than that, and the relation ⩽l is compatible with the multiplication, i.e., for every elements x,y,u, and v in C such that x⩽ly and u⩽lv; then, xu⩽lyv. We will prove this last statement using two lemmas. First, let us define a R4-Ehresmann category.Definition 9.
AnR4-Ehresmann category is an Ehresmann category with R4E:(5)⩽l=⩽r∘⩽l=⩽l∘⩽r.
To deduce Corollary1, which is another version of Ehresmann’s theorem, but now, for an range restriction semigroups, we need the following lemmas. The first lemma is proved just to prove the second lemma.Lemma 2.
(see Lemma 4.4 in [1]). Let C be an Ehresmann category:(1)
Ife⩽dx and f⩽e, then x|f⩽lx|e⩽lx(2)
Ife⩽rx and f⩽e, then f|x⩽re|x⩽rx
For anyR4-Ehresmann category C, we can define a pseudoproduct as follows. Let x,y∈C:(6)x⊗y=x|ee|y,wheree=dx∧ry.
The following lemma is similar to what is proved in Proposition 5.2 in [1], which states that, from a de Barros category, we can construct a de Barros semigroup.Lemma 3.
LetC be an R4-Ehresmann category. Let x,y, and u be elements in C:(1)
Ifx⩽ly, then x⊗u⩽ly⊗u(2)
Ifx⩽ly, then u⊗x⩽lu⊗y
That is, the relation⩽l is compatible with the multiplication.Proof.
Let us prove (1). Similarly, we can prove (2). First, we will prove (1) only for an elemente in C0, i.e., if x⩽ly, then x⊗e⩽ly⊗e, for some e∈C0. Since x⩽ly, then dx⩽dy; thus, dx∧e⩽dy∧e from the lemma above y|dx∧e⩽ly|dy∧e. Now, it is clear that x⊗e⩽ly⊗e.
Letu be any element in C. Therefore,(7)x⊗u=x|ee|u,wheree=dx∧ru,y⊗u=y|ff|u,wheref=dy∧ru.
Fromx⩽ly, we obtain x⊗ru⩽ly⊗ru which is just x|e⩽ly|f. From x⩽ly, we have dx⩽dy so e⩽f⩽ru. However, then by the previous lemma, we obtain e|u⩽rf|u. However, ⩽r⊆⩽l so that e|u⩽lf|u. From the fact that dx|e=re|u=e and dy|f=rf|u=f, we may conclude that(8)x|ee|u⩽ly|ff|u,which is just x⊗u⩽y⊗u.Corollary 1.
The category of range restriction semigroups and their homomorphisms is isomorphic to the category ofR4-Ehresmann categories and strongly ordered functors.


## 3.1. Ehresmann Semigroups

This section follows Lawson [1] and Gomes and Gould [7]. Let S be a semigroup and let U⊆ES be a fixed, nonempty subset of the idempotents of S. Before starting our definitions, we define two binary relations:(1)x,y∈ℒ˜,if for alle∈U,we havexe=x⟺ye=y,x,y∈ℛ˜,if for alle∈U,we haveex=x⟺ey=y.Remark 1.
The binary relationsℒ˜ and ℛ˜ are equivalence relations on S.Notation 1.
Letx be an element in S. We denote the classes ℒ˜− and ℛ˜−classes containing the element x by Lx˜ and Rx˜, respectively.Definition 4.
LetS be a semigroup. An equivalence relation ρ on S is a right (resp. left) congruence if and only if every a,b∈ρ and c∈S imply ac,bc∈ρ (resp. ca,cb∈ρ).Definition 5.
LetS be a semigroup. Then, S,U is an Ehresmann semigroup if U is a commutative subsemigroup of S and if each ℛ˜-class and each ℒ˜-class contains at least one element of U. If ℒ˜ is a right congruence and ℛ˜ is a left congruence, in this case, we say that S has the congruence condition.
Now, we provide a useful proposition to establish the equivalence between Ehresmann semigroups and range prerestriction semigroups.Proposition 1.
LetS,U be an Ehresmann semigroup. Then,(1)
Eachℒ˜ and ℛ˜ class contains a unique element in U. Let x be in S. We denote by x¯ (resp.x^) the unique element of U∩Lx˜ (resp. U∩Rx˜).(2)
For eache∈U, both e¯=e and e^=e.(3)
The congruence condition inS is equivalent to these two axioms:(2)xy¯=x¯y¯,xy^=xy^^.Proof.
(1)
We will prove the statement forℒ˜. The proof for ℛ˜ is similar. Let e,f∈U such that eℒ˜f. Then, ef=e and fe=f. As U is a commutative sub-semigroup, e=ef=fe=f.(2)
Too trivial because each class contains one idempotent element inU.(3)
Suppose thatS satisfies the congruence condition. Let x be in S; then, xℒ˜x¯. Then, for any y∈S, we have xyℒ˜x¯y; therefore, xy¯=x¯y¯.
Now, suppose thatxy¯=x¯y¯ holds for all x,y∈S. Suppose that xℒ˜y, and let z be in S. Then, x¯=y¯ so x¯z=y¯z; hence, xz¯=x¯z¯=y¯z¯=yz¯. Therefore, xzℒ˜yz. The second axiom is dual.


## 3.2. Range Prerestriction Semigroups

Manes, in [8], introduced guarded semigroups as a restriction category with one object and without necessarily an identity. Restriction categories were defined by Cockett and Lack in [9] to capture the notion of partiality on the domain of partial functions. Following similar ideas, we define range restriction semigroup as a range restriction category (defined by Cockett and Guo [4] to capture the notion of partiality not only on the domain but also on the range) with one object and without necessarily the identity.In 2005, Manes and Cockett defined prerestriction categories to capture the notion of partiality on the domain of partial relations in the category of sets and partial relations. Following similar ideas to Manes and Cockett, we now define range prerestriction semigroups, which have the almost the same axioms as range restriction semigroups, except for one conditionR4, which we weaken.Recall from [4] that a range restriction semigroup is a semigroup with two unary operations ¯ and ^ which satisfy the following axioms:(i)
R1xx¯=x(ii)
R2x¯y¯=y¯x¯(iii)
R3xy¯¯=x¯y¯(iv)
R4x¯y=yxy¯(v)
RR1x^¯=x^(vi)
RR2x^x=x(vii)
RR3y¯x^=y¯x^(viii)
RR4yx^^=yx^Example 1.
(i)
LetX be a set and PfnX the semigroup of all partial functions on X. It is a range restriction semigroup. Indeed, for any element f in PfnXf:X⟶X, the restriction structure is given by f¯=1domf and the range structure f^=1ranf.(ii)
Any monoid with identitye has trivial restriction structure and trivial range structure given by(3)f¯=f^=e.
However, the problem is thatR4 is not valid in RelX. So, instead of R4, we introduce pR4xy¯=x¯y¯. This axiom is true in RelX.
Now, we can define officially a range prerestriction semigroup as follows.Definition 6.
A semigroupS is a range prerestriction semigroup if and only if there are two unary operations ¯ and ^ satisfying the following axioms:(i)
R1xx¯=x(ii)
R2x¯y¯=y¯x¯(iii)
R3xy¯¯=x¯y¯(iv)
pR4xy¯=x¯y¯(v)
RR1x^¯=x^(vi)
RR2x^x=x(vii)
RR3y¯x^=y¯x^(viii)
RR4yx^^=yx^Proposition 2.
A range restriction semigroup is a range prerestriction semigroup.Proof.
x¯y¯=yxy¯¯=y¯xy¯=xy¯y¯=xyy¯¯=xy¯ using R4, R2, R3, and R1.
From this proposition, we can deduce that all the proprieties for range prerestriction semigroups remain true for range restriction semigroups. Some basic properties of range prerestriction semigroups are recorded in the following proposition.Proposition 3.
LetS be a range prerestriction semigroup. Then, for every element x and y in S, we have the following:(1)
x¯ is an idempotent element(2)
x¯yx¯=yx¯(3)
x¯¯=x¯(4)
y¯x¯¯=y¯x¯(5)
xy¯=x⇔x¯=x¯y¯(6)
x^y^=y^x^(7)
x^y¯=y¯x^(8)
x^y^=x^y^(9)
x^ is an idempotent element(10)
x^^=x^(11)
x¯^=x¯(12)
xy^x^=xy^(13)
x^y^^=x^y^(14)
x^y=y⇔y^=x^y^Proof.
(1)
ByR3 and R1, we have x¯x¯=xx¯¯=x¯.(2)
ByR2, R3, and R1, we have x¯yx¯=yx¯x¯=yxx¯¯=yx¯.(3)
By (1),R3, R2, and R1,we havex¯¯=x¯x¯¯=x¯¯x¯=x¯x¯¯=x¯.(4)
ByR3,y¯x¯¯ = y¯¯x¯ = y¯x¯.(5)
Suppose thatxy¯=x. By R3, we get x¯=xy¯¯  =  x¯y¯. Suppose that x¯=x¯y¯. And, by R1, we have xx¯=x. Therefore, xx¯y¯=x; thus, xy¯=x.(6)
ByRR1 and R2, x^y^=x^¯y^¯=y^¯x^¯=y^x^.(7)
ByRR1 and R2, x^y¯=x^¯y¯=y¯x^¯=y¯x^.(8)
ByRR1 and RR3, x^y^=x^¯y^=x^¯y^=x^y^.(9)
ByRR1 and (1), x^x^=x^¯x^¯=x^¯=x^.(10)
ByRR2, RR4, and (9), x^^=x^x^^=x^x^=x^.(11)
FromRR2, we have x^x=x for each x∈S. In particular, we get x¯^x¯=x¯. Now, we can use (7) x¯x¯^=x¯^x¯=x¯. Thus, using RR3, x¯x¯^=x¯x¯^=x¯. An application of (1) gives x¯^=x¯.(12)
By (6), (8), andRR2, xy^x^=x^xy^=x^xy^=xy^.(13)
ByRR1, RR2, and (10), x^y^^=x^¯y^^=x^¯y^^=x^y^.(14)
Suppose thatx^y=y. By (8), we get y^=x^y^=x^y^. Suppose now that y^=x^y^. By RR2, we have y^y=y. Therefore, x^y^y=y; thus, x^y=y.
We shall use the properties established in Proposition3 without further comment.


## 3.3. Ehresmann Semigroups Are Equivalent to Range Prerestriction Semigroups

LetS be a range prerestriction semigroup. To distinguish between idempotent elements and the elements of the form x¯ and x^, where x∈S, we will introduce the following definition.Definition 7.
LetS be range prerestriction (range restriction) semigroup. An element s in S is a restriction idempotent if there is an element x in S such that s=x¯.Theorem 1.
In a given range prerestriction (range restriction) semigroupS, the following statements are equivalent:(1)
s is a restriction idempotent(2)
s=s^(3)
There is an elementx in S such that s=x^(4)
s=s¯Proof.
Ifs is a restriction idempotent, then s=x¯ for some x in S. Hence,(4)s^=x¯^=x¯=s.
Thus, (1) implies (2). The remaining implications are trivial.
An idempotent and commutative semigroup is called a semilattice.Lemma 1.
In range prerestriction (range restriction) semigroups, the set of all restriction idempotent forms a semilattice.Proof.
The proof of this theorem is trivial because, for every two restriction idempotents and t in S, we have that s=s¯ and t=t¯, and these two elements commute using R2. The fact that they form a semigroup is again obvious using R3.Theorem 2.
LetS be a semigroup. Then, S is an Ehremann semigroup if and only if S is a range prerestriction semigroup.Proof.
LetS be a range prerestriction semigroup and let U be the set of all restriction idempotents. By Lemma 1, U is a commutative subsemigroup of S. Each ℒ˜-class contains at least one element; indeed, x,x¯∈ℒ˜ because, for all e∈U, we have xe=x⟺x¯e=x¯. Dually, we can prove x,x^∈ℛ˜ too, i.e., we can prove that, for all e∈U, we have ex=x⟺ex^=x^. By Proposition 1 (3), we conclude that any range prerestriction semigroup is Ehresmann semigroup by taking the set U to be all the restriction idempotents.
Conversely, suppose now thatS,U is an Ehresmann semigroup. Therefore, for every element x in S, there exist unique elements (using Proposition 1) x¯ and x^ in U such that xx¯=x and x^x=x; thus, R1 and RR2 are verified. Now, from Proposition 1 (2), for every element e in U, e¯=e and e^=e. In particular, when e=x^, we get x^¯=x^; thus, RR1 is satisfied. From Proposition 1 (3), S satisfies the congruence condition, so xy¯=x¯y¯ and xy^=xy^^. Therefore, pR4 and RR4 hold.
Now, for all elementsx and y in S, x¯ and y¯ are in U, so by definition, they commute; thus, R2 is verified.
To proveR3, we have to use a particular case of pR4xy¯=x¯y¯, namely, xy¯¯=x¯y¯¯. Then, x¯y¯¯=x¯y¯ because x¯y¯ belongs to U (U is a semigroup). So, by Proposition 1 (2), for every element e in U, e¯=e thus R3. It remains to show RR3y¯x^=y¯x^; for that, we are going to use RR4. By RR4, we have y¯x^=y¯x^^ which is equal to y¯x^ (U is a semigroup, and by Proposition 1 (2), for every element e in U, e^=e).
Now, we have obtained this connection between Ehresmann semigroups and range prerestriction semigroups, and we can redefine Ehresmann semigroups by eight axioms. In particular, Ehresmann semigroups form a variety of semigroups with two unary operations, so free Ehresmann semigroups exist.
We can easily pass to the categorical level and generalize several theorems. In this paper, we will give the generalizations for the following theorem:Theorem 3.
(see Theorem 4.24 in [1]). The category of Ehresmann semigroups and admissible homomorphisms in the sense of Lawson [1] is isomorphic to the category of Ehresmann categories and strongly ordered functors.Remark 2.
From the fact that Ehresmann semigroups are range prerestriction semigroups, we can easily check the following result: admissible homomorphisms are equivalent to semigroups’ homomorphisms preserving the two unary operations¯ and ^.Definition 8.
We define an Ehresmann category to be a categoryC equipped with two relations ⩽l and ⩽r, satisfying the following axioms:E1C,⩽r is an ordered category with corestrictionsE2C,⩽l is an ordered category with restrictionsE3 If e,f∈C0, then e⩽rf⟺e⩽lf, and put ⩽=⩽r=⩽l on C0E4C0,⩽ is a meet semilatticeE5⩽r∘⩽l=⩽l∘⩽r, andwe define the relation ⩽e equal to ⩽r∘⩽lE6 If f⩽ry and f∈C0, then f|df∧f⩽ry|dy∧fE7 If f⩽ly and f∈C0, then rf∧f|f⩽lry∧f|y
As mentioned above, a range restriction semigroup is exactly an Ehresmann semigroup (range prerestriction semigroup) with an extra condition which isR4. Therefore, in the next section, we will give Ehresmann’s theorem for range restriction semigroups using this later fact.


## 3.4. Range Restriction Semigroups

We will state, as a corollary of Theorem3, the isomorphism between range restriction semigroups and R4-Ehresmann categories which are Ehresmann categories with a condition, which we will define later.Now, we can define three different orders on any range prerestriction (range restriction) semigroup following Lawson (Proposition 3.13 in [1]):(i)x⩽ey⟺, and there exist e,f∈U such that x=eyf(ii)x⩽ry⟺, and there exists e∈U such that x=ey(iii)x⩽ly⟺, and there exists f∈U such that x=yfLetS be an Ehresmann (range Prerestriction) semigroup. Then, R4 (which is equivalent to Lawson’s ICr condition [1]) holds if and only if ⩽r⊆⩽l. Indeed, let x⩽ry. Then, there exists e∈U such that x=ey. By applying the R4 condition, we get that x=ey=yey¯; thus, x⩽ly. The converse is clear; see also Lemma 3.15 in [1]. Therefore, we have to add this condition to our definition of Ehresmann category to get an R4-Ehresmann category C. And, we have more than that, and the relation ⩽l is compatible with the multiplication, i.e., for every elements x,y,u, and v in C such that x⩽ly and u⩽lv; then, xu⩽lyv. We will prove this last statement using two lemmas. First, let us define a R4-Ehresmann category.Definition 9.
AnR4-Ehresmann category is an Ehresmann category with R4E:(5)⩽l=⩽r∘⩽l=⩽l∘⩽r.
To deduce Corollary1, which is another version of Ehresmann’s theorem, but now, for an range restriction semigroups, we need the following lemmas. The first lemma is proved just to prove the second lemma.Lemma 2.
(see Lemma 4.4 in [1]). Let C be an Ehresmann category:(1)
Ife⩽dx and f⩽e, then x|f⩽lx|e⩽lx(2)
Ife⩽rx and f⩽e, then f|x⩽re|x⩽rx
For anyR4-Ehresmann category C, we can define a pseudoproduct as follows. Let x,y∈C:(6)x⊗y=x|ee|y,wheree=dx∧ry.
The following lemma is similar to what is proved in Proposition 5.2 in [1], which states that, from a de Barros category, we can construct a de Barros semigroup.Lemma 3.
LetC be an R4-Ehresmann category. Let x,y, and u be elements in C:(1)
Ifx⩽ly, then x⊗u⩽ly⊗u(2)
Ifx⩽ly, then u⊗x⩽lu⊗y
That is, the relation⩽l is compatible with the multiplication.Proof.
Let us prove (1). Similarly, we can prove (2). First, we will prove (1) only for an elemente in C0, i.e., if x⩽ly, then x⊗e⩽ly⊗e, for some e∈C0. Since x⩽ly, then dx⩽dy; thus, dx∧e⩽dy∧e from the lemma above y|dx∧e⩽ly|dy∧e. Now, it is clear that x⊗e⩽ly⊗e.
Letu be any element in C. Therefore,(7)x⊗u=x|ee|u,wheree=dx∧ru,y⊗u=y|ff|u,wheref=dy∧ru.
Fromx⩽ly, we obtain x⊗ru⩽ly⊗ru which is just x|e⩽ly|f. From x⩽ly, we have dx⩽dy so e⩽f⩽ru. However, then by the previous lemma, we obtain e|u⩽rf|u. However, ⩽r⊆⩽l so that e|u⩽lf|u. From the fact that dx|e=re|u=e and dy|f=rf|u=f, we may conclude that(8)x|ee|u⩽ly|ff|u,which is just x⊗u⩽y⊗u.Corollary 1.
The category of range restriction semigroups and their homomorphisms is isomorphic to the category ofR4-Ehresmann categories and strongly ordered functors.


## 4. Range Prerestriction Categories

We will give in this section some helpful definitions and propositions that will lead us to prove our main theorem in this paper (generalization of Ehresmann’s Theorem).We start this section by giving a categorical generalization of range prerestriction semigroups.Definition 10.
LetC be a small category. A range Prerestriction structure on a category C is an assignment to each arrow x:dx⟶rx, two arrows x¯:dx⟶dx and x^:rx⟶rx, such that the following eight conditions are satisfied:(R1) xx¯=x for each map x(R2) x¯y¯=y¯x¯ with dx=dy(R3) yx¯¯=y¯x¯ with rx=ry(pR4) yx¯=y¯x¯ with rx=dy(RR1) x^¯=x^ for each map x(RR2) x^x=x for each map x(RR3) y¯x^=y¯x^ with rx=dy(RR4) yx^^=yx^ with rx=dy
If a categoryC has a range prerestriction structure, we call it a range prerestriction category.
IfC satisfies the conditions R1, R2, R3, and pR4, we call this category a prerestriction category, see [10]. This notion is due to Manes and Cockett. As we mentioned in the preliminaries, we will omit the set of objects in any category, and this gives us an equivalent definition which is easy to work with. The properties below come from Proposition 1 for range prerestriction semigroups, which are still true for range prerestriction categories. We are not going to prove these properties because we can apply the same arguments as before.Proposition 4.
LetC be a range prerestriction category and x,y arrows in C. We have the following:(1)
x¯ is idempotent(2)
x¯yx¯=yx¯ if rx=dy(3)
x¯¯=x¯(4)
y¯x¯¯=y¯x¯ if dx=dy(5)
xy¯=x⇔x¯=x¯y¯ if dx=dy(6)
x^y^=y^x^ if rx=ry(7)
x^y¯=y¯x^ if rx=dy(8)
x^y^=x^y^ if rx=ry(9)
x^x^=x^ for each map x(10)
x^^=x^ for each map x(11)
x¯^=x¯ for each map x(12)
yx^y^=yx^ if rx=dy(13)
y^x^^=y^x^ if rx=ry(14)
x^y=y⇔y^=x^y^ if rx=ryDefinition 11.
LetC be a range prerestriction (or restriction) category. An arrow x in C is called a restriction idempotent if there is an arrow y in C such that x=y¯.
The following theorem, inspired from the work on function systems in [5], describes various equivalent versions of this definition.Theorem 4.
In a given range prerestriction categoryC, the following statements are equivalent:(1)
x is a restriction idempotent(2)
x=x^(3)
There is an elementy in C such that x=y^(4)
x=x¯Proof.
Ifx is a restriction idempotent, then x=y¯ for some arrow y in C. Hence,(9)x^=y¯^=y¯=x.
Thus, (1) implies (2). The remaining implications are trivial.Notation 2.
LetC be a range prerestriction (or range restriction) category, and α and β two identities. We denote(1)
Homα,β=x∈C|dx=αandrx=β(2)
RIHomα,α=x∈Homα,α|such thatxis a restriction idempotent(3)
RIC=RIHomα,α|α∈C0, where C0 is the set of all identities in CTheorem 5.
In a range prerestriction (or range restriction) category, the setRIHomα,α is a idempotent commutative semigroup.Proof.
The proof of this theorem is trivial because for every two restriction idempotentsx and y in C, we have that x=x¯ and y=y¯ and these two elements commute using R2. The fact that it forms a semigroup is again obvious using Proposition 4 (4).
Now, we introduce three relations which will help us to investigate the structure of range prerestriction and restriction categories. We remark that all the definitions and propositions for range prerestriction categories are still true for range restriction categories, but there are some properties which are true for range restriction but not for range prerestriction categories.
The following definition comes from Lawson [1], but is now formulated for range prerestriction categories.Definition 12.
LetC be range prerestriction (or range restriction) category. We can define three relations on any Homα,β, where α and β are identities as follows:(i)x⩽ey⇔x=x^yx¯(ii)x⩽ry⇔x=x^y(iii)x⩽ly⇔x=yx¯Proposition 5.
On a range prerestriction categoryC, the relations ⩽e,⩽r and ⩽l are partial orders on Homα,β, where α and β are identities. Furthermore, the three orders are the same on RIHomα,α, which will be denoted ⩽. Moreover, RIHomα,α with this order forms a meet semilattice with the top element α. And, if we have(i)x⩽ry and x^=y^, then x=y(ii)x⩽ly and x¯=y¯, then x=yProof.
We will prove that⩽e is a partial order on Homα,β. The proof for the other relations is similar.
Reflexivity: clearlyx=x^xx¯ so that x⩽ex, where x is an arrow in C.
Antisymmetry: letx⩽ey and y⩽ex. Then, x=x^yx¯ and y=y^xy¯; thus,(10)x¯=x^yx¯¯=x^y¯x¯=x^y^xy¯¯x¯=x^y^x¯y¯x¯=x^y^x¯x¯y¯=x^y^xx¯¯y¯=y^x^x¯y¯=y^x¯y¯=y^xy¯¯=y¯.
Similarly, we can provex^=y^. Therefore, x=x^yx¯=y^yy¯=y.
Transitivity: ifx⩽ey and y⩽eh, then x=x^yx¯ and y=y^hy¯; furthermore, x=x^yx¯ implies x¯=x^yx¯¯=x^y¯x¯=x^yy¯¯x¯=x^y¯y¯x¯=y¯x^y¯x¯=y¯x^yx¯¯=y¯x¯. Similar to this proof, we can show x^=x^y^. Thus, x=x^y^hy¯x¯=x^hx¯. So, x⩽eh.
Lete,j∈RIHomα,α for some identity α. If e⩽ej, then e=e^je¯=je=ej because the restriction idempotents commute (see Theorem 5). Therefore, ⩽e=⩽r=⩽l on RIHomα,α, which we denote by ⩽.
RIHomα,α with ⩽ forms a meet semilattice with a top element. In fact, let e,j∈RIHomα,α. Then, eje=je=ej, by commutativity. Thus, ej⩽e. Similarly, ej⩽j. Now, let i⩽e,j. Then, iej=iej=ij since i⩽e, and ij=i since i⩽j. Thus, i⩽ej. Hence, e∧j=ej. It follows that RIHomα,α,⩽ is a meet semilattice. The top element is α because if not, there is e in RIHomα,α such that α⩽e. Then, α=αe, but αe=e because α is the identity. Thus, α=e.
The last two implications are trivial.Proposition 6.
LetC be a range prerestriction category. Then,(i)x⩽ey⇔ there exist e1,e2∈RIC such that x=e1ye2(ii)x⩽ry⇔ there exist e1∈RIC such that x=e1y(iii)x⩽ly⇔ there exist e2∈RIC such that ⩽eProof.
We will prove the⩽e case. Let x=e1ye2. Then, e1x=x and xe2=x so that e1x^=x^ and x¯e2=x¯; hence,(11)x=x^xx¯=x^e1ye2x¯=x^yx¯.
The converse is clear.Proposition 7.
LetC be a range prerestriction category. Then,(12)⩽l∘⩽r=⩽e=⩽r∘⩽l.Proof.
Note first that⩽r⊆⩽e since if x⩽ry so that (x=x^y), then by R1, we get x=xx¯=x^yx¯. Similarly, ⩽l⊆⩽e. Thus, it is clear that ⩽l∘⩽r and ⩽r∘⩽l are contained in ⩽e.
Now, letx⩽ey. Then, x=x^yx¯. Put h=x^y. Then, by the previous proposition above, h⩽ry. Furthermore, x=hx¯ so x⩽lh. Thus, we have shown x⩽lh⩽ry giving ⩽l∘⩽r=⩽e. The other case follows by symmetry.
The following lemma is proved in Lawson [1] for Ehresmann semigroups. We extend it to range prerestriction categories.Lemma 4.
LetC be a range prerestriction category. Then, C is a range restriction category if and only if ⩽r⊆⩽l.Proof.
Suppose thatC is a range restriction category. Then, R4 is verified. If x⩽ry then x=x^y=x^¯y=yx^y¯=yx¯.
Suppose that⩽r⊆⩽l. Let x and y be two arrows. Let h=x¯y. Then, h⩽ry. Therefore, h⩽ly which gives h=yh¯=yx¯y¯=yxy¯.Proposition 8.
LetC be a range prerestriction category. Then,(i) Ifx⩽ry, then xh⩽ryh for all h∈C(ii) Ifx⩽ly, then hx⩽lhy for all h∈C(iii) IfC is a range restriction category (i.e., satisfies R4), then ⩽e is compatible with the multiplication in the sense that x⩽ey, and x′⩽ey′, rx′=dx, and ry′=dy imply xx′⩽eyy′Proof.
Ifx⩽ry, then x=x^y so that xh=x^yh giving xh⩽ryh. The proof of (2) is similar to (3).
By Proposition7,(13)⩽l∘⩽r=⩽e.
By Lemma4, if R4 holds, then ⩽r⊆⩽l so that(14)⩽l∘⩽r⊆⩽l∘⩽l⊆⩽l,and hence, ⩽l=⩽e. By part (2), we will show only that if x⩽ey, then xh⩽eyh for any arrow h. Let x⩽ly. Then, x=ye, where e is a restriction idempotent. Thus, xh=yeh=yheh¯. Hence, xh⩽lyh.Proposition 9.
LetC be a range prerestriction category.(1)
Ifx⩽ry, then x¯⩽y¯ and x^⩽y^(2)
Ifx⩽ly, then x¯⩽y¯ and x^⩽y^(3)
Ifx⩽ey, then x¯⩽y¯ and x^⩽y^(4)
Ifx⩽ry and u⩽rv and x¯=u^ and y¯=v^, then xu⩽ryv(5)
Ifx⩽ly and u⩽lv and x¯=u^ and y¯=v^, then xu⩽lyvProof.
(1)
Ifx⩽ry, then x=ey for some restriction idempotent e. Thus, x^=ey^=ey^. Therefore, x^⩽y^. And, x¯=ey¯=eyy¯¯=ey¯y¯=x¯y¯. Thus, x¯⩽y¯. The proof is similar for (2).(3)
Ifx⩽ey, then x=x^yx¯. Furthermore, x=x^yx¯ implies x¯=x^yx¯¯=x^y¯x¯=x^yy¯¯x¯=x^y¯y¯x¯=y¯x^y¯x¯=y¯x^yx¯¯=y¯x¯. Similarly, to this proof, we can show x^=x^y^. Thus, x¯⩽y¯ and x^⩽y^.(4)
Ifx⩽ry and u⩽rv and x¯=u^ and y¯=v^, then x=x^y and u=u^v so that xu=x^yu^v. Now, x^y=x so that x^yx¯=x^y. However, x¯=u^ so that x^yu^=x^y. Hence, xu=x^yv, giving xu⩽ryv.Proposition 10.
LetC be a range prerestriction category. And, let e be a restriction idempotent.(1)
Ife⩽x^, then there is a unique element e|x such that e|x^=e and e|x⩽rx(2)
Ife⩽x¯, then there is a unique element x|e such that x|e¯=e and x|e⩽lxProof.
We prove (1). We will showe|x=ex. Note that ex⩽rx and ex^=ex^=e. If there exists an arrow y such that y⩽rx and y^=e, then y=y^x=ex.


## 5. Generalized Ehresmann Categories

In this section, we generalize Lawson’s notion of an Ehresmann category [1] and establish some helpful properties towards a Generalized Ehresmann’s theorem.First, we define a generalized Ehresmann category.Definition 13.
Define a generalized Ehresmann category to be a categoryC equipped with two relations ⩽l and ⩽r on C satisfying the following axioms:(GE1) C,⩽r is an ordered category with corestriction.(GE2) C,⩽l is an ordered category with restriction.(GE3) Let e and j be in C0. Then, e⩽rj⇔e⩽ljand in this case, we put ⩽=⩽r=⩽l.GE4C0,⩽ forms disjoint meet semilattices with a top element. Let e be in C0, we denote the meet semilattice containing e by MSLe.(GE5) ⩽r∘⩽l=⩽l∘⩽r, and we define ⩽e=⩽r∘⩽l.(GE6) If x⩽ry, then MSLdx=MSLdy, and let e be in MSLdx=MSLdy, then x|dx∧e⩽ry|dy∧e.(GE7) If x⩽ly, then MSLrx=MSLry, and let e be in MSLrx=MSLry, then rx∧e|x⩽lry∧e|y.Proposition 11.
IfC is a range prerestriction category, then define a category GEC to be(i) Identities: all the restriction idempotentsα,e,α of C, where α is an identity in C.(ii) Arrows: the arrows ofC, where x:α,e,α⟶β,e′,β⇔e=x¯ and e′=x^.(iii) Composition denoted by∗: the composition x∗y of two elements x and y in GEC is defined if y^=x¯. And, when it is defined is equal to xy.
GEC with ∗ composition is a generalized Ehresmann category.Remark 3.
Observe that the morphisms of a generalized Ehresmann category are maps between idempotents. Thus, these maps satisfy the general criterion of being in the Karoubi envelope (idempotent splitting completion) of a range prerestriction category (see [11]) so that x:e⟶e′ satisfies x:e⟶e′ as is easily seen.Proof.
First, we have to show thatGEC with ∗ is a category (i.e., GEC with the ∗-composition is a multiplicative set satisfying C1, C2, and C3). We begin by showing all restriction idempotents of C are identities of GEC,∗. Let e be a restriction idempotent and suppose that e∗x is defined for some arrow x. By the definition of the composition, we deduce that x^=e¯=e. Then, e∗x=x. Similarly, if x∗e is defined, then it is equal to x. We now check that the axioms C1, C2, and C3 hold.(i) AxiomC1 holds: suppose that x∗y∗h is defined. Then, x¯=y∗h^ and y∗h is defined. Hence, y∗h=yh and h^=y¯. Therefore, by RR4, we have x¯=yh^=yh^^=yy¯^=y^, so x∗y is defined. Also, x∗y¯=xy¯=x¯y¯=y^y¯=y¯=h^. Thus, x∗y∗h is defined. It is clear that x∗y∗h=x∗y∗h. A similar argument shows that if x∗y∗h exists, then x∗y∗h exists and they are equal.(ii) AxiomC2 holds: suppose that x∗y and y∗h are defined. We have that x¯=y^ and y¯=h^. Now,(15)y∗h^=yh^=yh^^=yy¯^=y^=x¯.(iii) Thus,x∗y∗h is defined.(iv)AxiomC3 holds: for each arrow x, we have that x∗x¯ is defined, and we have seen that restriction idempotents are identities in GEC,∗. Similarly, we have x^∗x is defined.
It is now clear thatGEC,∗ is a category.
Now,GEC,∗ is an ordered category. From Proposition 5, the relations ⩽r and ⩽l are partial orders, so OC1 is verified. And, OC4 is clear from the remark in the same Proposition. Now, by Proposition 3(1) and [12], the axiom OC2 holds and by (4) and (5) axiom OC3 holds. Proposition 4 implies the restriction and corestriction axioms. Thus, GE1 and GE2 are verified. By Proposition 5, the axiom GE3 holds. GE4 is consequence of Proposition 5. GE5 holds by Proposition 7. The axioms GE6 and GE7 holds by Proposition 8. In fact, if x⩽ry, then MSLx¯=MSLy¯, and let e be in MSLx¯=MSLy¯; then, by the Proposition 8, we have xe⩽rye; thus, x|x¯∧e⩽ry|y¯∧e. Similarly, we can show GE7.
Thus, from a range prerestriction category, we can construct a corresponding generalized Ehresmann category. In the remainder of this section, we will show that every generalized Ehresmann category gives rise to a range prerestriction category. LetC,⩽r,⩽l be a fixed generalized Ehresmann category.Lemma 5.
Letx and y to be in C. If y⩽ex, then dy⩽dx and ry⩽rx. Furthermore, the relation ⩽e is a partial order.Proof.
Lety⩽ex. Then, by GE5, ⩽e=⩽r∘⩽l so that there exists an arrow h such that y⩽rh and h⩽lx. However, then, by GE1,GE2, and GE3,(16)dy⩽dh,ry⩽rh,dh⩽dx,h⩽rx,so that dy⩽dx and ry⩽rx.
Reflexivity: from⩽r⊆⩽e, we have x⩽ex.
Antisymmetry: letx⩽ey and y⩽ex and let h be an arrow such that x⩽rh⩽ly. From the first part of the lemma, we have that dx=dh=dy and rx=rh=ry. However, x=rx|h so that, from rx=rh, we obtain x=h. In a similar way, h=y.
Transitivity: letx⩽ey and y⩽eh. Then, there exist arrows u and v such that(17)x⩽ru⩽ly⩽lv⩽rh.
Thus,(18)x⩽ru⩽lv⩽rh,and there exists an arrow u′ such that(19)x⩽lu′⩽rv⩽rh,but then, x⩽lu′⩽rh so that x⩽eh.Lemma 6.
Letx and y be arrows in C. Then,(1)
Ify⩽ex and dy=dx, then y⩽rx(2)
Ify⩽ex and ry=rx, then y⩽lx(3)
Ify⩽ex and dy=dx and ry=rx, then y=x(4)
The setMSLe for any identity e is an order ideal of C,⩽r (resp. C,⩽l and C,⩽e)(5)
The relation⩽e agrees with ⩽ on C0Proof.
(1)
ByGE5, there exists an arrow h such that y⩽rh⩽lx; thus, by Lemma 5,(20)dy⩽dh⩽dx,ry⩽rh⩽rx.(2)
dy=dh=dx. From h⩽lx, we have h=x|dh by GE2; since dh=dx, we obtain h=x giving y⩽rx.(3)
The proof of (2) is similar to (1).(4)
By (1), we have thaty⩽rx so that y=ry|x but ry=rx; hence, x=y.(5)
Letx⩽re, where e is in C0. Then, x=rx|e=rx.
The proof of (5) is straightforward.Lemma 7.
Letx be an arrow in C. Then,(1)
Ife⩽dx and j⩽e, then x|j⩽lx|e⩽lx(2)
Ife⩽rx and j⩽e, then j|x⩽re|x⩽rxProof.
We prove (1). Note first thatj⩽dx|e so that x|e|j is defined and x|ej⩽x|e. However, x|e|j⩽lx and dx|e|j=j so that x|e|j=x|j.Lemma 8.
Letx be an arrow in C.(1)
Lete∈C0 with e⩽dx. Then,(21)x|e=maxy:y⩽exsuch thatdy⩽e.(2)
Lete∈C0 with e⩽rx. Then,(22)e|x=maxy:y⩽exsuch thatry⩽e.Proof.
We will prove (1). ByGE2, we have x|e⩽lx and dx|e=e. By GE5, x|e⩽ex. Let y⩽ex with dy⩽ex. By GE5, there exists an arrow h such that y⩽rh⩽lx. By GE6,(23)y|dy∧e⩽rh|dh∧e,but, from dy⩽e, it follows that dy∧e=dy; thus, y=y|dy∧e. However, then(24)y⩽rh|dh∧e⩽lh⩽lx.
From the fact thath|dh∧e⩽lx and x|e⩽lx and dh∧e⩽e, we have h|dh∧e⩽lx|e by Lemma 7. Thus, y⩽ex|e as required.Lemma 9.
LetC be a generalized Ehresmann category. Let x and y be in C. If y⩽ex, then y=ry|x|dy=ry|x|dy.Proof.
By Lemma5,(25)dy⩽dx,ry⩽rx,so that x|dy⩽lx. Now, dy⩽dx so that, by Lemma 8,(26)y|dx=maxh:h⩽exanddh⩽dy.
However,y⩽ex and dy=dy so that y⩽ey|dx. Also, dy=dx|dy so that, by Lemma 6,(27)y⩽rx|dy⩽lx,and hence, y=ry|x|dy.
The proof of the other equality is similar.Lemma 10.
LetC be a generalized Ehresmann category. And, x,y∈C are two arrows.(1)
Letdy=rx and let e⩽dyx. Then,(28)yx|e=y|rx|ex|e.(2)
Letdy=rx and let e⩽ryx. Then,(29)e|yx=e|yde|y|x.Proof.
We will prove (1). The proof of (2) is similar.dyx=dx and e⩽dx so that the element x|e exists and x|e⩽lx. However, then rx|e⩽rx=dy so that the element y|rx|e exists and y|rx|e⩽ly. Now, y|rx|ex|e and yx exist so that we obtain y|rx|ex|e⩽lyx. However, yx|e⩽lyx and dy|rx|ex|e=dx|e=e=dyx|e so that, by the uniqueness,(30)yx|e=y|rx|ex|e.Proposition 12.
LetC be generalized Ehresmann category. Define a category RPC as follows:(i) Identities: the top elements of the meet semilattices inC.(ii) Arrows: ifx:dx⟶rx in C, then we define an arrow in RPC by x:Tdx⟶Trx, where Tdx and Trx are the top elements of the meet semilattices containing dx and rx, respectively.(iii) Composition: letx:Tdx⟶Trx assume that y:Tdy⟶Try in C, and dy and rx are in the same meet semilattice and e=dy∧rx. Therefore, e⩽dy and e⩽rx so that, by GE1 and GE2, the elements e|x and y|e exist. Furthermore,(31)y|e⩽ry,e|x⩽lx,dy|e=re|x=e.
Then, the composition inC of y|ee|x exists. We define a new composition ⊗ as(32)y⊗x=y|ee|x.
Then,RRC,⊗ is a range prerestriction category.
To prove this, we need to show some helpful lemmas.Lemma 11.
LetC be a generalized Ehresmann category. And, let x,y and h be arrows in C such that y⊗x⊗h exists. Then, y⊗x⊗h=y|e⊗e|y⊗h, where e=dy∧rx.Proof.
(33)y⊗x⊗h=y|ee|x⊗h=y|ee|x|ii|h,=y|e|re|y|ie|x|ii|h,wherei=de|x∧rhby the Lemma10.
Sincede|x∧rh=i(34)e|x|ii|h=e|x⊗h.
Now,e|x|i⩽le|x so that re|x|i⩽re|x=e which means that e∧re|x|i=re|x|i. Note also that(35)dx|e∧re|x⊗h=e∧re|x|i.
This now gives(36)y⊗x⊗h=y|e⊗e|x⊗h.Remark 4.
Lete and j be two elements of a meet semilattice of a generalized Ehresmann category. Then, if j⩽e, then there exists a unique element e|j such that e|j⩽e and de|j=j=dj. By the uniqueness, we get that e|j=j. The same holds for the dual, so e|j=j=j|e.Lemma 12.
LetC be a generalized Ehresmann category. And, let x be an arrow in C.(1)
Ife⩽dx, then x|e=x⊗e(2)
Ife⩽rx, then e|x=e⊗xProof.
We prove (1):(37)x⊗e=x|dx∧edx∧e|e=x|dx∧edx∧ex|dx∧e.
However,e⩽dx; thus, x⊗e=x|e.Lemma 13.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and y:Tdy⟶Try be two arrows such that y⊗e⊗x exists, where e in MSLrx=MSLdy. Then, y⊗e⊗x=y⊗e⊗x.Proof.
(38)y⊗e⊗xy⊗e|ii|x,=y⊗i|x,wherei=e∧rx,=y|jj|i|x,wherej=dx∧i.
However,j⩽i so that j|i|x=j|x; hence, y⊗e⊗x=x|jj|x:(39)y⊗e⊗x=y|nn|e⊗x=y|n⊗x,wheren=dy∧e=y|n|mm|x=y|mm|x,wherem=n∧rx,dy∧e∧rx=dy∧e∧rx.Lemma 14.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and y:Tdy⟶Try be two arrows such that Trx=Tdy. Then,(1)
ry⊗x=ry⊗rx(2)
dy⊗x=ddy⊗xProof.
We will prove (1):(40)ry⊗x=ry|ee|x=ry|e,wheree=dy∧rx,whereas(41)ry⊗rx=ry|ee|rx=ry|e.Lemma 15.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and e to be in MSLrx and j in MSLdx. Then,(1)
Ifu=e⊗x⊗j, then u⩽ex(2)
Ifv=e⊗x⊗j, then v⩽exProof.
We will prove (1). Putu′=e⊗x; then, u′=e⊗x=e∧rx|x so that u′⩽rx by GE1. Also, u=u′⊗j so that u=u′|du′∧j giving u⩽lu′ by GE2. Hence, u⩽lu′⩽rx, giving u⩽ex by GE5.Lemma 16.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and u:Tdu⟶Tru be arrows. Then,(1)
Ifu⩽ex, then u=ru|x|du(2)
Ifu⩽ex, then u=ru|x|duProof.
We will prove (1). Fromu⩽ex, we obtain du⩽dx and ru⩽rx by Lemma 5. Thus, x|du is defined and x|du⩽lx. By Lemma 8, u⩽ex|du and du=dx|du; thus, by Lemma 6, u⩽rx|du, where u=ru|x|du.Lemma 17.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and y:Tdy⟶Try be arrows such that Tdx⟶Trx and Tdy⟶Try. Then,(1)
Ify⩽rx, then y⊗h⩽rx⊗h, where h:Tdh⟶Trh and Trh=Tdx(2)
Ify⩽lx, then h⊗y⩽lh⊗x, where h:Tdh⟶Trh and Tdh=TrxProof.
We will prove (1):(42)y⊗h=y|ee|h,wheree=dy∧rh,x⊗h=x|jj|h,wherej=dx∧rh.
Sincey⩽rx, so, by GE6, y|dy∧rh⩽rx|dx∧rh so that y|e⩽rx|j. However, since e⩽j, we have e|h⩽rj|h, by Lemma 7. Now, apply OC3 with the order ⩽r giving y|ee|h⩽rx|jj|h.Proposition 13.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and let e be in MSLrx and j be in MSLdx. Then, e⊗x⊗j=e⊗x⊗j.Proof.
Putu=e⊗x⊗j. By Lemma 15, we have u⩽ex. By Lemma 16, u=ru|x|du. By Lemma 12, u=ru⊗x⊗du. Now,(43)u=e⊗x⊗j=e∧rx|x⊗j=e∧rx|x|de∧rx|x∧j⩽le∧rx|xso that ru⩽e∧rx⩽e. Also, du=de∧rx|x∧j⩽j.
And,du⩽j, and we obtain x⊗du⩽lx⊗j for(44)x⊗du=x|dx∧du,x⊗j=x|dx∧j,and du⩽j implies dx∧du⩽dx∧j so that, by Lemma 7,(45)x|dx∧du⩽lx|dx∧j.
By Lemma17, from x⊗du⩽lx⊗j, we have(46)ru⊗x⊗du⩽lru⊗x⊗j.
However,ru⩽e so that ru⊗x⊗j⩽re⊗x⊗j by Lemma 7. Thus, we have shown that e⊗x⊗j⩽ee⊗x⊗j.
We may similarly show by symmetry thate⊗x⊗j⩽ee⊗x⊗j and then apply the fact that ⩽e is a partial order.Lemma 18.
Ife,j are in a meet semilattice of some generalized Ehresmann category, then e⊗j=e∧j=j⊗e.Proof.
e⊗j=e|e∧je∧j|j=e∧j. □Lemma 19.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and y:Tdy⟶Try be arrows such that Trx=Tdy. And, let e be in MSLrx. Then,(1)
re⊗x⊗y=e⊗rx⊗y(2)
dx⊗y⊗e=dx⊗y⊗eProof.
We prove (1):(47)re⊗x⊗y=re⊗x⊗ryby Lemma14=re⊗x⊗ryby Proposition13=re⊗rx⊗ryby Lemma14=re⊗rx⊗yby Lemma14=e⊗rx⊗ysincee⊗rx⊗yis an identity by Lemma18.Lemma 20.
LetC be a generalized Ehresmann category. Let x:Tdx⟶Trx and y:Tdy⟶Try be arrows such that Trx=Tdy. And, let e be in MSLrx. Then, e⊗x⊗y=e⊗x⊗y.Proof.
Note thate⊗x⩽rx so that, by Lemma 17,(48)e⊗x⊗y⩽rx⊗y,and hence, e⊗x⊗y=re⊗x⊗y⊗x⊗y applying Lemma 12. By Lemma 19,(49)re⊗x⊗y=e⊗rx⊗y.
Thus,(50)e⊗x⊗y=e⊗rx⊗y⊗x⊗z=e⊗rx⊗y⊗x⊗z=e⊗x⊗yby Lemma13.
The proof of Proposition12.Proof.
First, we have to show thatRRC,⊗ is a category. Let us prove that top elements are identities with this new composition defined between arrows. Let x:Tdx⟶Trx, so Trx⊗x=Trx|ee|x, where e=Trx∧rx=rx because Trx∧rx=rx by Lemma 18. So, Trx⊗x=Trx|ee|x=rxrx|x=x. Similarly, we can prove that x⊗Tdx=x. It suffices to show that ⊗ is associative. Let C be a generalized Ehresmann category. And, let x,y, and h be arrows in C such that y⊗x⊗h exists. Put first e=dx∧dy and note that(51)x|e=x⊗ry,e|y=dx⊗y,x⊗y⊗h=x|e⊗e|y⊗hby Lemma11=x⊗ry⊗dx⊗y⊗hby the note above=x⊗ry⊗dx⊗y⊗hby Lemma20=x⊗ry⊗dx⊗y⊗hby Lemma13=x⊗ry⊗dx⊗y⊗hby Lemma13=x⊗dx⊗ry⊗y⊗hby Lemma18=x⊗dx⊗ry⊗y⊗hby Lemma13=x⊗ry⊗y⊗h=x⊗ry⊗y⊗hby Lemma13=x⊗ry⊗y⊗hby Lemma20=x⊗y⊗h.
Now, for every arrowx in C, we will write x¯=dx and x^=rx. We have to prove the eight axioms for range prerestriction category:(R1) x⊗x¯=x for each map x(R2) x¯⊗y¯=y¯⊗x¯ for all maps x,y with x¯ and y¯ are in the same meet semilattice(R3) y⊗x¯¯=y¯⊗x¯ for all maps x,y with x¯ and y¯ are in the same meet semilattice(pR4) y⊗x¯=y¯⊗x¯ for all maps x,y with x^ and y¯ are in the same meet semilattice(RR1) x^¯=x^ for each map x(RR2) x^⊗x=x for each map x(RR3) y¯⊗x^=y¯⊗x^ for all maps x,y with x^ and y¯ are in the same meet semilattice(RR4) y⊗x^^=y⊗x^ for all maps x,y with x^ and y¯ are in the same meet semilattice
R1 and RR2 are immediate. To prove R2, we use Lemma 18. We can prove pR4 and RR4 using Lemma 14. RR1 comes from the definition of restriction idempotent. To prove R3, we have to use a particular case of pR4x⊗y¯=x¯⊗y¯ (which is already proved) when y=y¯; thus, x⊗y¯¯=x¯⊗y¯¯. Then, x¯⊗y¯¯ = x¯⊗y¯ because x¯⊗y¯=x¯∧y¯ by Lemma 18. However, x¯∧y¯ is a restriction idempotent; thus, x¯∧y¯=x¯∧y¯¯. Therefore, R3 holds. It remains to show RR3y¯⊗x^=y¯⊗x^, for which we are going to use RR4. By RR4, we have y¯⊗x^=y¯⊗x^^ which is equal to y¯⊗x^ for the same reason as above.


## 6. A Generalization of Ehresmann’s theorem

Now, we will define a functor between range prerestriction categories.Definition 14.
LetS and T be two range prerestriction categories. Let θ be functor from S to T. Then, θ is called a functor of range prerestriction categories if θx¯=θx¯ and θx^=θx^.Remark 5.
IfC is a generalized Ehresmann category, then RPC is a range prerestriction category. In addition, the orders ⩽l and ⩽r in RPC coincide with the generalized Ehresmann category orders.Proof.
Suppose thatx⩽ly in RPC. Then, for some e in C0, we have x=y⊗e which is equals y|dy∧e. We have shown therefore that x⩽ly in C.
Conversely, letx⩽ly in C. Then, x=y|dx which equals y⊗dx. Thus, we have shown that x⩽ly in RPC.
Now, we define a functor between generalized Ehresmann categories.Definition 15.
LetC and D be two generalized Ehresmann categories. Let F be a functor between C to D. Then, F is called a strongly ordered functor if(1)
Ifx⩽ly (resp. x⩽ry), then Fx⩽lFy (resp. Fx⩽rFy)(2)
Ife,j∈C0 and e and j are in the same meet semilattice, then Fe∧j=Fe∧FjLemma 21.
Letθ:S⟶T be a functor of range prerestriction categories. Define GEθ:GES⟶GET by GEθx=θx, where x is an arrow in GES. Then, GEθ is a strongly ordered functor.Proof.
The proof is straightforward.Lemma 22.
LetF:C⟶D be a strongly ordered functor. Define RPF:RPC⟶RPD by RPFx=Fx, where x is an arrow in RPC. Then, RPF is a functor of range prerestriction categories.Proof.
Letx,y∈RPC such that Try=Tdx; then, by definition(52)RPFx⊗y=Fx⊗y=Fx|ee|y=Fx|eFe|y,wheree=dx∧rybecauseFis a functor.
Now,x|e⩽lx, since F is strongly ordered, and we have Fx|e⩽lFx. Moreover, dx|e=e; therefore, dFx|e=Fdx|e=Fe. Now, from the fact that Fx|e⩽lFx and dFx|e=Fe, we have Fe⩽dFx; thus, the element Fx|Fe exists. By the uniqueness, we get that Fx|e=Fx|Fe. Similarly, Fe|y=Fe|Fy. Thus,(53)RPFx⊗y=Fx|FeFe|Fy.
However,Fe=Fdx∧ry=dFx∧rFy, so(54)RPFx⊗y=Fx⊗Fy=RPFx⊗RPFy.
Finally,RPFx¯=Fx¯=Fdx=dFx=RPFx¯. Similarly, we can prove RPFx^=RPFx^.Theorem 6.
The category of range prerestriction categories and functors of range prerestriction categories is isomorphic to the category of generalized Ehresmann categories and strongly ordered functors.Proof.
We already proved that there is a functor from any range prerestriction category to generalized Ehresmann category and there is a functor opposite to the direction. To prove the isomorphism, we need to show thatRP∘GE and GE∘RP are identity functors in each case. This now is straightforward.


## 7. Range Restriction Categories

In this section, we will present a special case of what we proved in Theorem1. We will start our section by an equivalent definition of range restriction category.Definition 16.
A range prerestriction category withR4 (y¯x=xyx¯, where rx=dy) is called a range restriction category.
It is easy to see that this definition is equivalent to Guo’s definition [4]. Therefore, the order ⩽r is included in the order ⩽l as proved in Lemma 4. From this remark, we will give another definition for generalized Ehresmann category called generalized R4-Ehresmann category which is defined as follows.Definition 17.
Define a generalizedR4-Ehresmann category to be an Ehresmann category with additional condition GE5′⩽l=⩽r∘⩽l=⩽l∘⩽r
We can deduce the proposition below which is another version of Ehresmann’s theorem but now for a range restriction category.Proposition 14.
The category of range restriction categories and functors of range restriction categories is isomorphic to the category of generalizedR4-Ehresmann categories and strongly ordered functors.Proof.
The proof is straightforward.


## 8. Inverse Categories

Inverse categories are defined in [9], see also [13]. In this section, we will give a proposition which is again a special case of Theorem 1, and a categorical version of the standard Ehresmann’s theorem for inverse semigroups. This result was first proved by Lawson in [14].Proposition 15.
LetC be a category. Then, the following are equivalent:(1)
C is an inverse category (i.e., every arrow x:α⟶β has a unique y:β⟶α with xyx=x and yxy=y).(2)
C is a range prerestriction category in which each map is a partial isomorphism (i.e., there exist x−1 such that x−1x=x¯ and xx−1=x−1¯). Moreover, C is range restriction.Proof.
(1) implies (2) is immediate.
(2) implies (1). First, we are going to show that if a range prerestriction category has a partial isomorphism, then C is range restriction category. It suffices to prove, in fact, y¯x=y−1yx=y−1yxf−1x=ff−1y−1yx=xyx−1yx=xyx¯.
It remains to thatx−1 is unique to prove the equivalence. Suppose that x has two inverses g and h; thus, hf=x¯,fh=h¯,gx=x¯, and xg=g¯. Therefore, g=gg¯=gxg=hxg=hxgxg=gxhxg=gxgxh=x¯h=hfh=h.
Now, if generalizedR4-Ehresmann category satisfies that every arrow has an inverse, then we shall call it a general ordered groupoid.Definition 18.
Define general ordered groupoid to be a categoryC equipped with a relation ⩽ satisfying the following axioms:GOG1C is an ordered category with corestriction.GOG2x⩽y⇒x−1⩽y−1GOG3∀x,y,uandv∈C such that x⩽y and u⩽v and also the products xu and yv are defined; then, xu⩽yv.GOG4C0,⩽ forms disjoint meet semilattices with a top element. Let e be in C0, we denote the meet semilattice containing e by MSLe.Proposition 16.
The category of the inverse categories and functors of range prerestriction categories is isomorphic to the category of general ordered groupoids and strongly ordered functors.Proof.
The proof is straightforward.



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