#### 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.

#### 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 () which will denote the set of all relations between the set (). 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 (). 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 set , equipped with a partial multiplication satisfying the following axioms:(i) is defined if and only if is defined in this case and they are equal(ii) and are defined and then is defined(iii) For each , there exists an identity and an identity such that and One can prove that the identity (resp. ) such that (resp. ) is unique and call the domain of , denoted by (resp. range or codomain of , denoted by ).

*Definition 2. *A category is ordered if it satisfies the following axioms: is a poset implies and If and and both and exist, then If and and , then (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. *Let be an ordered category and the set of identities:(1) has restrictions if every and , where , then there exists a unique element called the restriction of to , which will be denoted by , such that and (2) has corestrictions if every and , where , then there exists a unique element called the corestriction of to , which will be denoted by , such that and To a category theorist, a restriction condition in an ordered category is equivalent to saying that the map is a discrete fibration (we consider 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 Theorem 2. 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 be a semigroup and let be a fixed, nonempty subset of the idempotents of . Before starting our definitions, we define two binary relations:

*Remark 1. *The binary relations and are equivalence relations on .

*Notation 1. *Let be an element in . We denote the classes and classes containing the element by and , respectively.

*Definition 4. *Let be a semigroup. An equivalence relation on is a right (resp. left) congruence if and only if every and imply (resp. ).

*Definition 5. *Let be a semigroup. Then, is an Ehresmann semigroup if is a commutative subsemigroup of and if each -class and each -class contains at least one element of . If is a right congruence and is a left congruence, in this case, we say that has the congruence condition.

Now, we provide a useful proposition to establish the equivalence between Ehresmann semigroups and range prerestriction semigroups.

Proposition 1. *Let be an Ehresmann semigroup. Then,*(1)*Each and class contains a unique element in . Let be in . We denote by (resp.) the unique element of (resp. ).*(2)*For each , both and .*(3)*The congruence condition in is equivalent to these two axioms:*

*Proof. *(1)We will prove the statement for . The proof for is similar. Let such that . Then, and . As is a commutative sub-semigroup, .(2)Too trivial because each class contains one idempotent element in .(3)Suppose that satisfies the congruence condition. Let be in ; then, . Then, for any , we have ; therefore, .Now, suppose that holds for all . Suppose that , and let be in . Then, so ; hence, . Therefore, . 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 condition , 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) (ii) (iii) (iv) (v) (vi) (vii) (viii)

*Example 1. *(i)Let be a set and the semigroup of all partial functions on . It is a range restriction semigroup. Indeed, for any element in , the restriction structure is given by and the range structure .(ii)Any monoid with identity has trivial restriction structure and trivial range structure given byHowever, the problem is that is not valid in . So, instead of , we introduce . This axiom is true in .

Now, we can define officially a range prerestriction semigroup as follows.

*Definition 6. *A semigroup is a range prerestriction semigroup if and only if there are two unary operations and satisfying the following axioms:(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Proposition 2. *A range restriction semigroup is a range prerestriction semigroup.*

*Proof. * using , , , and .

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. *Let be a range prerestriction semigroup. Then, for every element and in , we have the following:*(1)* is an idempotent element*(2)*(3)**(4)**(5)**(6)**(7)**(8)**(9)** is an idempotent element*(10)*(11)**(12)**(13)**(14)*

*Proof. *(1)By and , we have .(2)By , , and , we have .(3)By (1), , , and .(4)By = = .(5)Suppose that . By , we get = . Suppose that . And, by , we have . Therefore, ; thus, .(6)By and , .(7)By and , .(8)By and , .(9)By and (1), .(10)By , , and (9), .(11)From , we have for each . In particular, we get . Now, we can use (7) . Thus, using , . An application of (1) gives .(12)By (6), (8), and , .(13)By , , and (10), .(14)Suppose that . By (8), we get . Suppose now that . By , we have . Therefore, ; thus, .We shall use the properties established in Proposition 3 without further comment.

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

Let be a range prerestriction semigroup. To distinguish between idempotent elements and the elements of the form and , where , we will introduce the following definition.

*Definition 7. *Let be range prerestriction (range restriction) semigroup. An element in is a restriction idempotent if there is an element in such that .

Theorem 1. *In a given range prerestriction (range restriction) semigroup , the following statements are equivalent:*(1)* is a restriction idempotent*(2)*(3)**There is an element in such that *(4)

*Proof. *If is a restriction idempotent, then for some in . Hence,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 idempotent and in , we have that and , and these two elements commute using . The fact that they form a semigroup is again obvious using .

Theorem 2. *Let be a semigroup. Then, is an Ehremann semigroup if and only if is a range prerestriction semigroup.*

*Proof. *Let be a range prerestriction semigroup and let be the set of all restriction idempotents. By Lemma 1, is a commutative subsemigroup of . Each -class contains at least one element; indeed, because, for all , we have . Dually, we can prove too, i.e., we can prove that, for all , we have . By Proposition 1 (3), we conclude that any range prerestriction semigroup is Ehresmann semigroup by taking the set to be all the restriction idempotents.

Conversely, suppose now that is an Ehresmann semigroup. Therefore, for every element in , there exist unique elements (using Proposition 1) and in such that and ; thus, and are verified. Now, from Proposition 1 (2), for every element in , and . In particular, when , we get ; thus, is satisfied. From Proposition 1 (3), satisfies the congruence condition, so and . Therefore, and hold.

Now, for all elements and in , and are in , so by definition, they commute; thus, is verified.

To prove , we have to use a particular case of , namely, . Then, because belongs to ( is a semigroup). So, by Proposition 1 (2), for every element in , thus . It remains to show ; for that, we are going to use . By , we have which is equal to ( is a semigroup, and by Proposition 1 (2), for every element in , ).

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 category equipped with two relations and , satisfying the following axioms: is an ordered category with corestrictions is an ordered category with restrictions If , then , and put on is a meet semilattice , andwe define the relation equal to If and , then If and , then As mentioned above, a range restriction semigroup is exactly an Ehresmann semigroup (range prerestriction semigroup) with an extra condition which is . 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 Theorem 3, the isomorphism between range restriction semigroups and -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) , and there exist such that (ii) , and there exists such that (iii) , and there exists such that

Let be an Ehresmann (range Prerestriction) semigroup. Then, (which is equivalent to Lawson’s condition [1]) holds if and only if . Indeed, let . Then, there exists such that . By applying the condition, we get that ; thus, . 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 -Ehresmann category . And, we have more than that, and the relation is compatible with the multiplication, i.e., for every elements , and in such that and ; then, . We will prove this last statement using two lemmas. First, let us define a -Ehresmann category.

*Definition 9. *An -Ehresmann category is an Ehresmann category with :To deduce Corollary 1, 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 be an Ehresmann category:*(1)*If and , then *(2)*If and , then **For any -Ehresmann category , we can define a pseudoproduct as follows. Let :**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. *Let be an -Ehresmann category. Let , and be elements in :*(1)*If , then *(2)*If , then **That is, the relation is compatible with the multiplication.*

*Proof. *Let us prove (1). Similarly, we can prove (2). First, we will prove (1) only for an element in , i.e., if , then , for some . Since , then ; thus, from the lemma above . Now, it is clear that .

Let be any element in . Therefore,From , we obtain which is just . From , we have so . However, then by the previous lemma, we obtain . However, so that . From the fact that and , we may conclude thatwhich is just .

Corollary 1. *The category of range restriction semigroups and their homomorphisms is isomorphic to the category of -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. *Let be a small category. A range Prerestriction structure on a category is an assignment to each arrow , two arrows and , such that the following eight conditions are satisfied: () for each map () with () with () with () for each map () for each map () with () with If a category has a range prerestriction structure, we call it a range prerestriction category.

If satisfies the conditions , , , and , 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. *Let be a range prerestriction category and arrows in . We have the following:*(1)* is idempotent*(2)* if *(3)*(4)** if *(5)* if *(6)* if *(7)* if *(8)* if *(9)* for each map *(10)* for each map *(11)* for each map *(12)* if *(13)* if *(14)* if *

*Definition 11. *Let be a range prerestriction (or restriction) category. An arrow in is called a restriction idempotent if there is an arrow in such that .

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 category , the following statements are equivalent:*(1)* is a restriction idempotent*(2)*(3)**There is an element in such that *(4)

*Proof. *If is a restriction idempotent, then for some arrow in . Hence,Thus, (1) implies (2). The remaining implications are trivial.

*Notation 2. *Let be a range prerestriction (or range restriction) category, and and two identities. We denote(1)(2)(3), where is the set of all identities in

Theorem 5. *In a range prerestriction (or range restriction) category, the set is a idempotent commutative semigroup.*

*Proof. *The proof of this theorem is trivial because for every two restriction idempotents and in , we have that and and these two elements commute using . 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. *Let be range prerestriction (or range restriction) category. We can define three relations on any , where and are identities as follows: (i) (ii) (iii)

Proposition 5. *On a range prerestriction category , the relations and are partial orders on , where and are identities. Furthermore, the three orders are the same on , which will be denoted . Moreover, with this order forms a meet semilattice with the top element . And, if we have* *(i) and , then * *(ii) and , then *

*Proof. *We will prove that is a partial order on . The proof for the other relations is similar.

Reflexivity: clearly so that , where is an arrow in .

Antisymmetry: let and . Then, and ; thus,Similarly, we can prove . Therefore, .

Transitivity: if and , then and ; furthermore, implies . Similar to this proof, we can show . Thus, . So, .

Let for some identity . If , then because the restriction idempotents commute (see Theorem 5). Therefore, on , which we denote by .

with forms a meet semilattice with a top element. In fact, let . Then, , by commutativity. Thus, . Similarly, . Now, let . Then, since , and since . Thus, . Hence, . It follows that is a meet semilattice. The top element is because if not, there is in such that . Then, , but because is the identity. Thus, .

The last two implications are trivial.

Proposition 6. *Let be a range prerestriction category. Then,* *(i) there exist such that * *(ii) there exist such that * *(iii) there exist such that *

*Proof. *We will prove the case. Let . Then, and so that and ; hence,The converse is clear.

Proposition 7. *Let be a range prerestriction category. Then,*

*Proof. *Note first that since if so that (), then by , we get . Similarly, . Thus, it is clear that and are contained in .

Now, let . Then, . Put . Then, by the previous proposition above, . Furthermore, so . Thus, we have shown giving . 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. *Let be a range prerestriction category. Then, is a range restriction category if and only if .*

*Proof. *Suppose that is a range restriction category. Then, is verified. If then .

Suppose that . Let and be two arrows. Let . Then, . Therefore, which gives .

Proposition 8. *Let be a range prerestriction category. Then,* *(i) If , then for all * *(ii) If , then for all * *(iii) If is a range restriction category (i.e., satisfies ), then is compatible with the multiplication in the sense that , and , , and imply *

*Proof. *If , then so that giving . The proof of (2) is similar to (3).

By Proposition 7,By Lemma 4, if holds, then so thatand hence, . By part (2), we will show only that if , then for any arrow . Let . Then, , where is a restriction idempotent. Thus, . Hence, .

Proposition 9. *Let be a range prerestriction category.*(1)*If , then and *(2)*If , then and *(3)*If , then and *(4)*If and and and , then *(5)*If and and and , then *

*Proof. *(1)If , then for some restriction idempotent . Thus, . Therefore, . And, . Thus, . The proof is similar for (2).(3)If , then . Furthermore, implies . Similarly, to this proof, we can show . Thus, and .(4)If and and and , then and so that . Now, so that . However, so that . Hence, , giving .

Proposition 10. *Let be a range prerestriction category. And, let be a restriction idempotent.*(1)*If , then there is a unique element such that and *(2)*If , then there is a unique element such that and *

*Proof. *We prove (1). We will show . Note that and . If there exists an arrow such that and , then .

#### 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 category equipped with two relations and on satisfying the following axioms: () is an ordered category with corestriction. () is an ordered category with restriction. () Let and be in . Then, and in this case, we put . forms disjoint meet semilattices with a top element. Let be in , we denote the meet semilattice containing by . () , and we define . () If , then , and let be in , then . () If , then , and let be in , then .

Proposition 11. *If is a range prerestriction category, then define a category to be* *(i) Identities: all the restriction idempotents of , where is an identity in .* *(ii) Arrows: the arrows of , where and .* *(iii) Composition denoted by : the composition of two elements and in is defined if . And, when it is defined is equal to .** 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 satisfies as is easily seen.

*Proof. *First, we have to show that with is a category (i.e., with the -composition is a multiplicative set satisfying , , and ). We begin by showing all restriction idempotents of are identities of . Let be a restriction idempotent and suppose that is defined for some arrow . By the definition of the composition, we deduce that . Then, . Similarly, if is defined, then it is equal to . We now check that the axioms , , and hold. (i) Axiom holds: suppose that is defined. Then, and is defined. Hence, and . Therefore, by , we have , so is defined. Also, . Thus, is defined. It is clear that . A similar argument shows that if exists, then exists and they are equal. (ii) Axiom holds: suppose that and are defined. We have that and . Now, (iii) Thus, is defined. (iv)Axiom holds: for each arrow , we have that is defined, and we have seen that restriction idempotents are identities in . Similarly, we have is defined.It is now clear that is a category.

Now, is an ordered category. From Proposition 5, the relations and are partial orders, so is verified. And, is clear from the remark in the same Proposition. Now, by Proposition 3(1) and [12], the axiom holds and by (4) and (5) axiom holds. Proposition 4 implies the restriction and corestriction axioms. Thus, and are verified. By Proposition 5, the axiom holds. is consequence of Proposition 5. holds by Proposition 7. The axioms and holds by Proposition 8. In fact, if , then , and let be in ; then, by the Proposition 8, we have ; thus, . Similarly, we can show .

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. Let be a fixed generalized Ehresmann category.

Lemma 5. *Let and to be in . If , then and . Furthermore, the relation is a partial order.*

*Proof. *Let . Then, by , so that there exists an arrow such that and . However, then, by , and ,so that and .

Reflexivity: from , we have .

Antisymmetry: let and and let be an arrow such that . From the first part of the lemma, we have that and . However, so that, from , we obtain . In a similar way, .

Transitivity: let and . Then, there exist arrows and such thatThus,and there exists an arrow such thatbut then, so that .

Lemma 6. *Let and be arrows in . Then,*(1)*If and , then *(2)*If and , then *(3)*If and and , then *(4)*The set for any identity is an order ideal of (resp. and )*(5)*The relation agrees with on *

*Proof. *(1)By , there exists an arrow such that ; thus, by Lemma 5,(2). From , we have by ; since , we obtain giving .(3)The proof of (2) is similar to (1).(4)By (1), we have that so that but ; hence, .(5)Let , where is in . Then, .The proof of (5) is straightforward.

Lemma 7. *Let be an arrow in . Then,*(1)*If and , then *(2)*If and , then *

*Proof. *We prove (1). Note first that so that is defined and . However, and so that .

Lemma 8. *Let be an arrow in .*(1)*Let with . Then,*(2)*Let with . Then,*

*Proof. *We will prove (1). By , we have and . By , . Let with . By , there exists an arrow such that . By ,but, from , it follows that ; thus, . However, thenFrom the fact that and and , we have by Lemma 7. Thus, as required.

Lemma 9. *Let be a generalized Ehresmann category. Let and be in . If , then *