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.
Highlights
Introduction is 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 (Rel(X)) which will denote the set of all relations between the set (X)
Summary
We begin these preliminaries with an equivalent definition of a category where we omit the set of objects. A category is a set C, equipped with a partial multiplication satisfying the following axioms:. One can prove that the identity e Fx x) is unique and call the domain of x, denoted by d(x) A category C is ordered if it satisfies the following axioms:. Let C be an ordered category and C0 the set of identities:. A restriction condition in an ordered category is equivalent to saying that the map d: C ⟶ C0 is a discrete fibration (we consider C as a poset category) [6]
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