Ações de categorias, sistemas e equivalência entre as categorias de sistemas e semigrupos inversos

Abstract

Mark V. Lawson, in the book "Inverse Semigroups: The Theory of partial symmetries", provides a very relevant study of the characteristics of inverse semigroups, including Wagner-Preston Theorem of Representation, which states that every inverse semigroup can be faithfully represented by a inverse semigroup of partial bijections on a set. A refinement of this theorem shows that every inverse semigroup is isomorphic to an inverse semigroup of all partial symmetries (of a specific type) of some structure specifies. These structures belong to a class of category actions on sets. In this work we study each stage of refinement and go further, as the article "Constructing inverse semigroups from category actions"of this author, Initially, we point out that based on the actions on a set of categories that satisfy the condition of the orbit we obtain an inverse semigroup with zero. Reciprocally, each inverse semigroup with zero we can obtain a category action that satisfies some conditions. Such actions, called systems, constitute the category SY S. Next, build functors between the categories and category SY S and the category INV of inverse semigroups with zero: Θ : SY S ! INV and : INV ! SY S, showing that every inverse semigroup S of INV , we have Θ(Ω(S)) isomorphic to S. However, for each system T of SY S, (Θ(T)) and T does not always are isomorphic. Still, it is possible to show that INV is equivalent to a proper quotient of the category SY S.