The relation of semiadjacency in transformative $\cap\,$-semigroups

TL;DR

This paper studies semigroups of transformations closed under intersection, introducing and characterizing the relations of semicompatibility and semiadjacency, which relate to how transformations interact via their domains and images.

Contribution

It provides abstract characterizations of semigroups of transformations with semicompatibility and semiadjacency relations, expanding understanding of their algebraic structure.

Findings

Defined semicompatibility and semiadjacency relations on transformation semigroups.

Provided abstract algebraic characterizations of these semigroup structures.

Enhanced the theoretical framework for semigroups closed under intersection.

Abstract

We consider semigroups of transformations (partial mappings defined on a set $A$) closed under the set-theoretic intersection of mappings treated as subsets of $A\times A$. On such semigroups we define two relations: the relation of semicompatibility which identifies two transformations at the intersection of their domains and the relation of semiadjacency when the image of one transformation is contained in the domain of the second. Abstract characterizations of such semigroups are presented.