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

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.