The relation of semiadjacency of cap-semigroups of transformations

We consider two relations on a $\cap$-semigroup of partial functions of a given set: the inclusion of domains and the semiadjacencity (i.e., the inclusion of the image of the first function into the domain of the second), which characterized with an abstract point of view using the elementary system of axioms, i.e., system conditions, recorded in the language narrow predicate calculus with equality.