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We consider two relations on a ∩-semigroup of partial functions on a given set: the inclusion of domains and semiadjacency (i.e., the inclusion of the image of the first function in the domain of the second). These are characterized from an abstract point of view via a system of elementary axioms, i.e., conditions expressed in the language of pure predicate calculus with equality.

Investigation of partial multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for n-place functions is an (n + 1)-ary superposition [ ], but there are some other naturally defined operations, which are also worth of consideration. In this paper we consider binary Mann's compositions ⊕ 1 , . . . , ⊕ n for partial n-place functions, which have many important applications for the study of binary and n-ary operations. We present methods of representations of such algebras by n-place functions and find an abstract characterization of the set of n-place functions closed with respect to the set-theoretic inclusion.