Valentin S. Trokhimenko

Representations of Menger (2, n)-Semigroups by Multiplace Functions

ABSTRACT Investigation of partial multiplace functions by algebraic methods plays an important role in modern mathematics, where 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 article we consider binary Manns 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. Communicated by V. Artamonov.

FUNCTIONAL MENGER -ALGEBRAS

ABSTRACT Investigation of 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 functions is superposition (composition), but there are some other naturally defined operations, which are also worth of consideration. For example, the operation of set-theoretic intersection and the operation of projections. In this paper we find an abstract characterization of the set of multiplace functions which are closely related to these three operations.

The relation of semiadjacency on ∩-semigroups of transformations

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.

De Morgan (2, n)-Semigroups of n-Place Functions

An abstract characterization of sets of some partial functions from An to A closed with respect to the Mengers and Manns superpositions and a quasicomplementation is presented.

Stabilizers of Functional Menger Systems

A functional Menger system is a set of n-place functions containing n projections and closed under the so-called Mengers composition of n-place functions. We give the abstract characterization for subsets of these functional systems which contain functions having one common fixed point.

REPRESENTATIONS OF (2;n)-SEMIGROUPS BY MULTIPLACE FUNCTIONS

We describe the representations of (2;n)-semigroups, i.e., sets with n binary associative operations, by partial n-place functions and prove that any such representation is a union of some family of representations induced by Schein’s determining pairs.

Subtraction Menger algebras

We give an abstract characterization of algebras of partial functions from An to A endowed with the operations of the Menger superposition and the set-theoretic difference of functions as subsets of An+1.

Menger Algebras of n-Place Functions

It is a survey of the main results on abstract characterizations of algebras of

Menger algebras of idempotent n-ary operations

n

On σ-commutativity in Menger algebras of n-place functions

-place functions obtained in the last 40 years. A special attention is paid to those algebras of