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 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.

A Generalization ofn-Ary Algebraic Systems

“…In this article Dörnte observed that any n-ary groupoid G f of the form f x 1 x n = x 1 · x 2 · · x n , where G · is a group, an n-ary group, but for every n > 2, there are n-ary groups which are not of this form. A short review of basic results on n-ary groups one can find in Dudek and Głazek (2007), Dudek (2007), and Post (1940), also see Borowiec et al (2006) and Dudek and Trokhimenko (2006). A construction of all homomorphisms of an algebra with one n-ary operation (and finite number of operations) into algebra of the same type is described in Novotny (1996Novotny ( , 2002.…”

A Characterization of (2,n)-Semigroup ofn-Place Functions and De Morgan (2,n)-Semigroup ofn-Place Functions

“…Abstract algebras isomorphic to some sets of operations closed with respect to these compositions are described in [15]. The sets of partial functions closed with respect to these compositions and some additional operations are characterized in [4,6,7]. Also the set of partial binary functions closed with respect to the these compositions and one quasi-complementation operation is characterized in [12].…”

Stabilizers of Functional Menger Systems

“…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 (see for example [1,2,3,6,7]). The central role in these study play sets of functions with fixed points.…”