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 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.

Communicated by V. Artamonov.

Key Words:

• (2, n)-semigroup
• Menger algebra
• Multiplace function

Mathematics Subject Classification:

• 20N15
• 08N05

Notes

¹In the case of universal algebras, the term “partial functions” is replaced by the term “partial operations”. Instead of “full functions” we say also “operations”.

² f ⊂ g means that the domain of f is contained in the domain of g and f(x) = g(x) for all x from the domain of f.