Theory of restrictive Menger algebras

Abstract

In modern algebra, much attention is being paid at present to the study of superpositions of multiplace functions, which is explained by their applications to mathematics, cybernatics, programming, and other domains of science [I, 2]. In this connection, it is necessary to axiomatize certain concrete classes of algebras of multiplace functions. Side by side with the operation of superposition in function algebras, we also quite often consider other naturally defined operations, which enable us to express various dependences between the functions in convenient manner [3]. One of these operations is the operation of restrictive multiplication of functions, i.e., the operation of restricting one function to the domain of another function, considered for the first time by Vagner for one-place functions [4]. A joint study of this operation with the operation of superposition, and also with the operation of set-theoretic intersection in the case of multiplace functions, has been carried out in [5, 6], in which the restrictive Menger algebras and the restrictive Palgebras of multiplace functions have been characterized by finite systems of elementary axioms. In the consideration of sets of reversible multiplace functions, i.e., of functions that are one-to-one with respect to each variable separately, earlier only restrictive Palgebras could be characterized [6]. No characterization of the restrictive Menger algebras of reversible n-place functions has yet been found for n ~ 2; it is obtained in the present article.