Partial Menger algebras and their weakly isomorphic representation

Abstract

As generalization of semigroups, Karl Menger introduced in the 1940th algebras of multiplace operations. Such algebras satisfy the superassociative law, a generalization of the associative law. Menger algebras are defined as models of this superassociative law. Cayley’s theorem for semigroups says that any model of the associative law is isomorphic to a transformation semigroup. R. M. Dicker proved in 1963 that every Menger algebra of rank n is isomorphic to a Menger algebra of n-ary operations on some set. The composition of terms in which each variable occurs at most r-times, so-called r-terms, leads to partial algebras where the superassociative law is satisfied as a weak identity. In this paper, we introduce the concepts of a partial Menger algebra, a unitary partial Menger algebra and of a generalized partial Menger algebra. We prove that r-terms of some type form a generalized partial Menger algebra with infinitely many nullary operations. Using weak identities and weak isomorphisms, Dicker’s result will be extended to partial Menger algebras and to unitary partial Menger algebras.

Keywords

• n-ary operation
• n-ary term
• superposition of n-ary operations and n-ary terms
• r-term
• Menger algebra
• partial Menger algebra