Partial Menger algebras and their weakly isomorphic representation

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.