Narrow Results

Multiplace functions, which are also called functions of many variables, and their algebras called Menger algebras have been studied in various fields of mathematics. Based on the theory of many-sorted algebras, the primary aim of this paper is to present the ideas of Menger systems and Menger systems of full multiplace functions which are natural generalizations of Menger algebras and Menger algebras of $n$-ary operations, respectively. Two specific types of $n$-ary operations, which are called idempotent cyclic and weak near-unanimity generated by cyclic and weak near-unanimity terms, are provided. The Menger algebras under consideration have a two-element universe, the elements of which are two specific $n$-ary operations. Additionally, we provide necessary and sufficient conditions in which the abstract Menger algebra and the Menger algebras of these two $n$-ary operations are isomorphic. An abstract characterization of unitary Menger systems via systems of idempotent cyclic and weak near-unanimity multiplace functions is generally investigated. A strong connection between clone of terms and Menger systems of full multiplace functions is also investigated.

It is a survey of the main results on abstract characterizations of algebras of n-place functions obtained in the last 40 years. A special attention is paid to those algebras of n-place functions which are strongly connected with groups and semigroups, and to algebras of functions closed with respect natural relations defined on their domains.