Menger systems of idempotent cyclic and weak near-unanimity multiplace functions

Abstract

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 [Formula: see text]-ary operations, respectively. Two specific types of [Formula: see text]-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 [Formula: see text]-ary operations. Additionally, we provide necessary and sufficient conditions in which the abstract Menger algebra and the Menger algebras of these two [Formula: see text]-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.

The partial many-sorted algebras of terms and formulas with fixed variables count

“…Menger hyperalgebras generalizing Menger algebras were introduced by the authors in [20]. Algebras of weak-near unanimity functions were recently studied by the authors in [21]. A nice connection of Menger algebras and terms was mentioned in [10,35].…”

“…As mentioned in [21], a Menger system G of rank I is called unitary if it contains a complete collection of selectors of all arities, i.e., for every n, m ∈ I there are elements e 1m , . .…”