Menger algebras of $n$-place functions

TL;DR

This survey reviews 40 years of research on abstract algebraic structures of n-place functions, emphasizing connections with groups, semigroups, and natural relations on domains.

Contribution

It compiles and analyzes key results on the characterization of algebras of n-place functions, highlighting their links with group and semigroup theory.

Findings

Characterizations of algebras of n-place functions

Connections with groups and semigroups

Algebras closed under natural relations

Abstract

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.