A characterization of $n$-associative, monotone, idempotent functions on an interval that have neutral elements

TL;DR

This paper characterizes monotone, idempotent, n-ary functions with neutral elements on intervals, generalizing existing theorems and showing such functions form quasitrivial semigroups.

Contribution

It extends Czogala-Drewniak Theorem to n-ary functions and fully characterizes these functions with neutral elements on intervals.

Findings

Monotone, idempotent, n-ary semigroups are quasitrivial.

Provides a complete description of such functions with neutral elements.

Generalizes existing theorems to n-ary functions.

Abstract

We investigate monotone idempotent $n$-ary semigroups. One of the main result of this article is the generalisation of Czogala-Drewniak Theorem, which describes the idempotent monotone associative functions having neutral element. Furthermore we present the full characterisation of idempotent, monotone, $n$-associative functions on an interval which have neutral elements. %{\bl defined}. Our description provides that monotone, idempotent, $n$-ary semigroups are quasitrivial.