The weakest nontrivial idempotent equations

TL;DR

This paper identifies a fundamental nontrivial equational condition that is implied by all other nontrivial idempotent equational conditions, revealing a core property in algebraic structures.

Contribution

It introduces a unique minimal nontrivial equational condition that underpins all other nontrivial idempotent conditions.

Findings

Established a single nontrivial equational condition implied by all others

Provided a theoretical foundation for understanding idempotent equational conditions

Clarified the structure of algebraic conditions in universal algebra

Abstract

An equational condition is a set of equations in an algebraic language, and an algebraic structure satisfies such a condition if it possesses terms that meet the required equations. We find a single nontrivial equational condition which is implied by any nontrivial idempotent equational condition.