Definability of the variety generated by a commutative monoid in the lattice of commutative semigroup varieties

TL;DR

This paper presents a first-order formula that explicitly defines the variety generated by a given commutative monoid within the lattice of commutative semigroup varieties, advancing the understanding of algebraic structure definability.

Contribution

It introduces a precise first-order formula for defining the variety generated by any commutative monoid in the lattice of commutative semigroup varieties.

Findings

Explicit first-order formula for variety definition

Enhanced understanding of algebraic structure definability

Framework applicable to other algebraic structures

Abstract

Let M be a commutative monoid. We provide an explicit first-order formular that defines the variety generated by M in the lattice of commutative semigroup varieties.