Implicit operations in varieties of commutative monoids

TL;DR

This paper characterizes implicit operations in varieties of commutative monoids, showing they can be interpolated by terms precisely when the monoids are inverse monoids, using advanced algebraic and combinatorial methods.

Contribution

It provides a characterization of implicit operations in commutative monoids, linking them to inverse monoids through a novel algebraic and combinatorial framework.

Findings

Implicit operations can be interpolated by terms in inverse monoids.

The class of inverse monoids is characterized by a specific algebraic condition.

The methods extend Isbell's Zigzag Theorem to all equational classes of commutative monoids.

Abstract

An implicit operation of a class of similar algebras $\mathsf{K}$ is a collection of first order definable partial functions on the members of $\mathsf{K}$ that is globally preserved by homomorphisms. For instance, "taking inverses" can be viewed as a unary implicit operation of the class of all monoids because its graph on a given monoid is defined by the equation $xy \thickapprox 1 \thickapprox yx$ and monoid homomorphisms preserve existing inverses. As this example demonstrates, the implicit operations of a class $\mathsf{K}$ need not be given by a term of $\mathsf{K}$. We show that an equational class of commutative monoids can be expanded with enough implicit operations so that every implicit operation can be interpolated by a family of terms if and only, in each of its members, for every $a$ there exists some $b$ such that $a = a^2b$, i.e., the class consists of inverse monoids. Our methods build on the interaction of the theory of implicit operations with Grillet's description of finitely generated subdirectly irreducible commutative semigroups and the combinatorics deriving from an extension of Isbell's Zigzag Theorem to all equational classes of commutative monoids.