A note on idempotent semirings

TL;DR

This paper investigates the properties of S-algebras over a commutative semiring S, demonstrating that certain subvarieties defined by specific identities are closed under non-empty colimits, with implications for Boolean rings and distributive lattices.

Contribution

It establishes a general result that subvarieties of S-algebras defined by particular identities are closed under non-empty colimits, unifying known cases like Boolean rings and distributive lattices.

Findings

Subvarieties defined by 1+2x=1 and x^2=x are colimit-closed.

Boolean rings form a colimit-closed subcategory.

Distributive lattices form a colimit-closed subcategory.

Abstract

For a commutative semiring S, by an S-algebra we mean a commutative semiring A equipped with a homomorphism from S to A. We show that the subvariety of S-algebras determined by the identities 1+2x=1 and x^2=x is closed under non-empty colimits. The (known) closedness of the category of Boolean rings and of the category of distributive lattices under non-empty colimits in the category of commutative semirings both follow from this general statement.