Domain Semirings United

TL;DR

This paper explores conditions under which two different axiomatisations of domain operations on semirings are equivalent, unifying previously separate approaches in algebraic structures.

Contribution

It identifies classes of semirings where the two axiomatisations of domain operations coincide, unifying different theoretical frameworks.

Findings

Identifies classes of semirings where the two domain axiomatizations coincide

Provides conditions for the equivalence of the two approaches

Enhances understanding of algebraic structures involving domain operations

Abstract

Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an endofunction on a semiring that induces a distributive lattice bounded by the two units as its image. This note presents classes of semirings where these approaches coincide.