When does a semiring become a residuated lattice?

TL;DR

This paper investigates the conditions under which a semiring can be characterized as a residuated lattice, establishing that complete distributivity is key for the conversion, with specific results for complete and idempotent cases.

Contribution

It provides necessary and sufficient conditions for a semiring to be a residuated lattice, extending the understanding of their structural relationship.

Findings

Complete distributivity enables the conversion from semiring to residuated lattice.

Characterization of semirings associated with complete residuated lattices satisfying the double negation law.

Results for semirings related to idempotent residuated lattices.

Abstract

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.