Cayley’s and Holland’s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices

Abstract

We extend Cayley's and Holland's representation theorems to idempotent semirings and residuated lattices, and provide both functional and relational versions. Our analysis allows for extensions of the results to situations where conditions are imposed on the order relation of the representing structures. Moreover, we give a new proof of the finite embeddability property for the variety of integral residuated lattices and many of its subvarieties.

Distributive residuated frames and generalized bunched implication algebras

“…Distributive residuated lattices appear naturally and also have a simpler representation [7] than general ones. However, some useful methods and techniques already developed do not apply to the distributive case.…”

“…However, some useful methods and techniques already developed do not apply to the distributive case. In particular, relation semantics, known as residuated frames and introduced in [8], have turned out to be a very useful tool and provide a very natural setting for the investigation of both algebraic and logical properties in the area [7,6]. We develop such frames in the distributive case and use them to obtain various results in logic and in algebra.…”