Varieties of unary-determined distributive $\ell$-magmas and bunched implication algebras

TL;DR

This paper characterizes unary-determined distributive $ ext{l}$-magmas, explores their algebraic properties, and provides Kripke semantics, offering structural insights and computational tools for bunched implication logic.

Contribution

It introduces simple conditions for algebraic operations to be associative, commutative, and idempotent, and connects these to models of bunched implication logic.

Findings

Characterizes unary-determined distributive $ ext{l}$-magmas.

Provides Kripke semantics for these algebras.

Identifies all subdirectly irreducible algebras up to size eight.

Abstract

A distributive lattice-ordered magma ($d\ell$-magma) $(A,\wedge,\vee,\cdot)$ is a distributive lattice with a binary operation $\cdot$ that preserves joins in both arguments, and when $\cdot$ is associative then $(A,\vee,\cdot)$ is an idempotent semiring. A $d\ell$-magma with a top $\top$ is unary-determined if $x{\cdot} y=(x{\cdot}\!\top\wedge y)$ $\vee(x\wedge \top\!{\cdot}y)$. These algebras are term-equivalent to a subvariety of distributive lattices with $\top$ and two join-preserving unary operations $\mathsf p,\mathsf q$. We obtain simple conditions on $\mathsf p,\mathsf q$ such that $x{\cdot} y=(\mathsf px\wedge y)\vee(x\wedge \mathsf qy)$ is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the distributive lattice is a Heyting algebra, it provides structural insight into unary-determined algebraic models of bunched implication logic. We also provide Kripke semantics for the algebras under consideration, which leads to more efficient algorithms for constructing finite models. We find all subdirectly irreducible algebras up to cardinality eight in which $\mathsf p=\mathsf q$ is a closure operator, as well as all finite unary-determined bunched implication chains and map out the poset of join-irreducible varieties generated by them.