Unary-Determined Distributive ℓ -magmas and Bunched Implication Algebras

Authors

Natanael Alpay, Chapman UniversityFollow
Peter Jipsen, Chapman UniversityFollow
Melissa Sugimoto, Chapman UniversityFollow

Document Type

Conference Proceeding

Publication Date

10-22-2021

Abstract

A distributive lattice-ordered magma (dℓ-magma) (A,∧,∨,⋅) is a distributive lattice with a binary operation ⋅ that preserves joins in both arguments, and when ⋅ is associative then (A,∨,⋅) is an idempotent semiring. A dℓ-magma with a top ⊤ is unary-determined if x⋅y=(x⋅⊤∧y)∨(x∧⊤⋅y). These algebras are term-equivalent to a subvariety of distributive lattices with ⊤ and two join-preserving unary operations p, q. We obtain simple conditions on p, q such that x⋅y=(px∧y)∨(x∧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.

Comments

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science, volume 13027, in 2021. The final publication may differ and is available at Springer via https://doi.org/10.1007/978-3-030-88701-8_2

Recommended Citation

Alpay N., Jipsen P., Sugimoto M. (2021) Unary-Determined Distributive ℓ-magmas and Bunched Implication Algebras. In: Fahrenberg U., Gehrke M., Santocanale L., Winter M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science, vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_2

Copyright

Springer