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.
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
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