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