A distributive lattice-ordered magma (dl-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 $p,q$. We obtain simple conditions on $p,q$ such that $x{\cdot} y=(px\wedge y)\vee(x\wedge qy)$ 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.