一句话：刻画一元确定分配 ℓ-magma 的簇结构，并将其应用于束蕴涵代数的模型与有限模型构造。
本文研究一元确定的分配格序广群（dℓ-magma）：其乘法由顶元素和两个保并一元运算 p,q 决定，并与带顶分配格及 p,q 项等价。作者给出 p,q 的简单条件，使 x·y=(p x∧y)∨(x∧q y) 满足结合、交换、幂等或含单位元，从而推广双幂等半环的结构结果；当底格为 Heyting 代数时，为束蕴涵逻辑的一元确定代数模型提供结构洞察。文章还给出 Kripke 语义，以更高效构造有限模型，并枚举势不超过八且 p=q 为闭包算子的次直不可约代数、所有有限一元确定束蕴涵链，刻画其生成的 join-irreducible 簇偏序。
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.
@article{paperbot2779,
title = {Varieties of unary-determined distributive $\ell$-magmas and bunched implication algebras},
author = {Natanael Alpay and Peter Jipsen and Melissa Sugimoto},
journal = {Logical Methods in Computer Science},
volume = {Volume 20, Issue 1},
year = {2024},
doi = {10.46298/lmcs-20(1:12)2024}
}