Varieties of unary-determined distributive $\ell$-magmas and bunched implication algebras
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-d...
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| Veröffentlicht in: | Logical methods in computer science Jg. 20, Issue 1 |
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| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Logical Methods in Computer Science e.V
07.02.2024
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| Schlagworte: | |
| ISSN: | 1860-5974, 1860-5974 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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. |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-20(1:12)2024 |