Representable idempotent commutative residuated lattices. (English) Zbl 1117.03070
A commutative residuated lattice is a lattice-ordered commutative monoid with a residuation operation \(\rightarrow\) satisfying \(x \cdot z \leq y\) iff \(z \leq x \rightarrow y\); it is called idempotent if \(x \cdot x = x\), and representable if it embeds into a direct product of linearly ordered commutative residuated lattices. Let RICRL denote the variety of all representable idempotent commutative residuated lattices.
This variety contains all ‘relative Stone algebras’ and all ‘Sugihara monoids’, but is shown to be strictly larger than the varietal join of these two varieties. The main result is that the size of any \(n\)-generated subdirectly irreducible member of RICRL is bounded above by \(3n+1\); consequently, RICRL is a locally finite variety. The structure of subdirectly irreducible members of RICRL is fully described. This description is used to obtain a finite axiomatization of the subvariety of RICRL of all positive Sugihara monoids.
The local finiteness of RICRL implies that every finitely axiomatized subquasivariety of RICRL has a decidable quasi-equational theory. In the context of relevance logic, this result says that every finitely based extension of the positive relevant logic R\(_+\) containing the mingle and Gödel-Dummett axioms has a solvable deducibility problem.
This variety contains all ‘relative Stone algebras’ and all ‘Sugihara monoids’, but is shown to be strictly larger than the varietal join of these two varieties. The main result is that the size of any \(n\)-generated subdirectly irreducible member of RICRL is bounded above by \(3n+1\); consequently, RICRL is a locally finite variety. The structure of subdirectly irreducible members of RICRL is fully described. This description is used to obtain a finite axiomatization of the subvariety of RICRL of all positive Sugihara monoids.
The local finiteness of RICRL implies that every finitely axiomatized subquasivariety of RICRL has a decidable quasi-equational theory. In the context of relevance logic, this result says that every finitely based extension of the positive relevant logic R\(_+\) containing the mingle and Gödel-Dummett axioms has a solvable deducibility problem.
Reviewer: Clint van Alten (Wits)
MSC:
| 03G25 | Other algebras related to logic |
| 03B47 | Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) |
| 06F05 | Ordered semigroups and monoids |
| 06B20 | Varieties of lattices |
| 08B26 | Subdirect products and subdirect irreducibility |
| 08C15 | Quasivarieties |
Keywords:
locally finite variety; residuation; residuated lattice; representable; idempotent; Sugihara monoid; relative Stone algebra; relevance logic; mingleReferences:
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