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Equational bases for joins of residuated-lattice varieties. (English) Zbl 1068.06007
Author’s abstract: Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over \(\mathbf{FL}^+\) is also finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product.

MSC:
06B20 Varieties of lattices
03G10 Logical aspects of lattices and related structures
06F05 Ordered semigroups and monoids
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
08B05 Equational logic, Mal’tsev conditions
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