Equational bases for joins of residuated-lattice varieties.

*(English)*Zbl 1068.06007Author’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.

Reviewer: Yuri Movsisyan (Yerevan)

##### 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 |