Varieties of residuated lattices

Abstract

A residuated lattice is an algebraic structure that has a lattice and a monoid reduct, such that multiplication is residuated with respect to the order. Residuated lattices generalize many well studied algebras including lattice-ordered groups, Brouwerian algebras and generalized MV-algebras. Moreover, they are connected to sub-structural logic, since they constitute algebraic semantics for the unbounded full Lambek calculus. Residuated lattices form a variety. We investigate the lattice of its subvarieties and concentrate on a number of interesting subvarieties. In particular, we construct a continuum of atomic varieties and prove that the join of two finitely based commutative residuated-lattice varieties is also finitely based. Moreover, we study the varieties of cancellative and of distributive residuated lattices and present a duality theory for the bounded members of the latter. Finally, we generalize standard MV-algebras and describe a representation theorem and a categorical equivalence about them. As a corollary we obtain the decidablility of their equational theory.