THE STRUCTURE OF COMMUTATIVE RESIDUATED LATTICES

Abstract

A commutative residuated lattice, is an ordered algebraic structure [Formula: see text], where (L, ·, e) is a commutative monoid, (L, ∧, ∨) is a lattice, and the operation → satisfies the equivalences [Formula: see text] for a, b, c ∊ L. The class of all commutative residuated lattices, denoted by [Formula: see text], is a finitely based variety of algebras. Historically speaking, our study draws primary inspiration from the work of M. Ward and R. P. Dilworth appearing in a series of important papers [9, 10, 19–22]. In the ensuing decades special examples of commutative, residuated lattices have received considerable attention, but we believe that this is the first time that a comprehensive theory on the structure of residuated lattices has been presented from the viewpoint of universal algebra. In particular, we show that [Formula: see text] is an "ideal variety" in the sense that its congruences correspond to order-convex subalgebras. As a consequence of the general theory, we present an equational basis for the subvariety [Formula: see text] generated by all commutative, residuated chains. We conclude the paper by proving that the congruence lattice of each member of [Formula: see text] is an algebraic, distributive lattice whose meet-prime elements form a root-system (dual tree). This result, together with the main results in [12, 18], will be used in a future publication to analyze the structure of finite members of [Formula: see text]. A comprehensive study of, not necessarily commutative, residuated lattices is presented in [4].