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].

Commutative idempotent residuated lattices

“…The following two statements about congruence lattices of CI residuated lattices are immediate consequences of results in [4] and [9]. The second sentence of Proposition 2.2 appears also in [8] (in a more general setting).…”

“…found in [4] and [10] and commutative residuated lattices were particularly studied in [9]. We will use the notation and terminology of these papers.…”

Modal operators on commutative residuated lattices

“…Next results are well known concerning to CRL ( [2,3,4,5,9,10]). We note the proofs of these results are the same as those of CRL with divisibility.…”

“…We recall a definition of bounded commutative integral residuated lattice ( [2,4,5]). An algebraic structure M = (M, ∧, ∨, , →, 0, 1) is called a bounded commutative integral residuated lattice if A binary operator ⊕ is defined by…”

Amalgamation and interpolation in ordered algebras

“…Indeed, it is shown in [25] that any such algebra satisfies the law (a → b)∨(b → a) = e, and hence by [32], is semilinear, that is, a subdirect product of totally ordered commutative integral GMV-algebras. Now any totally ordered commutative integral GMV-algebra is either the reduct of an MV-algebra or the negative cone of an -group (see Corollary 54 above).…”