A residuated lattice is an ordered algebraic structure [Formula: see text] such that <L,∧,∨> is a lattice, <L,·,e> is a monoid, and \ and / are binary operations for which the equivalences [Formula: see text] hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45–48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ℛℒ as a whole. In particular, we develop the notion of a normal subalgebra and show that ℛℒ is an "ideal variety" in the sense that it is an equational class in which congruences correspond to "normal" subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ℛℒC that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]).

We generalize the notion of an MV-algebra in the context of residuated lattices to include noncommutative and unbounded structures. We investigate a number of their properties and prove that they can be obtained from lattice-ordered groups via a truncation construction that generalizes the Chang-Mundici Γ functor. This correspondence extends to a categorical equivalence that generalizes the ones established by D. Mundici and A. Dvurečenskij. The decidability of the equational theory of the variety of generalized MV-algebras follows from our analysis.  2004 Elsevier Inc. All rights reserved.

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Abstract. Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups). Although cancellative monoids are defined by quasi-equations, the class CanRL of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of CanRL.We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by LG − . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of LG to the lattice of subvarieties of LG − .Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzie's characterization of categorically equivalent varieties.

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].