Representable Integral Residuated Lattices
by
C. J. van Alten
University of the Witwatersrand, Johannesburg, South Africa

A residuated lattice is a lattice-ordered monoid that is `residuated' in the sense that it has binary operations / and \ which satisfy

x ·z <= y     <===>     z <= x \y     <===>     x <= y / z.

An integral residuated lattice also satisfies x <= 1. The class of integral residuated lattices is a variety which we denote by IR. These algebras originate in the study of ideal lattices of rings and as algebraic models of linear logic and Lambek's calculus with weakening.

Our objective is the axiomatization of the class of all members of IR that may be represented as subalgebras of products of linearly ordered members of IR. Such integral residuated lattices are called representable; we shall prove that the class of all such algebras is axiomatized, relative to IR, by the identity (x \y) \/ (w / (w / (((y \x) \z) \z))) = 1. The identity uses only the operations /, \, \/ and 1 so we shall also obtain axiomatizations of the class of representable algebras in subreduct classes of IR whose languages contain /, \, \/ and 1.

The congruence lattice of A in IR is isomorphic to the lattice of `filters' of A (ie convex submonoids F of A satisfying [aF) = [Fa)). The method of proof utilizes the larger lattice of `prefilters' of A (ie convex submonoids of A). In commutative examples, prefilters and filters coincide so the proof presented is an extension of the known result for commutative integral residuated lattices, where the identity (x \y) \/ (y \x) = 1 axiomatizes the representable subclass. This result is also known in the context of BCK-algebras, which are the <\, 1 >-subreducts of commutative integral residuated lattices.

Date received: December 21, 2001