On the structure of residuated lattices
by
Kevin Blount
Vanderbilt University
Coauthors: Constantine Tsinakis

A residuated lattice is an ordered algebraic structure L=<L, /\ , \/ 1, ·, \, /> such that <L, /\ , \/ > is a lattice, <L, 1, ·> is a monoid and \ and / are binary operations such that the quasi-identies a·b <= c iff a <= c/b iff b <= a\c hold for all a, b, c in L. The class of all residuated lattices is denoted RL.

The study of such objects originated in the context of the theory of ring ideals in the early part of this century. 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 seminal papers published in the 1930's. Since that time there has been substantial research regarding some specific subclasses (including l-groups, and Heyting algebras), but we believe that this is the first time that some 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 congruences correspond to these normal subalgebras in the same way that ring congruences correspond to ring ideals. Furthermore, we produce an equational basis for both the class RL and the subclass generated by all residuated chains. In the process, we find that this subclass has additional structural properties that we believe could lead to some important decomposition theorems for its members.

Date received: February 16, 2000