Residuated Ordered Structures in Logic and Algebra
by
W. J. Blok
University of Illinois at Chicago

One of the central concepts of logic, that of implication, has been formalized in numerous ways with the aim of capturing certain intended interpretations; to mention a few, the implication of classical, intuitionistic and relevant logic, and, more recently, linear logic. An algebraic study of these systems leads to the consideration of ordered groupoids <G, ·, <= > in which the multiplication operation · is left and right residuated. The multiplication · corresponds to a notion of conjunction, the residuation operations represent the implications.

Many special classes of ordered residuated groupoids have been studied, and not only in algebraic work inspired by logic. The papers by Ward and Dilworth from the late thirties on residuated lattices were not motivated by logical considerations, but by interest in the structure of the ordered set of ideals of a ring, as a lattice with the operations of ideal multiplication and ideal quotients added.

In the talk, I will discuss and compare some important classes of ordered residuated groupoids, stemming both from logic and algebra, and highlight recent progress made by various authors. I will then focus on the question whether-and how-some of the classes are generated by their finite members. A positive answer to that question will establish decidability of the equational or even universal theory of the class, and be of particular interest if it concerned a class associated with a logical system one is curious about.

Date received: March 28, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-94.