On some classes of Residuation Algebras
by
Willem Blok
University of Illinois at Chicago
Coauthors: J. Berman

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 Lukasiewicz's many-valued logic, and more recently, relevance logic and linear logic. In the associated classes of algebras the implication can often be viewed as the residuation operation --> associated with a suitably defined conjunction ·, giving rise to a residuated ordered groupoid <A, ·, --> , 1, <= > with unit 1. We are interested in the residuation subreducts of the resulting classes of algebras, and in particular in the free spectra of some of the locally finite examples.

We focus on the case where 1 is the top element in the order; this ensures that the order is definable in terms of the implication --> alone. We first consider the pure case, where we start from ordered sets with top element 1 in which the order and --> are interdefinable. Next we consider the more general case of residuation subreducts of ROGs with a k-potent, commutative multiplication; Hilbert algebras fall into this category, as do the --> -subreducts of Lukasiewicz's matrices for k-valued logic, and more generally, the --> -subreducts of k-potent hoops. Our main result concerns a description of the finitely generated free algebras in some of these varieties.

Date received: May 16, 2003