Implication Algebra
by
Ivan Chajda
Palacky University Olomouc

By an implication algebra is called a groupoid satisfying the following identities:

contraction, quasi-commutativity and exchange. It was showed by J.C.Abbott in 1967 that this binary operation can be interpreted as a connective implication in the classical propositional logic (based on a Boolean algebra). Moreover, an implication algebra induces a join-semilattice with 1 where every interval [p,1] is a Boolean algebra.

Also conversely, having such a semilattice, it induces an implication algebra and this correspondence is one-to-one.

We study non-classical logics (based on orthomodular lattices or ortholattices or lattices with involutions) which rises e.g. in the logic of quantum mechanics and we assign to every of them an "implication algebra". To these algebras can be assigned a join-semilattice with 1 where every interval [p,1]is a lattice with some additional property (orthomodular or an ortholattice or a lattice with antitone involutions).

We show also conversely that each of these semilattices induces a corresponding "implication algebra". We will unify our approach to get a common theory of implication algebras.

Date received: January 16, 2003

Copyright © 2003 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 # cake-16.