Quasi-implication algebras, Part II: Structure theory

Abstract

The chief purpose of this paper is to show that the theory of quasi-implication algebras (QIA's) provides a complete characterization of the quasi-implication operation, as defined on orthomodular lattices (OML's) (on any OML,a→b=a ^⊥⋁(a⋀b)). This is accomplished by showing that every QIA can be embedded into an OML preserving quasi-implication. Besides yielding completeness, the embedding theorem provides the canonical class of QIA's, as follows. First of all, since QIA's are equationally defined, the class of QIA's is closed under the formation of subalgebras and homomorphic images. An orthomodular QIA is defined to be a QIA which is a quasi-implicational subalgebra of an OML. Together with the fact that QIA's are closed under the formation of homomorphic images, the embedding theorem entails that the class of orthomodular QIA's is canonical; specifically, every QIA is isomorphic to an orthomodular QIA.