Unorthodox Algebras and their associated Unorthodox Logics

This paper grew out of our investigation into a simple, but natural, question: Can 'F implies T' be distinct from F and T? To this end, we introduce five 'unorthodox' algebras that will play a major role, not only in providing a positive answer to the question, but also in their similarity to the 2-element Boolean algebra 2. Yet, they are remarkably dissimilar from 2 in many respects. In this paper, we will examine these five algebras both algebraically and logically. We define, and initiate an investigation into, a subvariety, called RUNO1, of the variety of De Morgan semi-Heyting algebras and show that RUNO1 is, in fact, the variety generated by the five algebras. Then several applications of this theorem are given. It is shown that RUNO1 is a discriminator variety and that all five algebras are primal. It is also shown that every subvariety of RUNO1 satisfies the Strong Amalgamation Property and the property that epimorphisms are surjective (ES). It is shown that the lattice of subvarieties of RUNO1 is a Boolean lattice of 32 elements. The bases for all the subvarieties of RUNO1 are also given. We introduce a new logic called mathcal{RUNO1} and show that it is algebraizable with the variety RUNO1 as its equivalent algebraic semantics. We then present axiomatizations for all 32 axiomatic extensions of mathcal{RUNO1} and deduce that all the axiomatic extensions are decidable. The paper ends with some open problems.