Unorthodox Algebras and their associated Unorthodox Logics
Hanamantagouda P. Sankappanavar

TL;DR
This paper introduces five novel unorthodox algebras and their associated logic, demonstrating their algebraic properties, logical applications, and the structure of their subvarieties, including decidability and axiomatizations.
Contribution
It defines a new subvariety RUNO1 generated by five unorthodox algebras, explores its properties, and develops a corresponding algebraizable logic with complete axiomatizations.
Findings
RUNO1 is a discriminator variety and primal.
The lattice of subvarieties of RUNO1 has 32 elements and is Boolean.
All axiomatic extensions of the logic are decidable.
Abstract
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…
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Taxonomy
TopicsAdvanced Algebra and Logic · Logic, Reasoning, and Knowledge · Logic, programming, and type systems
