Simple Logics for Basic Algebras

Ja̅nis Cı̅rulis

doi:10.18778/0138-0680.44.3.4.01

Authors

Ja̅nis Cı̅rulis Institute of Mathematics and Computer Science, University of Latvia, Raina b., 29, Riga LV-1459, Latvia

DOI:

https://doi.org/10.18778/0138-0680.44.3.4.01

Abstract

An MV-algebra is an algebra (A, ⊕, ¬, 0), where (A, ⊕, 0) is a commutative monoid and ¬ is an idempotent operation on A satisfying also some additional axioms. Basic algebras are similar algebras that can roughly be characterised as nonassociative (hence, also non-commutative) generalizations of MV-algebras. Basic algebras and commutative basic algebras provide an equivalent algebraic semantics in the sense of Blok and Pigozzi for two recent logical systems. Both are Hilbert-style systems, with implication and negation as the primitive connectives. We present a considerably simpler logic, Lʙ, for basic algebras, where implication and falsum are taken as primitives. We also consider some subvarieties of basic algebras known in the literature, discuss classes of implicational algebras term-equivalent to each of these varieties, and construct axiomatic extensions of Lʙ for which these classes serve as equivalent algebraic semantics.

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