Boolean-like algebras of finite dimension
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We introduce Boolean-like algebras of dimension n (nBA) having n constants e1,...,en, and an (n+1)-ary operation q (a "generalised if-then-else") that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBAs share many remarkable properties with the variety of Boolean algebras and with primal varieties. Exploiting the concept of central element, we extend the notion of Boolean power to that of semiring power and we prove two representation theorems: (i) Any pure nBA is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator. This gives a new proof of Foster's theorem on primal varieties. The nBAs provide the algebraic framework for generalising the classical propositional calculus to the case of n - perfectly symmetric - truth-values. Every finite-valued tabular logic can be embedded into such an n-valued propositional logic, nCL, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in nCL, and, via the embeddings, in all the finite tabular logics.
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