5 research outputs found
On Noncommutative Generalisations of Boolean Algebras
Skew Boolean algebras (SBA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed SBAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use SBAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed SBAs in terms of nBAs of n-partitions
Sequent calculi of finite dimension
In recent work, the authors introduced the notion of n-dimensional Boolean
algebra and the corresponding propositional logic nCL. In this paper, we
introduce a sequent calculus for nCL and we show its soundness and
completeness.Comment: arXiv admin note: text overlap with arXiv:1806.0653
Classical logic with n truth values as a symmetric many-valued logic
We introduce Boolean-like algebras of dimension n (nBA s) 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 nBA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The nBA s 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 a 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