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B-varieties with normal free algebras

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Abstract

The starting point for the investigation in this paper is the following McKinsey-Tarski's Theorem: if f and g are algebraic functions (of the same number of variables) in a topological Boolean algebra (TBA) and if C(f)∩C(g) vanishes identically, then either f or g vanishes identically. The present paper generalizes this theorem to B-algebras and shows that validity of that theorem in a variety of B-algebras (B-variety) generated by SCI B -equations implies that its free Lindenbaum-Tarski's algebra is normal. This is important in the semantical analysis of SCI B (the Boolean strengthening of the sentential calculus with identity, SCI) since normal B-algebras are just models of this logic. The rest part of the paper is concerned with relationships between some closure systems of filters, SCI B -theories, B-varieties and closed sets of SCI B -equations that have been derived both from the semantics of SCI B and from the semantics of the usual equational logic.

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Authors and Affiliations

  1. Department of Mathematics, Pedagogical University, Siedlce, Poland

    Bronisław Tembrowski

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  1. Bronisław Tembrowski
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To the memory of Jerzy Słupecki

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Tembrowski, B. B-varieties with normal free algebras. Studia Logica 48, 555–564 (1989). https://doi.org/10.1007/BF00370207

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  • DOI: https://doi.org/10.1007/BF00370207

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