B-varieties with normal free algebras

✍ Scribed by Bronisław Tembrowski

• Publisher: Springer Netherlands
• Year: 1989
• Tongue: English
• Weight: 576 KB
• Volume: 48
• Category: Article
• ISSN: 0039-3215
• DOI: 10.1007/bf00370207

✦ Synopsis

The starting point for the investigation in this paper is the following 1VlcI(insey-Tarski's Theorem: if f and 9 are algebraic functions (of the same number of variables) in a topological Boolean algebra (TBA) and if C(f)n C(O) vanishes identically, then either f or 9 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 SCIB-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, SCIB-theories, B-varieties and closed sets of SCIn-equations that have been derived both from the semantics of SCI B and from the semantics of the usual equational logic.