B-varieties with normal free algebras

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.To the memory of Jerzy Supecki