Relative congruence formulas and decompositions in quasivarieties

摘要

AbstractQuasivarietal analogues of uniform congruence schemes are discussed, and their relationship with the equational definability of principalrelativecongruences (EDPRC) is established, along with their significance for relative congruences on subalgebras of products. Generalizing the situation in varieties, we prove that a quasivariety is relatively ideal iff it has EDPRC; it is relatively filtral iff it is relatively semisimple with EDPRC. As an application, it is shown that a finitary sentential logic, algebraized by a quasivarietyK, has a classical inconsistency lemma if and only ifKis relatively filtral and the subalgebras of its nontrivial members are nontrivial. A concrete instance of this result is exhibited, in whichKis not a variety. Finally, for quasivarieties$${sf{M} subseteq sf{K}}$$M?K, we supply some conditions under whichMis the restriction toKof a variety, assuming thatKhas EDPRC.]]>