Relative principal congruences in congruence-modular quasivarieties
For a quasivariety \(Q\) of type \(\tau \) and an algebra \(A\) of the same type, \(\text{Con}_QA\) is defined to be the lattice of all congruences \(\theta \) on \(A\) such that \(A/\theta \in Q\). The members of \(\text{Con}_QA\) are called relative congruences. The concept of relative congruence modularity for quasivarieties is introduced in the paper (it is equivalent to the condition that the lattice \(\text{Con}_QF(\omega)\) is modular, where \(F(\omega)\) is the free algebra of \(Q\) with countable many generators). In any quasivariety relative principal congruences are defined by means of a possibly infinite disjunction of existentially parametrized conjunctions of finite sets of equations; this property is denoted by LEDPC with respect to the family \(P\) of finite set of equations. A quasivariety \(Q\) has equationally definable relative principal congruences iff \(Q\) has LEDPC with respect to a one-element family. The purpose of the paper is to study the relationship between the LEDPC and relative congruence modularity.
scientific article; zbMATH DE number 1407091