Abstract
Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated. For any integer \({m \geqslant 3}\) and a quasivariety Q, the notion of an m-triangularily meet-irreducible Q-congruence in the algebras of Q is defined. In Section 2, some characterizations of finitely generated quasivarieties involving this notion are provided. Section 3 deals with quasivarieties with equationally definable m-triangular meets of relatively principal congruences. References to finitely based quasivarieties and varieties are discussed.
Highlights
Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated
Any class of algebras defined by a set of quasi-identities is called a quasivariety
If Q is a quasivariety, any set Γ of quasi-identities defining Q is called a base for Q; we write Q = Mod(Γ )
Summary
Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated. It follows from Theorem 3.3(2) that every RCD quasivariety Q has m-EDTPM for all m 3, Q need not be finitely generated. The following conditions are equivalent: (1) Q is generated by a finite class of algebras each of which has at most m − 1 elements.
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