Definable principal subcongruences. (Q1771906)

A variety \(\mathcal V\) has definable principal congruences if there is a finite list of congruence schemes sufficient to compute any principal congruence. \(\mathcal V\) has definable principal subcongruences if there is a finite list of congruence schemes sufficient to reach a principal congruence that can be fully computed by a predetermined finite list of congruence schemes. It is shown that if a locally finite variety \(\mathcal V\) of finite type has definable principal subcongruences then \(\mathcal V\) has a finite equational base if and only if the class of subdirectly irreducible members is finitely axiomatizable. Hence, a finite algebra \(\mathcal A\) of finite type whose variety \(\mathcal V(\mathcal A)\) is congruence distributive has definable principal subcongruences. On the other hand, there exist varieties of groups which do not have definable principal subcongruences.