Abstract.
For varieties of algebras, we present the property of having "definable principal subcongruences" (DPSC), generalizing the concept of having definable principal congruences. It is shown that if a locally finite variety V of finite type has DPSC, then V has a finite equational basis if and only if its class of subdirectly irreducible members is finitely axiomatizable. As an application, we prove that if A is a finite algebra of finite type whose variety V(A) is congruence distributive, then V(A) has DPSC. Thus we obtain a new proof of the finite basis theorem for such varieties. In contrast, it is shown that the group variety V(S 3 ) does not have DPSC.
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Received May 9 2000; accepted in final form April 26, 2001.
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Basker, K., Wang, J. Definable principal subcongruences . Algebra univers. 47, 145–151 (2002). https://doi.org/10.1007/s00012-002-8180-5
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DOI: https://doi.org/10.1007/s00012-002-8180-5