A Mal’cev condition for congruence principal permutable varieties. (English) Zbl 0552.08006

An algebra is congruence principal iff the join (in the congruence lattice) of finitely many principal congruences is principal. R. W. Quackenbush [ibid. 14, 292-296 (1982; Zbl 0493.08006)] has proved that congruence principal varieties can be characterized by a Mal’cev condition. This paper proves that a congruence permutable variety is congruence principal iff there are 5-ary polynomials r and s and a 6-ary polynomial t such that the variety satisfies \(x=r(t(x,z,x,y,z,v),x,y,z,v)\) \(y=r(t(y,v,x,y,z,v),x,y,z,v)\) \(z=s(t(x,z,x,y,z,v),x,y,z,v)\) \(v=x(t(y,v,x,y,z,v),x,y,z,v).\)
Reviewer: E.Nelson


08B05 Equational logic, Mal’tsev conditions
08A30 Subalgebras, congruence relations


Zbl 0493.08006
Full Text: DOI


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