×

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

MSC:

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

Citations:

Zbl 0493.08006
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chajda I.,Lattice of compatible relations, Archiv. Math. (Brno)13 (1977), 89-96. · Zbl 0372.08002
[2] Quackenbush R. W.,Varieties with n-principal Compact Congruences, Algebra Univ. 14 (1982), 292-296. · Zbl 0493.08006 · doi:10.1007/BF02483933
[3] Werner H.,A Mal’cev condition for admissible relations, Algebra Univ.3 (1973), 263. · Zbl 0276.08004 · doi:10.1007/BF02945126
[4] Werner H.,Algebraic representation and model theoretic properties of algebras with ternary discriminator, Habilitationsschrift, Technische Hochschule Darmstadt, preprint Nr. 237, 1-100.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.