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Chajda, Ivan; Rachůnek, Jiří

Relational characterizations of permutable and n-permutable varieties. (English) Zbl 0545.08011

Czech. Math. J. 33(108), 505-508 (1983).
The paper rediscovers the two Mal’tsev-type theorems that a variety is permutable iff each reflexive algebraic relation is a congruence and that a variety is n-permutable for some \(n\geq 2\) iff every reflexive, transitive, algebraic relation is a congruence. Only the trivial variety has every symmetric algebraic relation being transitive.
Reviewer: H.Werner

Cited in 4 Documents

MSC:

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

Keywords:

Mal’tsev-type theorems; congruence; symmetric algebraic relation
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\textit{I. Chajda} and \textit{J. Rachůnek}, Czech. Math. J. 33(108), 505--508 (1983; Zbl 0545.08011)
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References:

[1] J. Hageman A. Mitschke: On \(n\)-permutable congruences. Algebra Univ. 3 (1973), 8-12. · Zbl 0273.08001
[2] G. Grätzer: Universal Algebra. N. Y. 1968. · Zbl 0182.34201
[3] A. I. Mal’cev: On the general theory of algebraic systems. (Russian), Math. Sb. 35 (1954), 3-20.
[4] J. Rachůnek: Quasi-orders of algebras. Časop. pěst. mat. 104 (1979), 327-337.
[5] H. Werner: A Mal’cev condition for admissible relations. Algebra Univ. 3 (1973), 263. · Zbl 0276.08004
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