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congruence distributivity; Triangular Scheme; variety of algebras; Jónsson terms
A diagrammatic scheme characterizing congruence distributivity of congruence permutable algebras was introduced by the first author in 2001. It is known under the name Triangular Scheme. It is known that every congruence distributive algebra satisfies this scheme and an algebra satisfying the Triangular Scheme which is not congruence distributive was found by E. K. Horváth, G. Czédli and the autor in 2003. On the other hand, it was an open problem if a variety of algebras satisfying the Triangular Scheme must be congruence distributive. We get a negative solution by presenting an example.
[1] Chajda I.: A note on the triangular scheme. East-West J. of Mathem. 3 (2001), 79–80. MR 1866645 | Zbl 1007.08002
[2] Chajda I., Halaš R.: On schemes for congruence distributivity. Central European J. of Mathem. 2, 3 (2004), 368–376. MR 2113537 | Zbl 1062.08002
[3] Chajda I., Horváth E. K.: A scheme for congruence semidistributivity. Discuss. Math., General Algebra and Appl. 23 (2003), 13–18. MR 2070042 | Zbl 1057.08001
[4] Chajda I., Horváth E. K.: A triangular scheme for congruence distributivity. Acta Sci. Math. (Szeged) 68 (2002), 29–35. MR 1916565 | Zbl 0997.08001
[5] Chajda I., Horváth E. K., Czédli G.: Trapezoid Lemma and congruence distributivity. Math. Slovaca 53 (2003), 247–253. MR 2025021 | Zbl 1058.08007
[6] Chajda I., Horváth E. K., Czédli G.: The Shifting Lemma and shifting lattice identities. Algebra Universalis 50 (2003), 51–60. MR 2026826 | Zbl 1091.08006
[7] Jónsson B.: Algebras whose congruence lattice are distributive. Math. Scand. 21 (1967), 110–121. MR 0237402
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