zbMATH — the first resource for mathematics

Trapezoid lemma and congruence distributivity. (English) Zbl 1058.08007
Let \({\mathcal V}\) be a variety and \({\mathcal A}=(A,F)\in {\mathcal V}\). \({\mathcal A}\) is said to satisfy the Rectangular Lemma if \(\alpha ,\beta ,\gamma \in {\operatorname {Con}}\, A\), \(\alpha \cap \beta \subseteq \gamma \), \((x,u),(y,v)\in \alpha \), \((x,y),(u,v)\in \beta \) and \((u,v)\in \gamma \) together imply \((x,y)\in \gamma \). If \(\alpha \) is only a tolerance on \({\mathcal A}\) then this condition is called the Rectangular Principle. Analogously, the Triangular Lemma (Principle) and the Trapezoid Lemma (Principle) are defined. All these conditions can be visualized by simple figures. The main result of the paper characterizes the congruence distributivity of \({\mathcal V}\) by means of the above conditions. Many years ago, H. P. Gumm achieved similar results on congruence modularity. Some parts of the results of the paper under review were announced by J. Duda in a similar but more complicated formulation.

08B10 Congruence modularity, congruence distributivity
08B05 Equational logic, Mal’tsev conditions
Full Text: EuDML
[1] CHAJDA L.-CZÉDLI G.- HORVÁTH E. K.: Shifting lemma and shifting lattice identities. Algebra Universalis · Zbl 1091.08006
[2] CHAJDA I.-GLAZEK K.: A Basic Course on Algebra. Technical University Press, Zielona Góra, Poland, 2000.
[3] CHAJDA I.-HORVÁTH E. K.: A triangular scheme for congruence distributivity. Acta Sci. Math. (Szeged) 68 (2002), 29-35. · Zbl 0997.08001
[4] CZÉDLI G.-HORVÁTH E. K.: Congruence distributivity and modularity permit tolerances. Acta Univ. Palack. Olomouc. Fac. Rerum Natur. Math. 41 (2002), 39-42. · Zbl 1043.08002
[5] CZÉDLI G.-HORVÁTH E. K.: All congruence lattice identities implying modularity have Mal’tsev conditions. Algebra Universalis · Zbl 1091.08007
[6] DAY A.: A characterization of modularity for congruence lattices of algebras. Canad. Math. Bull. 12 (1969), 167-173. · Zbl 0181.02302
[7] DUDA J.: The Upright Principle for congruence distributive varieties. Abstract of a seminar lecture presented in Brno, March, 2000.
[8] DUDA J.: The Triangular Principle for congruence distributive varieties. Abstract of a seminar lecture presented in Brno, March, 2000.
[9] FRASER G. A.-HORN A.: Congruence relations in direct products. Proc Amer. Math. Soc 26 (1970), 390-394. · Zbl 0241.08004
[10] FREESE R.-McKENZIE R.: Commutator Theory for Congruence Modular Varieties. Cambridge Univ. Press, Cambridge, 1987. · Zbl 0636.08001
[11] GUMM H. P.: Geometrical methods in congruence modular algebras. Mem. Amer. Math. Soc 45 no. 286 (1983), viii+79. · Zbl 0547.08006
[12] GUMM H. P.: Congruence modularity is permutability composed with distributivity. Arch. Math. (Basel) 36 (1981), 569-576. · Zbl 0465.08005
[13] JONSSON B.: Algebras whose congruence lattices are distributive. Math. Scand. 21 (1967), 110-121. · Zbl 0167.28401
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.