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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.

MSC:
08B10 Congruence modularity, congruence distributivity
08B05 Equational logic, Mal’tsev conditions
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References:
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