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A scheme for congruence semidistributivity. (English) Zbl 1057.08001

A lattice \(L\) is \(\wedge\)-semidistributive if it satisfies the following implication for all \(x,y,z\in L:\) \[ SD_{\wedge }:x\wedge y=x\wedge z\Rightarrow x\wedge (y\vee z)=x\wedge y. \] More generally, for \(n\geq 2,\) \(M\subseteq \{0,1,\dots,n-1\}\) such that \(\left| M\right| \geq 2\), let the generalized meet semidistributive law \(SD_{\wedge }(n,M)\) for lattices be defined as follows: \[ \begin{split} SD_{\wedge}(n,M) : \text{ for all }x,y_{0},y_{1},\dots,y_{n-1},\\ x\wedge y_{0}=x\wedge y_{1}=\dots=x\wedge y_{n-1} \Rightarrow x\wedge y_{0}=x\wedge \Bigl(\bigwedge_{I\subseteq M} \bigvee_{i\in I}y_{i}\Bigr). \end{split} \] In this paper a diagrammatic statement is developed for the generalized semidistributive law in case of single algebras assuming that their congruences are permutable. Without permutable congruences, a diagrammatic statement is developed for the \(\wedge\)-semidistributive law.

MSC:

08A30 Subalgebras, congruence relations
08B10 Congruence modularity, congruence distributivity
06D99 Distributive lattices
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