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Infinite \(\theta\)-decompositions in upper continuous lattices. (English) Zbl 0719.06003

A complete lattice L is called upper continuous iff, for every \(a\in L\) and for every chain \(C\subseteq L\), \(a\wedge \bigvee C=\bigvee (a\wedge c:\;c\in C).\) Let \(\theta\) be a congruence relation in L. A representation of an element as a \(\theta\)-join of elements of the lattice L is said to be a \(\theta\)-decomposition of the element.
The author studies infinite \(\theta\)-decompositions of the unit elements of an upper continuous lattice. The main results are described for modular lattices and for the case that there are two decompositions.
Reviewer: B.F.Smarda (Brno)

MSC:

06B35 Continuous lattices and posets, applications
06C05 Modular lattices, Desarguesian lattices
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