Walendziak, Andrzej Infinite \(\theta\)-decompositions in upper continuous lattices. (English) Zbl 0719.06003 Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 29, No. 2, 313-324 (1990). 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) Cited in 2 Documents MSC: 06B35 Continuous lattices and posets, applications 06C05 Modular lattices, Desarguesian lattices Keywords:congruence relation; \(\theta \) -decomposition; upper continuous lattice; modular lattices PDFBibTeX XMLCite \textit{A. Walendziak}, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 29, No. 2, 313--324 (1990; Zbl 0719.06003)