A converse to the Kurosh-Ore theorem
The author mentions a result of Dilworth, cf. Theorem 7.7 in \textit{P. Crawley} and \textit{R. P. Dilworth's} book [Algebraic theory of lattices (1973; Zbl 0494.06001)], from which it follows that any modular DCC lattice has the \(\wedge\)-KORP and the \(\vee\)-KORP. His main result (Thm. 6) is a partial converse of this for semimodular algebraic lattices in which for every interval \([a, b]\) there exists a lower cover of \(b\). For such lattices DCC, \(\wedge\)-KORP and \(\vee\)-KORP imply modularity. A complete lattice is said to have \(\wedge\)-KORP (\(\wedge\)-Kurosh-Ore replacement property) iff any \(b\in L\) admits an irredundant \(\wedge\)-decomposition and given two such decompositions \(b= \bigwedge Q= \bigwedge R\) any \(q\in Q\) may be replaced by some \(r\in R\) to give again an irredundant \(\wedge\)-decomposition.
scientific article; zbMATH DE number 878125
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