Stern, M. A converse to the Kurosh-Ore theorem. (English) Zbl 0856.06007 Acta Math. Hung. 70, No. 3, 177-184 (1996). The author mentions a result of Dilworth, cf. Theorem 7.7 in P. Crawley and 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. Reviewer: H.Szambien (Garbsen) Cited in 1 ReviewCited in 1 Document MSC: 06C10 Semimodular lattices, geometric lattices 06C20 Complemented modular lattices, continuous geometries 06B15 Representation theory of lattices Keywords:semimodular lattice; Kurosh-Ore replacement property; algebraic lattices; modularity Citations:Zbl 0494.06001 PDFBibTeX XMLCite \textit{M. Stern}, Acta Math. Hung. 70, No. 3, 177--184 (1996; Zbl 0856.06007) Full Text: DOI References: [1] G. Birkhoff,Lattice Theory (3rd edition) (Providence R. I., 1967). · Zbl 0153.02501 [2] K. P. Bogart, R. Freese and J. P. S. Kung (Editors),The Dilworth Theorems, Birkhäuser (Boston, 1990). [3] P. Crawley, The isomorphism theorem in compactly generated lattices,Bull. Amer. Math. Soc.,65 (1959), 377–379. · Zbl 0094.01702 · doi:10.1090/S0002-9904-1959-10384-0 [4] P. Crawley, Decomposition theory for nonsemimodular lattices,Trans. Amer. Math. Soc.,99 (1961), 246–254. · Zbl 0098.02601 · doi:10.1090/S0002-9947-1961-0120173-8 [5] P. Crawley and R. P. Dilworth,Algebraic Theory of Lattices, Prentice Hall (Englewood Cliffs, New Jersey, 1973). · Zbl 0494.06001 [6] R. P. Dilworth, Lattices with unique irreducible decompositions,Ann. of Math.,41 (1940), 771–777. · Zbl 0025.10202 · doi:10.2307/1968857 [7] R. P. Dilworth, The arithmetic theory of Birkhoff lattices,Duke Math. J.,8 (1941), 286–299. · Zbl 0025.10203 · doi:10.1215/S0012-7094-41-00822-0 [8] R. P. Dilworth,Background to Chapter 3 of [2]. [9] U. Faigle, Geometries on partially ordered sets,J. Combin. Theory, Ser B8 (1980), 26–51. · Zbl 0359.05018 [10] G. Grätzer,General Lattice Theory, Birkhäuser Verlag (Basel, 1978). · Zbl 0436.06001 [11] B. Jónsson,Dilworth’s work on decomposition in semimodular lattices, in [2], pp. 187–191. [12] J. P. S. Kung, Matchings and Radon transforms in lattices. I. Consistent lattices,Order,2 (1985), 105–112. · Zbl 0582.06008 · doi:10.1007/BF00334848 [13] A. G. Kurosh, Durchschnittsdarstellungen mit irreduziblen Komponenten in Ringen und in sogenannten Dualgruppen,Math. Sb.,42 (1935), 613–616. · Zbl 0013.19504 [14] O. Ore, On the foundations of abstract algebra II,Ann. of Math.,37 (1936), 265–295. · JFM 62.1099.08 · doi:10.2307/1968442 [15] K. Reuter, The Kurosh-Ore exchange property,Acta Math. Hungar.,53 (1989), 119–127. · Zbl 0675.06003 · doi:10.1007/BF02170062 [16] M. Stern,Semimodular Lattices, Teubner Verlag (Stuttgart/Leipzig, 1991). [17] M. Stern, On the Kurosh-Ore replacement property in semimodular lattices,Algebra Universalis,30 (1993), 352–353. · Zbl 0808.06011 · doi:10.1007/BF01190445 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.