Kung, Joseph P. S. Matchings and Radon transforms in lattices. II: Concordant sets. (English) Zbl 0626.06008 Math. Proc. Camb. Philos. Soc. 101, 221-231 (1987). [Part I cf. Order 2, 105-112 (1985; Zbl 0582.06008).] A sufficient condition for the existence of a matching of a subset of a finite lattice into another subset of the same lattice (i.e. an injection of the first one into the second with the property that the image of any element is not smaller than the element itself) is that the first subset is ”concordant” with the second one. Using the finite Radon transform, the author proves that the incidence matrix of concordant subsets has maximum rank which implies the existence of a matching. The notion of concordance is further used to derive several rank and covering inequalities in finite lattices among which are generalizations of the Dowling-Wilson inequalities and Dilworth’s covering theorem to semimodular lattices and a refinement of Dilworth’s covering theorem for modular lattices. Reviewer: M.Pirlot Cited in 2 ReviewsCited in 5 Documents MSC: 06B05 Structure theory of lattices 06C10 Semimodular lattices, geometric lattices 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 44A15 Special integral transforms (Legendre, Hilbert, etc.) 06C05 Modular lattices, Desarguesian lattices Keywords:Möbius function; covering inequality; matching; finite lattice; finite Radon transform; incidence matrix; concordant subsets; semimodular lattices; Dilworth’s covering theorem Citations:Zbl 0582.06008 PDFBibTeX XMLCite \textit{J. P. S. Kung}, Math. Proc. Camb. Philos. Soc. 101, 221--231 (1987; Zbl 0626.06008) Full Text: DOI References: [1] DOI: 10.2307/1969639 · Zbl 0056.26203 · doi:10.2307/1969639 [2] DOI: 10.1007/BF00334848 · Zbl 0582.06008 · doi:10.1007/BF00334848 [3] Bolker, Proc. Conf. on Integral Geometry (1984) [4] Birkhoff, Lattice Theory 25 (1967) [5] Aigner, Combinatorial Theory (1979) · doi:10.1007/978-1-4615-6666-3 [6] DOI: 10.1007/BF00531932 · Zbl 0121.02406 · doi:10.1007/BF00531932 [7] DOI: 10.1007/BF02485838 · Zbl 0423.06008 · doi:10.1007/BF02485838 [8] DOI: 10.1007/BF00383603 · Zbl 0552.06003 · doi:10.1007/BF00383603 [9] Kurinnoi, Vestnik Char’kov Univ. 93 pp 11– (1973) [10] Kung, Combinatorics and Ordered Sets 57 pp 33– (1986) · doi:10.1090/conm/057/856232 [11] DOI: 10.1016/0097-3165(79)90059-1 · Zbl 0406.05023 · doi:10.1016/0097-3165(79)90059-1 [12] Ganter, Algebra Universalis 3 pp 348– (1973) [13] DOI: 10.2307/2039773 · Zbl 0297.05010 · doi:10.2307/2039773 [14] Doubilet, Stud. Appl. Math. 51 pp 377– (1972) · Zbl 0274.05008 · doi:10.1002/sapm1972514377 [15] DOI: 10.1007/BF01448979 · JFM 31.0211.01 · doi:10.1007/BF01448979 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.