## On meet matrices on posets.(English)Zbl 0870.15016

The author considers the meet matrices on posets as a generalization of the greatest common divisor (GCD) matrices. Some of the most important properties of GCD matrices are expressed in the language of meet matrices. The author proves a structure theorem for meet matrices, derives explicit expressions and bounds for the determinants of meet matrices, and finally considers the inverse of meet matrices.

### MSC:

 15B57 Hermitian, skew-Hermitian, and related matrices 11C20 Matrices, determinants in number theory 15A15 Determinants, permanents, traces, other special matrix functions 15A09 Theory of matrix inversion and generalized inverses
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### References:

 [1] Beslin, S.; Ligh, S., Greatest common divisor matrices, Linear Algebra Appl., 118, 69-76 (1989) · Zbl 0672.15005 [2] Beslin, S.; Ligh, S., Another generalisation of Smith’s determinant, Bull. Austral. Math. Soc., 40, 413-415 (1989) · Zbl 0675.10002 [3] Beslin, S.; Ligh, S., GCD-closed sets and the determinants of GCD matrices, Fibonacci Quart, 30, 157-160 (1992) · Zbl 0752.11012 [4] Bourque, K.; Ligh, S., On GCD and LCM matrices, Linear Algebra Appl., 174, 65-74 (1992) · Zbl 0761.15013 [5] Bourque, K.; Ligh, S., Matrices associated with classes of arithmetical functions, J. Number Theory, 45, 367-376 (1993) · Zbl 0784.11002 [6] Bourque, K.; Ligh, S., Matrices associated with arithmetical functions, Linear and Multilinear Algebra, 34, 261-267 (1993) · Zbl 0815.15022 [7] Gantmacher, F. R., (Matrix Theory, Vol. I (1977), Chelsea: Chelsea New York) [8] Haukkanen, P., Classical arithmetical identities involving a generalization of Ramanujan’s sum, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 68, 1-69 (1988) · Zbl 0651.10005 [10] Li, Z., The determinants of GCD matrices, Linear Algebra Appl., 134, 137-143 (1990) · Zbl 0703.15012 [11] Lindström, B., Determinants on semilattices, (Proc. Amer. Math. Soc., 20 (1969)), 207-208 · Zbl 0165.02902 [12] Mirsky, L., An Introduction to Linear Algebra (1972), Oxford U.P: Oxford U.P London · Zbl 0766.15001 [13] Pólya, G.; Szegö, G., (Aufgaben und Lehrsätze aus der Analysis, Vol. II (1971), Springer-Verlag: Springer-Verlag New York) · Zbl 0219.00003 [14] Rajarama Bhat, B. V., On greatest common divisor matrices and their applications, Linear Algebra Appl., 158, 77-97 (1991) · Zbl 0754.15012 [15] Smith, D. A., Bivariate function algebras on posets, J. Reine Angew. Math., 251, 100-109 (1971) · Zbl 0224.06002 [16] Smith, H. J.S., On the value of a certain arithmetical determinant, (Proc. London Math. Soc., 7 (1875-1876)), 208-212 · JFM 08.0074.03 [17] Stanley, R. P., (Enumerative Combinatorics, Vol. I (1986), Wadsworth and Brooks/Cole: Wadsworth and Brooks/Cole Monterey, Calif) · Zbl 0608.05001 [18] Wilf, H. S., Hadamard determinants, Möbius functions, and the chromatic number of a graph, Bull. Amer. Math. Soc., 74, 960-964 (1968) · Zbl 0172.01602
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