A lattice-theoretic approach to the Bourque–Ligh conjecture

References

• Rajarama Bhat BV. On greatest common divisor matrices and their applications. Linear Algebra Appl. 1991;158:77–97. doi: 10.1016/0024-3795(91)90051-W
   Web of Science ®Google Scholar
• Haukkanen P. On meet matrices on posets. Linear Algebra Appl. 1996;249:111–123. doi: 10.1016/0024-3795(95)00349-5
   Web of Science ®Google Scholar
• Korkee I, Haukkanen P. On meet and join matrices associated with incidence functions. Linear Algebra Appl. 2003;372:127–153. doi: 10.1016/S0024-3795(03)00497-X
   Web of Science ®Google Scholar
• Korkee I, Haukkanen P. Bounds for determinants of meet matrices associated with incidence functions. Linear Algebra Appl. 2001;329:77–88. doi: 10.1016/S0024-3795(01)00238-5
   Web of Science ®Google Scholar
• Altinisik E, Sagan BE, Tuglu N. GCD matrices, posets, and nonintersecting paths. Linear Multilinear Algebra. 2005;53:75–84. doi: 10.1080/03081080500054612
   Web of Science ®Google Scholar
• Hong S, Sun Q. Determinants of matrices associated with incidence functions on posets. Czechoslovak Math J. 2004;54(129):431–443. doi: 10.1023/B:CMAJ.0000042382.61841.0c
   Web of Science ®Google Scholar
• Korkee I, Haukkanen P. On meet matrices with respect to reduced, extended and exchanged sets. JP J Algebra Number Theory Appl. 2004;4:559–575.
   Google Scholar
• Smith HJS. On the value of a certain arithmetical determinant. Proc London Math Soc. 1875/76;7:208–212.
   Google Scholar
• Haukkanen P, Ilmonen P, Nalli A, et al. On unitary analogs of GCD reciprocal LCM matrices. Linear Multilinear Algebra. 2010;58:599–616. doi: 10.1080/03081080902816576
   Web of Science ®Google Scholar
• Haukkanen P, Sillanpää J. Some analogues of Smith's determinant. Linear Multilinear Algebra. 1996;41:233–244. doi: 10.1080/03081089608818478
   Google Scholar
• Korkee I. On meet and join matrices associated with incidence functions [PhD thesis]. Acta Universitatis Tamperensis 1149. Tampere: Tampere University Press; 2006.
   Google Scholar
• Haukkanen P, Wang J, Sillanpää J. On Smith's determinant. Linear Algebra Appl. 1997;258:251–269. doi: 10.1016/S0024-3795(96)00192-9
   Web of Science ®Google Scholar
• Sándor J, Crstici B. Handbook of number theory II. Dordrecht: Kluwer; 2004.
   Google Scholar
• Bourque K, Ligh S. On GCD and LCM matrices. Linear Algebra Appl. 1992;174:65–74. doi: 10.1016/0024-3795(92)90042-9
   Web of Science ®Google Scholar
• Hong S. On the Bourque-Ligh conjecture of least common multiple matrices. J Algebra. 1999;218:216–228. doi: 10.1006/jabr.1998.7844
   Web of Science ®Google Scholar
• Hong S. Nonsingularity of matrices associated with classes of arithmetical functions. J Algebra. 2004;281:1–14. doi: 10.1016/j.jalgebra.2004.07.026
   Web of Science ®Google Scholar
• Hong S. Nonsingularity of least common multiple matrices on gcd-closed sets. J Number Theory. 2005;113:1–9. doi: 10.1016/j.jnt.2005.03.004
   Web of Science ®Google Scholar
• Li M. Notes on Hong's conjectures of real number power LCM matrices. J Algebra. 2007;315:654–664. doi: 10.1016/j.jalgebra.2007.05.005
   Web of Science ®Google Scholar
• Hong S, Loewy R. Asymptotic behavior of eigenvalues of greatest common divisor matrices. Glasg Math J. 2004;46:551–569. doi: 10.1017/S0017089504001995
   Web of Science ®Google Scholar
• Hong S, Loewy R. Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r). Int J Number Theory. 2011;7:1681–1704. doi: 10.1142/S179304211100437X
   Web of Science ®Google Scholar
• Ilmonen P. On meet hypermatrices and their eigenvalues. Linear Multilinear Algebra. 2016;64(5):842–855. doi: 10.1080/03081087.2015.1059797
   Web of Science ®Google Scholar
• Mattila M, Haukkanen P, Mäntysalo J. Studying the singularity of LCM-type matrices via semilattice structures and their Möbius functions. J Combin Theory Ser A. 2015;135:181–200. doi: 10.1016/j.jcta.2015.05.002
   Web of Science ®Google Scholar
• Ovall JS. An analysis of GCD and LCM matrices via the LDLT-factorization. Electron J Linear Algebra. 2004;11:51–58. doi: 10.13001/1081-3810.1121
   Web of Science ®Google Scholar
• Birkhoff G. Lattice theory. Vol. 25. Rhode Island: American Mathematical Society Colloquium Publications; 1984.
   Google Scholar
• Rearick D. Semi-multiplicative functions. Duke Math J. 1966;33:49–53. doi: 10.1215/S0012-7094-66-03308-4
   Web of Science ®Google Scholar
• Sivaramakrishnan R. Classical theory of arithmetic functions. New York:Marcel Dekker, Inc.; 1989. (Monographs and Textbooks in Pure and Applied Mathematics; 126).
   Google Scholar
• Altinisik E, Tuglu N, Haukkanen P. Determinant and inverse of meet and join matrices. Int J Math Math Sci. 2007. Article ID 37580.
   Google Scholar
• Mattila M, Haukkanen P. Determinant and inverse of join matrices on two sets. Linear Algebra Appl. 2013;438:3891–3904. doi: 10.1016/j.laa.2011.12.014
   Web of Science ®Google Scholar
• Aigner M. Combinatorial theory. Berlin: Springer–Verlag; 1979.
   Google Scholar
• Stanley RP. Enumerative combinatorics. Vol. 1. Cambridge: Cambridge University Press; 1997. (Cambridge Studies in Advanced Mathematics; 49). (Corrected reprint of the 1986 original.)
   Google Scholar
• Mattila M, Haukkanen P. On the positive definiteness and eigenvalues of meet and join matrices. Discrete Math. 2014;326:9–19. doi: 10.1016/j.disc.2014.02.018
   Web of Science ®Google Scholar
• Heitzig J, Reinhold J. Counting finite lattices. Algebra Universalis. 2002;48:43–53. doi: 10.1007/PL00013837
   Web of Science ®Google Scholar