Determinants of matrices associated with incidence functions on posets

Shaofang Hong; Qi Sun

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 2, page 431-443
  • ISSN: 0011-4642

Abstract

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Let S = { x 1 , ⋯ , x n } be a finite subset of a partially ordered set P . Let f be an incidence function of P . Let [ f ( x i ∧ x j ) ] denote the n × n matrix having f evaluated at the meet x i ∧ x j of x i and x j as its i , j -entry and [ f ( x i ∨ x j ) ] denote the n × n matrix having f evaluated at the join x i ∨ x j of x i and x j as its i , j -entry. The set S is said to be meet-closed if x i ∧ x j ∈ S for all 1 ≤ i , j ≤ n . In this paper we get explicit combinatorial formulas for the determinants of matrices [ f ( x i ∧ x j ) ] and [ f ( x i ∨ x j ) ] on any meet-closed set S . We also obtain necessary and sufficient conditions for the matrices f ( x i ∧ x j ) ] and [ f ( x i ∨ x j ) ] on any meet-closed set S to be nonsingular. Finally, we give some number-theoretic applications.

How to cite

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Hong, Shaofang, and Sun, Qi. "Determinants of matrices associated with incidence functions on posets." Czechoslovak Mathematical Journal 54.2 (2004): 431-443. <http://eudml.org/doc/30873>.

@article{Hong2004,
abstract = {Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a finite subset of a partially ordered set $P$. Let $f$ be an incidence function of $P$. Let $[f(x_i\wedge x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the meet $x_i\wedge x_j$ of $x_i$ and $x_j$ as its $i,j$-entry and $[f(x_i\vee x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the join $x_i\vee x_j$ of $x_i$ and $x_j$ as its $i,j$-entry. The set $S$ is said to be meet-closed if $x_i\wedge x_j\in S$ for all $1\le i,j\le n$. In this paper we get explicit combinatorial formulas for the determinants of matrices $[f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$. We also obtain necessary and sufficient conditions for the matrices $f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$ to be nonsingular. Finally, we give some number-theoretic applications.},
author = {Hong, Shaofang, Sun, Qi},
journal = {Czechoslovak Mathematical Journal},
keywords = {meet-closed set; greatest-type lower; incidence function; determinant; nonsingularity; meet-closed set; greatest-type lower; incidence function; determinant; nonsingularity},
language = {eng},
number = {2},
pages = {431-443},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Determinants of matrices associated with incidence functions on posets},
url = {http://eudml.org/doc/30873},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Hong, Shaofang
AU - Sun, Qi
TI - Determinants of matrices associated with incidence functions on posets
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 2
SP - 431
EP - 443
AB - Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a finite subset of a partially ordered set $P$. Let $f$ be an incidence function of $P$. Let $[f(x_i\wedge x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the meet $x_i\wedge x_j$ of $x_i$ and $x_j$ as its $i,j$-entry and $[f(x_i\vee x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the join $x_i\vee x_j$ of $x_i$ and $x_j$ as its $i,j$-entry. The set $S$ is said to be meet-closed if $x_i\wedge x_j\in S$ for all $1\le i,j\le n$. In this paper we get explicit combinatorial formulas for the determinants of matrices $[f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$. We also obtain necessary and sufficient conditions for the matrices $f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$ to be nonsingular. Finally, we give some number-theoretic applications.
LA - eng
KW - meet-closed set; greatest-type lower; incidence function; determinant; nonsingularity; meet-closed set; greatest-type lower; incidence function; determinant; nonsingularity
UR - http://eudml.org/doc/30873
ER -

References

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  1. Combinatorial Theory, Springer-Verlag, New York, 1979. (1979) Zbl0415.05001MR0542445
  2. Greatest common divisor matrices, Linear Algebra Appl. 118 (1989), 69–76. (1989) MR0995366
  3. 10.1017/S0004972700017457, Bull. Austral. Math. Soc. 40 (1989), 413–415. (1989) MR1037636DOI10.1017/S0004972700017457
  4. 10.1080/03081089308818225, Linear and Multilinear Algebra 34 (1993), 261–267. (1993) MR1304611DOI10.1080/03081089308818225
  5. 10.1016/0024-3795(95)00349-5, Linear Algebra Appl. 249 (1996), 111–123. (1996) Zbl0870.15016MR1417412DOI10.1016/0024-3795(95)00349-5
  6. LCM matrix on an r -fold gcd-closed set, J.  Sichuan Univ., Nat. Sci. Ed. 33 (1996), 650–657. (1996) Zbl0869.11021MR1440627
  7. 10.1006/jabr.1998.7844, J.  Algebra 218 (1999), 216–228. (1999) Zbl1015.11007MR1704684DOI10.1006/jabr.1998.7844
  8. On the factorization of LCM matrices on gcd-closed sets, Linear Algebra Appl. 345 (2002), 225–233. (2002) Zbl0995.15006MR1883274
  9. 10.1215/S0012-7094-66-03308-4, Duke Math.  J. 33 (1966), 49–53. (1966) Zbl0154.29503MR0184897DOI10.1215/S0012-7094-66-03308-4
  10. On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875–1876), 208–212. (1875–1876) 

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