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

top
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

top

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

top
  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) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.