# Determinants of matrices associated with incidence functions on posets

Czechoslovak Mathematical Journal (2004)

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

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

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