Determinants of matrices associated with incidence functions on posets

被引:21
作者
Hong, SF [1 ]
Sun, Q [1 ]
机构
[1] Sichuan Univ, Coll Math, Chengdu 610064, Peoples R China
基金
中国国家自然科学基金;
关键词
meet-closed set; greatest-type lower; incidence function; determinant; nonsingularity;
D O I
10.1023/B:CMAJ.0000042382.61841.0c
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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) Lambda x(j))] denote the n x n matrix having f evaluated at the meet x(i) Lambda x(j) of x(i) and x(j) as its i, j-entry and [f (x(i) V x(j))] denote the n x n matrix having f evaluated at the join x(i) V 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) Lambda x(j) is an element of S for all 1 less than or equal to i, j less than or equal to n. In this paper we get explicit combinatorial formulas for the determinants of matrices [f (x(i) A x(j))] and [f (x(i) V x(j))] on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices f (x(i) A x(j))] and [f (x(i) V x(j))] on any meet-closed set S to be nonsingular. Finally, we give some numbertheoretic applications.
引用
收藏
页码:431 / 443
页数:13
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