DETERMINANTS OF MATRICES ASSOCIATED WITH INCIDENCE FUNCTIONS ON POSETS


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Abstract

Let S = {x1,…, xn} be a finite subset of a partially ordered set P. Let f be an incidence function of P. Let [f(xi ∧ xj)] denote the n × n matrix having f evaluated at the meet xi ∧ xj of xi and xj as its i, j-entry and [f(xi ∨ xj)] denote the n × n matrix having f evaluated at the join xi ∨ xj of xi and xj as its i, j-entry. The set S is said to be meet-closed if xi ∧ xj ∈ S for all 1 ≤ i, jn. In this paper we get explicit combinatorial formulas for the determinants of matrices [f(xi ∧ xj)] and [f(xi ∨ xj)] on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices f(xi ∧ xj)] and [f(xi ∨ xj)] on any meet-closed set S to be nonsingular. Finally, we give some number-theoretic applications.

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