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.
A set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ of $n$ distinct positive integers is said to be gcd-closed if $(x_{i},x_{j})\in \mathcal{S}$ for all $1\le i,j\le n $. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k(t)$ depending only on $t$, such that if $n\le k(t)$, then the power LCM matrix $([x_i,x_j]^t)$ defined on any gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ is nonsingular, but for $n\ge k(t)+1$, there exists a gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ such that the power LCM matrix $([x_i,x_j]^t)$ on $\mathcal{S}$ is singular. In 1996, Hong proved $k(1)=7$ and noted $k(t)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that $k(t)\ge 8$ for all $t\ge 2$. We further prove that $k(t)\ge 9$ iff a special Diophantine equation, which we call the LCM equation, has no $t$-th power solution and conjecture that $k(t)=8$ for all $t\ge 2$, namely, the LCM equation has $t$-th power solution for all $t\ge 2$.