Local compactness in approach spaces. II.

Lowen, R.; Verbeeck, C.

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Let $S = {x_{1}, \dots, x_{n}}$ be a finite subset of a partially ordered set $P$ . Let $f$ be an incidence function of $P$ . Let $[f (x_{i} \land x_{j})]$ denote the $n \times n$ matrix having $f$ evaluated at the meet $x_{i} \land x_{j}$ of $x_{i}$ and $x_{j}$ as its $i, j$ -entry and $[f (x_{i} \lor x_{j})]$ denote the $n \times n$ matrix having $f$ evaluated at the join $x_{i} \lor x_{j}$ of $x_{i}$ and $x_{j}$ as its $i, j$ -entry. The set $S$ is said to be meet-closed if $x_{i} \land x_{j} \in S$ for all $1 \leq i, j \leq n$ . In this paper we get explicit combinatorial formulas for the determinants of matrices $[f (x_{i} \land x_{j})]$ and $[f (x_{i} \lor x_{j})]$ on any meet-closed set $S$ . We also obtain necessary and sufficient conditions for...