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-kinase interacting protein 1 (AKIP1) has been shown to interact
with a broad range of proteins involved in various cellular
processes, including apoptosis, tumorigenesis, and oxidative
stress suggesting it might have multiple cellular functions. In this
study, we used an epitope-tagged AKIP1 and by combination of
immunochemical approaches, microscopic methods and reporter
assays we studied its properties. Here, we show that various
levels of AKIP1 overexpression in HEK-293 cells affected not only
its subcellular localization but also resulted in aggregation. While
highly expressed AKIP1 accumulated in electron-dense
aggregates both in the nucleus and cytosol, low expression of
AKIP1 resulted in its localization within the nucleus as a free,
non-aggregated protein. Even though AKIP1 was shown to
interact with p65 subunit of NF-κB and activate this transcription
factor, we did not observe any effect on NF-κB activation
regardless of various AKIP1 expression level.