Some properties of row-adjusted meet and join matrices

Abstract

Let $(P,⪯)$ be a lattice, $S$ a finite subset of $P$ and $f_1,f_2,...,f_n$ complex-valued functions on $P$. We define row-adjusted meet and join matrices on $S$ by $(S{)}_{f_1,...,f_n}=(f_i(x_i\wedge x_j))$ and $[S{]}_{f_1,...,f_n}=(f_i(x_i\vee x_j))$. In this paper we determine the structure of the matrix $(S{)}_{f_1,...,f_n}$ in general case and in the case when the set $S$ is meet closed we give bounds for $\text{rank}(S{)}_{f_1,...,f_n}$ and present expressions for $\det (S{)}_{f_1,...,f_n}$ and $(S{)}_{f_1,...,f_n}^{-1}$. The same is carried out dually for row-adjusted join matrix of a join closed set $S$.