Some properties of row-adjusted meet and join matrices

TL;DR

This paper investigates the structure, rank bounds, determinants, and inverses of row-adjusted meet and join matrices on finite subsets of lattices, providing general and special case results.

Contribution

It characterizes the structure of row-adjusted meet and join matrices and derives bounds and explicit formulas for their rank, determinant, and inverse in various cases.

Findings

Determined the structure of row-adjusted meet matrices in general.

Provided bounds for the rank of these matrices when the set is meet closed.

Derived formulas for determinants and inverses of the matrices.

Abstract

Let $(P,\preceq)$ 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 $S$.