Determinant and inverse of join matrices on two sets

TL;DR

This paper derives formulas for the determinant and inverse of join matrices on two sets within a lattice, extending previous results to cases where the function is not semimultiplicative, and applies these to various special matrices.

Contribution

It provides new and generalized formulas for the inverse and determinant of join matrices, especially when the function is not semimultiplicative, covering broader cases.

Findings

Derived formulas for determinants and inverses of join matrices.

Extended existing formulas to non-semimultiplicative functions.

Applied formulas to LCM, GCD, MAX, and MIN matrices.

Abstract

Let $(P,\preceq)$ be a lattice and $f$ a complex-valued function on $P$. We define meet and join matrices on two arbitrary subsets $X$ and $Y$ of $P$ by $(X,Y)_f=(f(x_i\wedge y_j))$ and $[X,Y]_f=(f(x_i\vee x_j))$ respectively. Here we present expressions for the determinant and the inverse of $[X,Y]_f$. Our main goal is to cover the case when $f$ is not semimultiplicative since the formulas presented earlier for $[X,Y]_f$ cannot be applied in this situation. In cases when $f$ is semimultiplicative we obtain several new and known formulas for the determinant and inverse of $(X,Y)_f$ and the usual meet and join matrices $(S)_f$ and $[S]_f$. We also apply these formulas to LCM, MAX, GCD and MIN matrices, which are special cases of join and meet matrices.