# Determinant and inverse of join matrices on two sets

Mathematics – Number Theory

Scientific paper

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## Details Determinant and inverse of join matrices on two sets Determinant and inverse of join matrices on two sets

Scientific paper

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.

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