Inversion of matrices over a~pseudocomplemented lattice

E. E. Marenich ; V. G. Kumarov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 139-154

Résumé

We compute the greatest solutions of systems of linear equations over a lattice $(P,\leq)$. We also present some applications of the obtained results to lattice matrix theory. Let $(P,\leq)$ be a pseudocomplemented lattice with $\tilde0$ and $\tilde1$ and let $A=\|a_{ij}\|_{n\times n}$, where $a_{ij}\in P$ for $i,j=1,\dots,n$. Let $A^*=\|a'_{ij}\|_{n\times n}$ and $a'_{ij}=\bigwedge\limits_{\substack{r=1\\ r\ne j}}^na_{ri}^*$ for $i,j=1,\dots,n$, where $a^*$ is the pseudocomplement of $a\in P$ in $(P,\leq)$. A matrix $A$ has a right inverse over $(P,\leq)$ if and only if $A\cdot A^*=E$ over $(P,\leq)$. If $A$ has a right inverse over $(P,\leq)$, then $A^*$ is the greatest right inverse of $A$ over $(P,\leq)$. The matrix $A$ has a right inverse over $(P,\leq)$ if and only if $A$ is a column orthogonal over $(P,\leq)$. The matrix $D=A\cdot A^*$ is the greatest diagonal such that $A$ is a left divisor of $D$ over $(P,\leq)$. Invertible matrices over a distributive lattice $(P,\leq)$ form the general linear group $\mathrm{GL}_n (P,\leq)$ under multiplication. Let $(P,\leq)$ be a finite distributive lattice and let $k$ be the number of components of the covering graph $\Gamma(\operatorname{join}(P,\leq)-\{\tilde0\},\leq)$, where $\operatorname{join}(P,\leq)$ is the set of join irreducible elements of $(P,\leq)$. Then $\mathrm{GL}_n(P,\leq)\cong S_n^k$. We give some further results concerning inversion of matrices over a pseudocomplemented lattice.