(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 3, PAGES 139-154

Inversion of matrices over a pseudocomplemented lattice

E. E. Marenich
V. G. Kumarov

Abstract

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We compute the greatest solutions of systems of linear equations over a lattice $\left(P,$£). We also present some applications of the obtained results to lattice matrix theory. Let $\left(P,$£) be a pseudocomplemented lattice with $\tilde 0$ and $\tilde 1$ and let $A=||a$ij||n ´ n, where $a$ij Î P for $i,j = 1,$¼,n. Let $A$* = ||a'ij||n ´ n and $a\text{'}$ij = \bigwedger = 1, r ≠ jn a*ri for $i,j = 1,$¼,n, where $a$* is the pseudocomplement of $a$Î P in $\left(P,$£). A matrix $A$ has a right inverse over $\left(P,$£) if and only if $A$× A* = E over $\left(P,$£). If $A$ has a right inverse over $\left(P,$£), then $A$* is the greatest right inverse of $A$ over $\left(P,$£). The matrix $A$ has a right inverse over $\left(P,$£) if and only if $A$ is a column orthogonal over $\left(P,$£). The matrix $D=A$× A* is the greatest diagonal such that $A$ is a left divisor of $D$ over $\left(P,$£).

Invertible matrices over a distributive lattice $\left(P,$£) form the general linear group $GL$n(P, £) under multiplication. Let $\left(P,$£) be a finite distributive lattice and let $k$ be the number of components of the covering graph $\Gamma \left(join\left(P,$£) − {\tilde 0}, £), where $join\left(P,$£) is the set of join irreducible elements of $\left(P,$£). Then $GL$n (P, £) @ Snk.

We give some further results concerning inversion of matrices over a pseudocomplemented lattice.

Location: http://mech.math.msu.su/~fpm/eng/k05/k053/k05310h.htm