FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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 $(P,$£). We also present some applications of the obtained results to lattice matrix theory. Let $(P,$£) be a pseudocomplemented lattice with $\$\; \backslash tilde\; 0\; \$$ and $\$\; \backslash tilde\; 1\; \$$ 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 = \bigwedge_{r = 1, r ≠ j}ⁿ a^*_ri for $i,j\; =\; 1,$¼,n, where $a$* is the pseudocomplement of $a$Î P in $(P,$£). A matrix $A$ has a right inverse over $(P,$£) if and only if $A$× A^* = E over $(P,$£). If $A$ has a right inverse over $(P,$£), then $A$* is the greatest right inverse of $A$ over $(P,$£). The matrix $A$ has a right inverse over $(P,$£) if and only if $A$ is a column orthogonal over $(P,$£). The matrix $D=A$× A^* is the greatest diagonal such that $A$ is a left divisor of $D$ over $(P,$£).

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

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
Last modified: September 14, 2005