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