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

Inversion of matrices over a pseudocomplemented lattice

E. E. Marenich
V. G. Kumarov


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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 $ \tilde 0 $ and $ \tilde 1 $ and let A=||aij||n ´ n, where aij Î P for i,j = 1,¼,n. Let A* = ||a'ij||n ´ n and a'ij = \bigwedger = 1, r ≠ jn 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 GLn(P, £) under multiplication. Let (P, £) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P, £) − {\tilde 0}, £), where join(P, £) is the set of join irreducible elements of (P, £). Then GLn (P, £) @ Snk.

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

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Last modified: September 14, 2005