Inversion of matrices over a pseudocomplemented lattice

Author: Marenich E.   Kumarov V.  

Publisher: Springer Publishing Company

ISSN: 1072-3374

Source: Journal of Mathematical Sciences, Vol.144, Iss.2, 2007-07, pp. : 3968-3979

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Abstract

We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with and and let A = ‖A ij n×n , where A ij P for i, j = 1,..., n. Let A* = ‖A ij n×n and for i, j = 1,..., n, where A* is the pseudocomplement of AP 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 Γ(join(P,≤) − , ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL A (P, ≤) ≅ = S n k .We give some further results concerning inversion of matrices over a pseudocomplemented lattice.