Inversion of matrices over a pseudocomplemented lattice

  • E. E. Marenich
  • V. G. Kumarov
Article

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 \(\widetilde0\) and \(\widetilde1\) and let A = ‖a ij ‖ n×n , where a ij ∈ P for i, j = 1,..., n. Let A* = ‖a ij ′ ‖ n×n and \( a_{ij} ' = \mathop \Lambda \limits_{r = 1r \ne j}^n a_{ri}^* \) for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ 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 Γ(join(P,≤) − \(\{ \widetilde0\} \), ≤), 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.

Keywords

Distributive Lattice Orthogonal Matrix Permutation Matrix General Linear Group Great Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • E. E. Marenich
    • 1
  • V. G. Kumarov
    • 1
  1. 1.Murmansk State Pedagogical UniversityRussia

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