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 k n .
We give some further results concerning inversion of matrices over a pseudocomplemented lattice.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005.
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Marenich, E.E., Kumarov, V.G. Inversion of matrices over a pseudocomplemented lattice. J Math Sci 144, 3968–3979 (2007). https://doi.org/10.1007/s10958-007-0250-y
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DOI: https://doi.org/10.1007/s10958-007-0250-y