Inversion of matrices over a pseudocomplemented lattice

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