Multiplication of Matrices Over Lattices
Kamilla Kátai-Urbán and Tamás Waldhauser
We study the multiplication operation of square matrices over lattices. If the underlying lattice is distributive, then matrices form a semigroup; we investigate idempotent and nilpotent elements and the maximal subgroups of this matrix semigroup. We prove that matrix multiplication over nondistributive lattices is antiassociative, and we determine the invertible matrices in the case when the least or the greatest element of the lattice is irreducible.
Keywords: Semigroups of matrices over lattices, semigroup of binary relations, Green’s relations, idempotent matrix, nilpotent matrix, invertible matrix, distributive lattice, nonassociative operation, associative spectrum