The Semigroup of Hall Matrices over Distributive Lattices

In this paper, the semigroup Hn(L) of Hall matrices over a complete and completely distributive lattice L is studied. A Hall matrix is a matrix which is greater (for the order associated with the lattice structure) than an invertible matrix. Some necessary and sufficient conditions for a Hall matrix to be regular in the semigroup Hn(L) are given and Green's relations of the semigroup Hn(L) are described. Also, the sandwich semigroup of Hall matrices over the lattice L is studied.