The semigroup of Hall matrices over distributive lattices
Let \(L\) be a complete and completely distributive lattice with the least and greatest elements \(0\) and \(1\), and let \(M_n(L)\) be the set of all \(n\times n\) matrices over \(L\). Clearly, \(M_n(L)\) is a semigroup with respect to a naturally defined product of matrices (where the meet stands for multiplication and join for addition). Matrices that are greater (under the naturally defined order) than invertible matrices are called Hall matrices, and form a subsemigroup \(H_n(L)\) of \(M_n(L)\). The author describes Green's relations in \(H_n(L)\), considers regular elements of \(H_n(L)\) and also looks at certain derived semigroups of \(H_n(L)\).
scientific article; zbMATH DE number 1523496