On solvability of some types of matrix equations over lattices (Q1760496)

A partially ordered set \(L\) with top and bottom elements 0 and 1 and the meet and join operations is considered here. A matrix with entries in such a lattice \(L\) is called a matrix over \(L\). The set of all such \(m\) by \(n\) matrices is partially ordered by elementwise entry order and addition and multiplication of such matrices is defined via lattice join and lattice meet, respectively. The paper considers the matrix equations \(AX = B\) and \(XA = B\) for matrices over a distributive or Boolean lattice \(L\) and finds the greatest solution in either of the four cases. More specific results are proved for matrices over \(L\) whose entries, rows or columns form an orthogonal system.