Covering in the set of principal ideals in the semigroup of binary relations (Q1322587)

This paper deals with coverings in the poset of \(D\)-classes of the semigroup \(B_ n\) of \(n\times n\) Boolean matrices. A cover of the class of a matrix \(B\) by a matrix \(A\) is said to be permanent if we still have a covering after adjoining any number of zero rows and columns to \(A\), \(B\), in the final places. The author previously showed a cover is permanent if and only if \(r(A) = r(B) + 1\). Here he shows that if \(A\) covers \(B\) and \(B\) comes from \(B_ k\), \(k\) less than the size of \(B\) by adjoining zero rows and columns, that \(A\) permanently covers \(B\), and derives inequalities on row and column rank. Base is the transpose of the relation cover, and the author characterizes permanence of bases and gives a sharp inequality on the size of permanent bases.