Transitive reduction of a rectangular Boolean matrix (Q798324)

This paper deals with simplification of rectangular Boolean matrices of zeros and ones. For x,y whose values are zero or one, the following operations are defined \(x+y=\max (x,y), x\Theta y=\max (0,x-y)\) and for \(m\times n\) Boolean matrices A,B, \(A\Theta\) B and \(A\leq B\) are defined elementwise. An \(n\times n\) Boolean matrix R such that \(R^ 2\leq R\) is said to be transitive and a Boolean matrix, all of whose diagonal elements are zero, is said to be irreflexive. The main result is as follows: Let R, S, P be transitive matrices. If P is irreflexive and \(P\leq R\), then \(S(A\Theta SAP)R=SAR.\) The author also shows some similar relationships and remarks on an application of the results to information retrieval models.