A comment on some recent results concerning the reverse order law for \(\{ 1, 3, 4 \}\)-inverses (Q708139)

A generalized inverse of an \(m \times n\) matrix \(A\) is an \(n \times m\) matrix \(X\) satisfying some of the following four Penrose equations: \[ (1)\quad AXA=A, \qquad (2)\quad XAX=X,\qquad (3)\quad (AX)^*=AX, \qquad (4)\quad (XA)^*=XA. \] Let \(\emptyset\neq \rho\subseteq \{1, 2, 3, 4\}\). Then \(A\rho\) denotes the set of all matrices \(X\) which satisfy (i) for all \(i \in \rho\). Motivated by a result of \textit{D. Liu} and \textit{H. Yang} [Appl. Math. Comput. 215, No.~12, 4293--4303 (2010; Zbl 1187.15005)], the authors prove that \((AB)\{1,3,4\} \subseteq B\{1,3,4\} \cdot A\{1,3,4\}\) is equivalent to \((AB)\{1,3,4\}= B\{1,3,4\} \cdot A\{1,3,4\}\). They show that \((AB)\{1,3,4\} \subseteq B\{1,3,4\} \cdot A\{1,3,4\}\) can only be possible if \(n \leq m\) and in this case, they present purely algebraic necessary and sufficient conditions for this inclusion to hold. In addition they give some characterizations of \(B\{1,3,4\} \cdot A\{1,3,4\}\subseteq (AB)\{1,3,4\}\).