Linear preservers of extremes of matrix pairs over nonbinary Boolean algebra (Q1941001)

Let \(M_{m,n}(\mathbb{B}_{k})\) denote the set of all \(m\times n\) matrices with entries from a (finite) Boolean algebra \(\mathbb{B}_{k}\). Throughout the paper, the authors assume that \(m\leq n\) and that \(\mathbb{B}_{k}\) is not a binary (i.e. zero-one) Boolean algebra. If \(\mathbb{B}_{k}\) is not a binary Boolean algebra, it is called a nonbinary Boolean algebra. The Boolean rank \(b(A)\) of a matrix \(A\in M_{m,n}(\mathbb{B}_{k}),\) \(A\neq0\), is defined as the least positive \(r\) such that \(A=BC\) for some \(B\in M_{m,r}(\mathbb{B}_{k})\) and \(C\in M_{r,n}(\mathbb{B}_{k})\) (\(b(0)\) is defined to be \(0).\) Let us recall two inequalities which hold for Boolean rank: \(b(A+B)\leq b(A)+b(B)\), \[ b(A+B)\geq\begin{cases} b(A) & \text{if }B=0,\\ b(B) & \text{if }A=0,\\ 1 & \text{ if }A\neq0\text{ and }B\neq0. \end{cases} \] The following sets of matrices arise as extremal cases of these inequalities: \(\mathcal{R}_{SA}(\mathbb{B}_{k})=\{(X,Y)\in M_{m,n}(\mathbb{B}_{k})^{2}:b(X+Y)=b(X)+b(Y)\}\), \(\mathcal{R}_{S1}(\mathbb{B}_{k})=\{(X,Y)\in M_{m,n}(\mathbb{B}_{k})^{2}:b(X+Y)=1\}\). We say that a map \(T\) preserves a set \(\mathcal{P}\) if \(X\in\mathcal{P}\) implies \(T(X)\in\mathcal{P}\) for every \(X\in\mathcal{P}\), or if \(\mathcal{P}\) is the set of ordered pairs such that \((X,Y)\in\mathcal{P}\) implies \((T(X),T(Y))\in\mathcal{P}\) for every \((X,Y)\in\mathcal{P}\). In the paper, the authors characterize linear transformations \[ T:M_{m,n}(\mathbb{B}_{k})\rightarrow M_{m,n}(\mathbb{B}_{k}) \] which preserve \(\mathcal{R}_{SA}(\mathbb{B}_{k})\) or \(\mathcal{R}_{S1}(\mathbb{B}_{k})\).