Regular matrices in the semigroup of Hall matrices
Let \(H_ n\) denote the set of all \(n\)-square matrices \(A\) over the Boolean algebra of order two such that \(\text{per}(A)>0\) (semigroup of Hall matrices). Moreover, let, for any positive integer \(s\), \(H_ n(s)= \{A\in H_ n: \text{ per}(A)\geq s\}\). A matrix \(A\in H_ n\) is called regular if there exists a generalized inverse of \(A\) in the same set; \(A\) is called semiinvertible in \(H_ n\) if there exists a semiinverse of \(A\) in the same set. The paper presents a variety of characterizations of regular matrices in \(H_ n\) and \(H_ n(s)\) in terms of idempotent matrices, semiinvertible matrices, adjoint matrices, and identifying permutation matrices.
scientific article; zbMATH DE number 459079