Regular matrices in the semigroup of Hall matrices. (English) Zbl 0787.05018
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.
Reviewer: A.R.Kräuter (Leoben)
MSC:
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 15A30 | Algebraic systems of matrices |
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