# zbMATH — the first resource for mathematics

Boolean-rank-preserving operators and Boolean-rank-1 spaces. (English) Zbl 0536.20044
The authors prove that a Boolean linear operator from the set of $$n\times m$$ Boolean matrices to itself $$(m,n>1)$$ preserves Boolean rank iff if preserves ranks 1, 2 iff it has the form UAV or $$UA^ tV$$ for U, V permutation matrices. They also find the maximum cardinality and dimension of rank 1 subspaces of $$n\times m$$ Boolean matrices.
Reviewer: K.-H.Kim

##### MSC:
 20M20 Semigroups of transformations, relations, partitions, etc. 15A04 Linear transformations, semilinear transformations
##### Keywords:
Boolean linear operator; Boolean matrices; Boolean rank
Full Text:
##### References:
 [1] Beasley, L.B., Linear transformations which preserve fixed rank, Linear algebra appl., 40, 183-187, (1981) · Zbl 0467.15003 [2] Courant, R., Differential and integral calculus, Vol. 1, (1937), Blackie London · JFM 63.1067.01 [3] de Caen, D.; Gregory, D.A., Primes in the semigroup of Boolean matrices, Linear algebra appl., 37, 119-134, (1981) · Zbl 0457.05001 [4] de Caen, D.; Gregory, D.A.; Pullman, N.J., The Boolean rank of zero-one matrices, Proceedings of the third Caribbean conference on combinatorics and computing, 169-173, (1981), Cave Hill, Barbados · Zbl 0496.20052 [5] Kim, K.H., Boolean matrix theory and applications, () [6] Lautemann, C., Linear transformations on matrices: rank preservers and determinant preservers, Linear and multilinear algebra, 10, 343-345, (1981) · Zbl 0484.15004 [7] Marcus, M.; Moyls, B., Linear transformations on algebras of matrices, Canad. J. math., 11, 61-66, (1959) · Zbl 0086.01703 [8] Marcus, M.; Moyls, B., Transformations on tensor product spaces, Pacific J. math., 9, 1215-1221, (1959) · Zbl 0089.08902 [9] Orlin, J., Contentment in graph theory, Indag. math., 39, 406-424, (1977) · Zbl 0374.05041 [10] Pullman, N.J., A property of infinite products of Boolean matrices, SIAM J. appl. math., 15, 871-873, (1967) · Zbl 0153.02801 [11] Rao, K.; Rao, P., On generalized inverses of Boolean matrices, Linear algebra appl., 11, 135-153, (1975) · Zbl 0322.15011 [12] Rao, K.; Rao, P., On generalized inverses of Boolean matrices, II, Linear algebra appl., 42, 133-144, (1982) · Zbl 0479.15003 [13] Richman, D.J.; Schneider, H., Primes in the semigroup of nonnegative matrices, Linear and multilinear algebra, 2, 135-140, (1974) [14] Schein, B.M., Relation algebras and function semigroups, Semigroup forum, 1, 1-62, (1970) · Zbl 0197.29404 [15] Westwick, R., Transformations on tensor spaces, Pacific J. math., 23, 613-620, (1967) · Zbl 0178.03502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.