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Linear Representations of Semigroups of Boolean Matrices
Ki Hang Kim and Fred W. Roush
Proceedings of the American Mathematical Society
Vol. 63, No. 2 (Apr., 1977), pp. 203-207
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2041789
Page Count: 5
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Let $B_n$ be the multiplicative semigroup of $n \times n$ matrices over the semiring 0, 1 under the operations "or" and "and". We show that the least possible degree of a faithful representation of $B_n$ over a field is $2^n - 1$ by studying representations of a subsemigroup of $B_n$. By different methods we answer the same question for the subsemigroups of Boolean matrices greater than or equal to some permutation matrix (Hall matrices) and greater than or equal to the identity (reflexive Boolean matrices). We prove every representation of the latter semigroup can be triangularized.
Proceedings of the American Mathematical Society © 1977 American Mathematical Society