Linear representations of semigroups of Boolean matrices
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- by Ki Hang Kim and Fred W. Roush PDF
- Proc. Amer. Math. Soc. 63 (1977), 203-207 Request permission
Abstract:
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.References
- Kim Ki Hang Butler, The semigroup of Hall relations, Semigroup Forum 9 (1974/75), no. 3, 253–260. MR 376910, DOI 10.1007/BF02194854
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- Štefan Schwarz, The semigroup of fully indecomposable relations and Hall relations, Czechoslovak Math. J. 23(98) (1973), 151–163. MR 316612
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 63 (1977), 203-207
- MSC: Primary 20M30
- DOI: https://doi.org/10.1090/S0002-9939-1977-0444823-9
- MathSciNet review: 0444823