A note on Green’s relations on the semigroup $N_{n}.$
HTML articles powered by AMS MathViewer
- by D. J. Hartfiel, C. J. Maxson and R. J. Plemmons PDF
- Proc. Amer. Math. Soc. 60 (1976), 11-15 Request permission
Abstract:
Solvability criteria for nonnegative matrix equations are applied in characterizing the first three of the four Green’s relations $\mathcal {L},\mathcal {R},\mathcal {D}$ and $\mathcal {J}$ on the semigroup ${N_n}$ of all $n \times n$ nonnegative matrices. For the relation $\mathcal {J}$, it is shown that $\mathcal {D} = \mathcal {J}$ when the relation is restricted to the regular matrices in ${N_n}$ although on the entire semigroup ${N_n},n \geqslant 3,\mathcal {D} \ne \mathcal {J}$.References
- I. Borosh, D. J. Hartfiel, and C. J. Maxson, Answers to questions posed by Richman and Schneider, Linear and Multilinear Algebra 3 (1975/76), no. 4, 255–258. MR 404295, DOI 10.1080/03081087608817117
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- Chandler Davis, Theory of positive linear dependence, Amer. J. Math. 76 (1954), 733–746. MR 64011, DOI 10.2307/2372648
- H. K. Farahat, The semigroup of doubly-stochastic matrices, Proc. Glasgow Math. Assoc. 7 (1966), 178–183 (1966). MR 202730
- Peter Flor, On groups of non-negative matrices, Compositio Math. 21 (1969), 376–382. MR 257115
- J. S. Montague and R. J. Plemmons, Doubly stochastic matrix equations, Israel J. Math. 15 (1973), 216–229. MR 321952, DOI 10.1007/BF02787568
- Robert J. Plemmons, Regular nonnegative matrices, Proc. Amer. Math. Soc. 39 (1973), 26–32. MR 313274, DOI 10.1090/S0002-9939-1973-0313274-6
- R. J. Plemmons and R. E. Cline, The generalized inverse of a nonnegative matrix, Proc. Amer. Math. Soc. 31 (1972), 46–50. MR 285541, DOI 10.1090/S0002-9939-1972-0285541-5 W. T. Ransone, Green’s relations on semigroups of nonnegative matrices, M. S. Thesis, Univ. of Tennessee, August, 1973.
- Daniel J. Richman and Hans Schneider, Primes in the semigroup of non-negative matrices, Linear and Multilinear Algebra 2 (1974), 135–140. MR 354739, DOI 10.1080/03081087408817050
- Štefan Schwarz, On the structure of the semigroup of stochastic matrices, Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 297–311 (1965) (English, with Russian summary). MR 188337
- Štefan Schwarz, A note on the structure of the semigroup of doubly-stochastic matrices, Mat. Časopis Sloven. Akad. Vied 17 (1967), 308–316 (English, with Russian summary). MR 241451
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 60 (1976), 11-15
- MSC: Primary 20M10
- DOI: https://doi.org/10.1090/S0002-9939-1976-0419655-7
- MathSciNet review: 0419655