A note on Green's relations on the semigroup
Authors:
D. J. Hartfiel, C. J. Maxson and R. J. Plemmons
Journal:
Proc. Amer. Math. Soc. 60 (1976), 1115
MSC:
Primary 20M10
MathSciNet review:
0419655
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Abstract: Solvability criteria for nonnegative matrix equations are applied in characterizing the first three of the four Green's relations and on the semigroup of all nonnegative matrices. For the relation , it is shown that when the relation is restricted to the regular matrices in although on the entire semigroup .
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 [1]
 I. Borosh, D. J. Hartfiel and C. J. Maxson, Answers to questions posed by Richman and Schneider, Linear and Multilinear Algebra (in press). MR 0404295 (53:8097)
 [2]
 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. 1, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961. MR 24 #A2627. MR 0132791 (24:A2627)
 [3]
 C. Davis, Theory of positive linear dependence, Amer. J. Math. 76 (1954), 733746. MR 16, 211. MR 0064011 (16:211e)
 [4]
 H. K. Farahat, The semigroup of doublystochastic matrices, Proc. Glasgow Math. Assoc. 7 (1966), 178183. MR 34 #2590. MR 0202730 (34:2590)
 [5]
 P. Flor, On groups of nonnegative matrices, Compositio Math. 21 (1969), 376382. MR 41 #1769. MR 0257115 (41:1769)
 [6]
 J. S. Montague and R. J. Plemmons, Doubly stochastic matrix equations, Israel J. Math. 15 (1973), 216228. MR 48 #317. MR 0321952 (48:317)
 [7]
 R. J. Plemmons, Regular nonnegative matrices, Proc. Amer. Math. Soc. 39 (1973), 2632. MR 47 #1829. MR 0313274 (47:1829)
 [8]
 R. J. Plemmons and R. E. Cline, The generalized inverse of a nonnegative matrix, Proc. Amer. Math. Soc. 31 (1972), 4650. MR 44 #2759. MR 0285541 (44:2759)
 [9]
 W. T. Ransone, Green's relations on semigroups of nonnegative matrices, M. S. Thesis, Univ. of Tennessee, August, 1973.
 [10]
 D. J. Richman and H. Schneider, Primes in the semigroup of nonnegative matrices, Linear and Multilinear Algebra 2 (1974), 135140. MR 0354739 (50:7216)
 [11]
 S. Schwarz, On the structure of the semigroup of stochastic matrices, Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 297311. MR 32 #5776. MR 0188337 (32:5776)
 [12]
 , A note on the structure of the semigroup of doublystochastic matrices, Mat. Časopis Sloven. Akad. Vied 17 (1967), 308316. MR 39 #2791. MR 0241451 (39:2791)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197604196557
PII:
S 00029939(1976)04196557
Keywords:
Green's relations,
matrix equations,
nonnegative matrices,
regularity
Article copyright:
© Copyright 1976 American Mathematical Society
