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), 11-15

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 .

**[1]**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**0404295****[2]**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****[3]**Chandler Davis,*Theory of positive linear dependence*, Amer. J. Math.**76**(1954), 733–746. MR**0064011****[4]**H. K. Farahat,*The semigroup of doubly-stochastic matrices*, Proc. Glasgow Math. Assoc.**7**(1966), 178–183 (1966). MR**0202730****[5]**Peter Flor,*On groups of non-negative matrices*, Compositio Math.**21**(1969), 376–382. MR**0257115****[6]**J. S. Montague and R. J. Plemmons,*Doubly stochastic matrix equations*, Israel J. Math.**15**(1973), 216–229. MR**0321952****[7]**Robert J. Plemmons,*Regular nonnegative matrices*, Proc. Amer. Math. Soc.**39**(1973), 26–32. MR**0313274**, 10.1090/S0002-9939-1973-0313274-6**[8]**R. J. Plemmons and R. E. Cline,*The generalized inverse of a nonnegative matrix*, Proc. Amer. Math. Soc.**31**(1972), 46–50. MR**0285541**, 10.1090/S0002-9939-1972-0285541-5**[9]**W. T. Ransone,*Green's relations on semigroups of nonnegative matrices*, M. S. Thesis, Univ. of Tennessee, August, 1973.**[10]**Daniel J. Richman and Hans Schneider,*Primes in the semigroup of non-negative matrices*, Linear and Multilinear Algebra**2**(1974), 135–140. MR**0354739****[11]**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**0188337****[12]**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 Loose Russian summary). MR**0241451**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0419655-7

Keywords:
Green's relations,
matrix equations,
nonnegative matrices,
regularity

Article copyright:
© Copyright 1976
American Mathematical Society