Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A natural partial order for semigroups


Author: H. Mitsch
Journal: Proc. Amer. Math. Soc. 97 (1986), 384-388
MSC: Primary 20M10; Secondary 06F05
DOI: https://doi.org/10.1090/S0002-9939-1986-0840614-0
MathSciNet review: 840614
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A partial order on a semigroup $ (S, \cdot )$ is called natural if it is defined by means of the multiplication of $ S$. It is shown that for any semigroup $ (S, \cdot )$ the relation $ a \leq b$ iff $ a = xb = by$, $ xa = a$ for some $ x$, $ y \in {S^1}$, is a partial order. It coincides with the well-known natural partial order for regular semigroups defined by Hartwig [4] and Nambooripad [10]. Similar relations derived from the natural partial order on the regular semigroup $ ({T_X}, \circ )$ of all maps on the set $ X$ are investigated.


References [Enhancements On Off] (What's this?)

  • [1] T. Blyth and G. Gomes, On the compatibility of the natural partial order on a regular semigroup, Proc. Roy. Soc. Edinburgh 94 (1983), 79-84. MR 700501 (84h:20063)
  • [2] W. Burgess and R. Raphael, On Conrad's partial order relation on semiprime rings and on semigroups, Semigroup Forum 16 1978, 133-140. MR 0491395 (58:10651)
  • [3] G. Gomes, On left quasinormal orthodox semigroups, Proc. Roy. Soc. Edinburgh 95 (1983), 59-71. MR 723097 (85b:20086)
  • [4] R. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980), 1-13. MR 571255 (81i:06014)
  • [5] J. Hickey, Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc. 26 (1983), 371-382. MR 722568 (85d:20068)
  • [6] J. Howie, An introduction to semigroup theory, Academic Press, New York, 1976. MR 0466355 (57:6235)
  • [7] G. Kowol and H. Mitsch, Naturally ordered full transformation semigroups, Monatsh. Math. (to appear). MR 861935 (88b:20102)
  • [8] M. Lawson, The natural partial order on an abundant semigroup, preprint, Univ. of York, England, 1984. MR 892688 (88g:20132)
  • [9] D. McAlister, Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups, J. Austral. Math. Soc. 31 (1981), 325-336. MR 633441 (84d:20062)
  • [10] K. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc. 23 (1980), 249-260. MR 620922 (82g:20092)
  • [11] V. Vagner, Generalized groups, Dokl. Akad. Nauk SSSR 84 (1952), 1119-1122. (Russian) MR 0048425 (14:12b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20M10, 06F05

Retrieve articles in all journals with MSC: 20M10, 06F05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0840614-0
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society