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The natural partial order on an abundant semigroup

Published online by Cambridge University Press:  20 January 2009

Mark V. Lawson
Mark V. Lawson
Affiliation:
Department of Mathematics, University of York, Heslington, York YO1 5DD
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In this paper we will study the properties of a natural partial order which may bedefined on an arbitrary abundant semigroup: in the case of regular semigroups werecapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistroviˇ [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 2 , June 1987 , pp. 169 - 186
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Abian, A., Direct product decomposition of commutative semisimple rings, Proc. Amer. Math. Soc. 24 (1970), 502–507.Google Scholar
2.Abian, A., Direct-sum decomposition of atomic and orthogonally complete rings, J. Austral.Math. Soc. 11 (1970), 357–361.CrossRefGoogle Scholar
3.Armstrong, S., The structure of type A semigroups, Semigroup Forum 29 (1984), 319–336.CrossRefGoogle Scholar
4.Armstrong, S., The Structure of Concordant Semigroups (D.Phil, thesis, York, 1985).Google Scholar
5.Burgess, W. D., Completions of semilattices of cancellative semigroups, Glasgow Math. J. 21 (1980), 29–37.CrossRefGoogle Scholar
6.Burmistrovic, I. E., Commutative bands of cancellative semigroups, Sibirsk. Mat. Zh. 6 (1965), 284–299.Google Scholar
7.Chacron, M., Direct products of division rings and a paper of Abian, Proc. Amer. Math. Soc. 29 (1971), 259–262.CrossRefGoogle Scholar
8.Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups (Amer. Math. Soc, Math. Surveys No. 7, vols. 1 and 2).Google Scholar
9.Dorofeeva, M. P., Hereditary and semi-hereditary monoids, Semigroup Forum 4 (1972), 301–311.CrossRefGoogle Scholar
10.El-Qallali, A., Structure Theory for Abundant and Related Semigroups (D.Phil, thesis, York, 1980).Google Scholar
11.El-Qallali, A. and Fountain, J. B., Idempotent-connected abundant semigroups, Proc. Roy. Soc. Edinburgh Sect. A 91 (1981), 79–90.CrossRefGoogle Scholar
12.Fountain, J. B., Right PP monoids with central idempotents, Semigroup Forum 13 (1977), 229–237.CrossRefGoogle Scholar
13.Fountain, J. B., A class of right PP monoids, Quart. J. Math. 28 (1977), 285–300.CrossRefGoogle Scholar
14.Fountain, J. B., Adequate semigroups, Proc. Edinburgh Math. Soc. 22 (1979), 113–125.CrossRefGoogle Scholar
15.Fountain, J. B., Abundant semigroups, Proc. London Math. Soc. 44 (1982), 103–129.CrossRefGoogle Scholar
16.Hall, T. E., Some properties of local subsemigroups inherited by larger subsemigroups, Semigroup Forum 25 (1982), 35–49.CrossRefGoogle Scholar
17.Howie, J. M., An Introduction to Semigroup Theory (Academic Press, 1976).Google Scholar
18.Kilt, M., Commutative monoids all of whose principal ideals are projective, Semigroup Forum 6 (1973), 334–339.Google Scholar
19.Lawson, M. V., Abundant Rees matrix semigroups, submitted.Google Scholar
20.Lawson, M. V., Rees matrix covers for a class of abundant semigroups, submitted.Google Scholar
21.Lawson, M. V., The structure of type A semigroups, Quart. J. Math., accepted for publication.Google Scholar
22.Meakin, J., The structure mapping approach to regular and inverse semigroups, The Algebraic Theory of Semigroups (Coll. Math. Soc. János Bolyai 20, Szeged, 1976), 371–382.Google Scholar
23.Nambooripad, K. S. S., The structure of regular semigroups, Mem. Amer. Math. Soc. 22, No. 224 (1979).Google Scholar
24.Nambooripad, K. S. S., The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc. 23 (1980), 249–260.CrossRefGoogle Scholar
25.Sussman, I. A., A generalisation of Boolean rings, Math. Ann. 136 (1958), 326–338.CrossRefGoogle Scholar