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Semigroups in Which each Ideal is a Retract

Published online by Cambridge University Press:  09 April 2009

Edward J. Tully Jr
Edward J. Tully Jr
Affiliation:
University of CaliforniaDavis, California
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A non-empty subset I of a semigroup S is called an ideal if ab, ba ∈ I whenever a ∈I, b∈S. A subset R of S will be called a retract if there exsists a retraction of S onto R, that is a homomorphism of S onto R which leaves each element of R fixed. The purpose of this paper is to study semigroups in which every ideal is a retract. For convenience we shall call such semigroups retractable. Such semigroups seem to arise naturally; for example, it is easy to show that if the lattice of congruence relations on S is a complemented lattice then S is retractable.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 1-2 , February 1969 , pp. 239 - 245
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Clifford, A., [1941] ‘Semigroups admitting relative inverses’, Ann. of. Math. 2 (1941), 1037–1049Google Scholar
[2]Clifford, A. and Preston, G. B., [1961] The algebraic theory of semigroups, Volume I (Math. Surveys No. 7, Amer. Math. Soc., Providence, R. I., 1961).CrossRefGoogle Scholar
[3]Green, J., [1951] ‘On the structure of semigroups’, Ann. of Math. 54 (1951), 163–172.Google Scholar
[4]Rees, D., [1940] ‘On semigroups’, Proc. Cambridge Philos. Soc. 36 (1940), 387–400.Google Scholar
[5]Suschkewitsch, A. K., [1928] ‘Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit’, Math. Ann. 99 (1928), 30–50.CrossRefGoogle Scholar