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Adequate Semigroups

Published online by Cambridge University Press:  20 January 2009

John Fountain
John Fountain
Affiliation:
Department of Mathematics, University of York, Heslington, York Y01 5DD
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A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 22 , Issue 2 , June 1979 , pp. 113 - 125
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

REFERENCES

(1) Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. II (Math. Surveys No. 7, Amer. Math. Soc, 1967).Google Scholar
(2) Dorofeeva, M. P., Hereditary and semi-hereditary monoids, Semigroup Forum 4 (1972), 301–311.Google Scholar
(3) Fountain, J. B., Right PP monoids with central idempotents, Semigroup Forum 13 (1977), 229–237.CrossRefGoogle Scholar
(4) Fountain, J. B., A class of right PP monoids, Quart. J. Math. Oxford to appear.Google Scholar
(5) Hall, T. E., On orthodox semigroups and uniform and anti-uniform bands, J. Algebra 16 (1970), 204–217.CrossRefGoogle Scholar
(6) Howie, J. M., The maximum idempotent-separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. (2) 14 (1964), 71–79.Google Scholar
(7) Howie, J. M., An introduction to semigroup theory (Academic Press, 1976).Google Scholar
(8) Kil'P, M., Commutative monoids all of whose principal ideals are projective, Semigroup Forum 6 (1973), 334–339.Google Scholar
(9) McAlister, D. B., One-to-one partial right translations of a right cancellative semigroup, J. Algebra 43 (1976), 231–251.Google Scholar
(10) Munn, W. D., Uniform semilattices and bisimple inverse semigroups, Quart. J. Math. Oxford (2) 17 (1966), 151–159.Google Scholar
(11) Pastijn, F., A representation of a semigroup by a semigroup of matrices over a group with zero, Semigroup Forum 10 (1975), 238–249.Google Scholar