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Proper left type-A monoids revisited

  • John Fountain (a1) and Gracinda M. S. Gomes (a2)

Extract

The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.

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References

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1.Fountain, J., A class of right PP monoids, Quart. J. Math. Oxford, 28 (2), (1977), 285330.
2.Fountain, J., Adequate semigroups, Proc. Edinb. Math. Soc., 22 (1979), 113125.
3.Fountain, J. and Gomes, Gracinda M. S., Left proper E-dense monoids, J. Pure & Appl. Alg., 80(1) (1992), 127.
4.Margolis, S. W. and Pin, J.-E., Graphs, inverse semigroups and languages, Proc. 1984 Marquette Conf. on Semigroups,(Marquette University, Milwaukee 1985) 85112.
5.Margolis, S. W. and Pin, J.-E., Inverse semigroups and extensions of groups by semilattices, J. Algebra, 110 (1987), 277297.
6.McAlister, D. B., Groups, semilattices and inverse semigroups, Trans. Amer. Math. Soc., 192 (1974) 227244.
7.McAlister, D. B., Groups, semilattices and inverse semigroups II, Trans. Amer. Math. Soc., 196 (1974), 351370.
8.O'Carroll, L., Embedding theorems for proper inverse semigroups, J. Algebra,42 (1976), 2640.

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