Semigrupos-F Generalizados - Semigrupos-F
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2010
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Resumo
Nesta dissertação são apresentados alguns dos resultados que consideramos com
mais interesse sobre semigrupos-F generalizados e semigrupos-F, obtidos por Emília
Giraldes, Paula Marques-Smith e Heinz Mitsch.
De nimos um semigrupo-F generalizado como um semigrupo em que existe uma
congruência de grupo, tal que a sua classe identidade tem um elemento máximo para
a ordem parcial natural do semigrupo. Prova-se que nestes semigrupos a congruência
da de nição é única, mais precisamente, a sua congruência mínima de grupo. Apre-
sentamos caracterizações para semigrupos-F generalizados, usando o conceito de an-
ticone principal e, alternativamente, propriedades do seu conjunto de idempotentes e
de um seu majorante. Na sequência desta caracterização, apresentamos outras para
semigrupos-F generalizados com idempotente máximo, e dentro destes para monóides-
-F generalizados e semigrupos-F generalizados regulares.
Depois, de nimos semigrupos-F como semigrupos em que existe uma congruência
de grupo tal que as suas classes de congruência têm elemento máximo para a ordem
parcial natural do semigrupo. Estes semigrupos são, obviamente, uma subclasse dos
semigrupos-F generalizados e, portanto, a congruência da de nição também é única,
mais precisamente, a sua congruência mínima de grupo. Caracterizamos os semigrupos-
-F, utilizando o conceito de anticone principal com elemento máximo residuado e,
alternativamente, propriedades do seu conjunto de idempotentes e de um seu majorante
e dos residuais dos seus elementos. Na sequência desta caracterização apresentamos
outras para semigrupos-F com idempotente máximo, monóides-F e semigrupos-F re-
gulares. Caracterizamos ainda os semigrupos-F, usando elementos maximais e uma
axiomática através de uma operação unária adicional. Por m, estabelecemos alguns resultados especí cos para semigrupos-F regulares
e para monóides-F, que permitem apresentar caracterizações interessantes para estes
semigrupos. Acabamos a dissertação com alguns exemplos das classes de semigrupos
estudados.
In this dissertation we present some of the results on generalised F-semigroups and on F-semigroups, obtained by Emília Giraldes, Paula Marques-Smith e Heinz Mitsch, that we consider more important. A semigroup S is called generalised F-semigroup if there exists a group-congruence on S, such that the identity-class contains a greatest element with respect to the natural partial order of S. It is shown that this congruence is the minimum group-congruence on S and is, therefore, unique. Characterisations of generalised F-semigroups in terms of principal anticones and by means of the structure of the set of idempotents and by the existence of one of its upper bounds are given. Following these characterisa- tions, generalized F-semigroups in the following classes are described: semigroups with greatest idempotent and, within these ones, monoids and regular semigroups. We proceed with the presentation of the de nition of F-semigroup: a semigroup for wich there exists a group-congruence such that each congruence-class contains a great- est element with respect to the natural partial order. These semigroups are, clearly, a subclass of the class of generalized F-semigroups and hence the congruence in the de nition is unique. We characterise the class of F-semigroups using the concept of principal anticone with residuated greatest element and, alternatively, the properties of the set of idempotents, one of its upper bounds and the residuals of its elements. Following these characterisations, F-semigroups in the following classes are also char- acterised: semigroups with greatest idempotent, monoids and regular semigroups. We also characterise F-semigroups in terms of maximal elements and of an axiomatic by an additional unary operation. Finally, we establish some speci c results for both F-regular semigroups and F--monoids, which allow interesting characterisations for these semigroups to be established. We conclude the dissertation with some examples of the classes of semigroups that we have studied.
In this dissertation we present some of the results on generalised F-semigroups and on F-semigroups, obtained by Emília Giraldes, Paula Marques-Smith e Heinz Mitsch, that we consider more important. A semigroup S is called generalised F-semigroup if there exists a group-congruence on S, such that the identity-class contains a greatest element with respect to the natural partial order of S. It is shown that this congruence is the minimum group-congruence on S and is, therefore, unique. Characterisations of generalised F-semigroups in terms of principal anticones and by means of the structure of the set of idempotents and by the existence of one of its upper bounds are given. Following these characterisa- tions, generalized F-semigroups in the following classes are described: semigroups with greatest idempotent and, within these ones, monoids and regular semigroups. We proceed with the presentation of the de nition of F-semigroup: a semigroup for wich there exists a group-congruence such that each congruence-class contains a great- est element with respect to the natural partial order. These semigroups are, clearly, a subclass of the class of generalized F-semigroups and hence the congruence in the de nition is unique. We characterise the class of F-semigroups using the concept of principal anticone with residuated greatest element and, alternatively, the properties of the set of idempotents, one of its upper bounds and the residuals of its elements. Following these characterisations, F-semigroups in the following classes are also char- acterised: semigroups with greatest idempotent, monoids and regular semigroups. We also characterise F-semigroups in terms of maximal elements and of an axiomatic by an additional unary operation. Finally, we establish some speci c results for both F-regular semigroups and F--monoids, which allow interesting characterisations for these semigroups to be established. We conclude the dissertation with some examples of the classes of semigroups that we have studied.
Descrição
Dissertação de Mestrado em Ensino da Matemática