F-monoids

• Autor(a) principal: Giraldes, E.
• Data de Publicação: 2007
• Outros Autores: Smith, M. Paula Marques, Mitsch, H.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/1822/11065
• topic: E-inversive; E-unitary; Group-congruence; Natural partial order; Monoid; E-inversive E-unitary; Science & Technology

A semigroup $S$ is called $F-monoid$ if $S$ has an identity and if there exists a group congruence $\rho$ on $S$ such that each $\rho$-class of $S$ contains a greatest element with respect to the natural partial order of $S$ (see Mitsch, 1986). Generalizing results given in Giraldes et al. (2004) and specializing some of Giraldes et al. (Submitted) five characterizations of such monoids $S$ are provided. Three unary operations $\star$, $\circ$ and $-$ on $S$ defined by means of the greatest elements in the different $\rho$-classes of $S$ are studied. Using their properties, a charaterization of $F$-monoids $S$ by their regular part $S^\circ=\{a^\circ:a\in S\}$ and the associates of elements in $S^\circ$ is given. Under the hypothesis that $S^\star=\{a^\star:a\in S\}$ is a subsemigroup it is shown that $S$ is regular, whence of a known structure (see Giraldes et al., 2004).