A congruence-free semigroup associated with an infinite cardinal number

Let X be a set with infinite cardinality m and let Qm be the semigroup of balanced elements in T (X ), as described by Howie. If I is the ideal {α ∈ Qm : |X α| < m} then the Rees factor Pm = Qm/I is 0 −bisimple and idempotent-generated. Its minimum non-trivial homomorphic image P∗m has both these properties and is congruence-free. Moreover, P∗m has depth 4, in the sense that [E(P∗m)]4 = P∗m and [E(P∗m )]3 ≠ P∗m

Matemática discreta

Publicação pedagógicaO presente livro é um texto de apoio à unidade curricular Matemática Discreta e tem assim o objectivo de ser uma apresentação simples, mas cuidada, de conceitos e resultados básicos da Teoria de Grafos e da Teoria de Números

Partial orders on transformation semigroups

In 1986, Kowol and Mitsch studied properties of the so-called 'natural partial order' less than or equal to on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), less than or equal to). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare less than or equal to with another 'natural' partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.Centro de Matemática da Universidade do Minho.Fundação para a Ciência e a Tecnologia (FCT)

The ideal structure of nilpotent-generated transformation semigroups

In1987 Sullivan determined the elements of the semigroup N (X ) generated by all nilpotent partial transformations of an infinite set X ; later in 1997 he studied subsemigroups of N (X ) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and the Green’s relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X . We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.Fundação para a Ciência e a Tecnologia (FCT)Centro de Matemática da Universidade do Minh

F-monoids

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).Fundação para a Ciência e a Tecnologia (FCT

F −semigroups

A semigroup S is called F−semigroup if there exists a group congruence ρ on S such that every ρ −class contains a greatest element with respect to the natural partial order ≤ of S . This generalizes the concept of F−inverse semigroups introduced by V. Wagner in 1961 and investigated by McFadden and O’Caroll in 1971. Five different characterisations of general F−semigroups S are given: by means of residuals, by special principal anticones, by properties of the set of idempotents, by the maximal elements in (S, ≤) and finally, an axiomatic one using an additional unary operation. Also, F−semigroups in special classes are considered; in particular, inflations of semigroups and strong semi- lattices of monoids are studied.Centro de Matemática da Universidade do MinhoFundação para a Ciência e a Tecnologia (FCT

Generalised F-semigroups

Fundação para a Ciência e a Tecnologia (FCT) - POCTI

Associate inverse subsemigroups of regular semigroups

By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x ∈ S, in relation to the natural partial order ≤S. We describe the structure of a regular semigroup with an associate inverse subsemigroup, satisfying two natural conditions. As a articular application, we obtain the structure of regular semigroups with an associate subgroup with medial identity element.Fundação para a Ciência e a Tecnologia (FCT

Some orthodox monoids with associate inverse subsemigroups

By an associate inverse subsemigroup of a regular semigroup $S$ we mean a subsemigroup $T$ of $S$ containing a least associate of each $x \in S$, in relation to the natural partial order $\leq_S$ in $S$. In this paper we investigate a class of orthodox monoids with an associate inverse subsemigroup and obtain a known description of uniquely unit regular orthodox semigroups as a corollary. Also, by considering a more general situation, we identify the homomorphic image of a kind of semidirect product of a band with identity by an inverse monoid, thus extending a known result for unit regular orthodox semigroups.Portuguese Foundation for Science and Technology (FCT) through the Research Centre CMA