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121 results on '"Gomes, Gracinda M. S."'

4. On the semigroup rank of a group

Author

Branco, Mário J. J., Gomes, Gracinda M. S., and Silva, Pedro V.

Subjects
Mathematics - Group Theory, 20E05, 20K15, 20F34, 20M05
Abstract

For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This is the starting point for the rest of the article, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of relatively free groups, for any variety of groups, is computed. For a finitely generated abelian group~$G$, it is proven that $G{\rm rk}S = G{\rm rk}G+1$ if and only if $G$ is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case. It is also proven that if $M$ is a connected closed surface, then $(\pi_1(M)){\rm rk}S = (\pi_1(M)){\rm rk}G+1$ if and only if $M$ is orientable.

Published
2017

5. Takahasi semigroups

Author

Branco, Mário J. J., Gomes, Gracinda M. S., and Silva, Pedro V.

Subjects
Mathematics - Group Theory, 20M99, 20M15, 37C25, 20M18
Abstract

Takahasi's theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations. In addition to the background on semigroups, proofs involve arguments over groups and finite automata.

Published
2015

12. A topological shuttle between inequalities and pseudoinequalities

Author

Hipólito Fernandes, Bernardo, Gomes, Gracinda M. S., Hipólito Fernandes, Bernardo, and Gomes, Gracinda M. S.

Abstract

Topological procedures to relate pseudoinequalities that define a pseudovariety of ordered algebras with inequalities that ultimately define it, and vice-versa, are presented., QC 20230825

Published
2021
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26. The cardinal and the idempotent number of various monoids of transformations on a finite chain

Author

Fernandes, Vítor H., Gomes, Gracinda M. S., and Jesus, Manuel M.

Subjects
monoid, transformation, order-preserving, orientation-preserving
Abstract

Bulletin of the Malaysian Mathematical Sciences Society, 2, 34 (1),(2011), p. 79–85 We consider various classes of monoids of transformations on a finite chain, in particular transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested in computing both their cardinals and the number of their idempotents. In this note we present a short survey on these questions which have been approached by various authors and close the problem by computing the number of idempotents of those monoids not considered before. Fibonacci and Lucas numbers play an essential role in the last computations.

Published
2011

27. Presentations for some monoids of injective partial transformations on a finite chain

Author

Fernandes, Vítor H., Gomes, Gracinda M. S., and Jesus, Manuel M.

Subjects
Orientation-preserving, Mathematics::Category Theory, Order-preserving, Monoid, Presentation, Transformation
Abstract

Communications in Algebra, 33 (2005), p. 587-604 In this paper we calculate presentations for some natural monoids of transformations on a chain Xn = 1 < 2 < · · · < n . First we consider n [ n], the monoid of all full [partial] transformations on Xn that preserve or reverse the order. Two other monoids of partial transformations on Xn we look at are n and n—the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.

Published
2004

42. Even Covers for Left Ample Semigroups.

Author

Gomes, Gracinda M. S., Szendrei, Mária B., and Márki, L.

Subjects
*SEMIGROUPS (Algebra), *MONOIDS, *GROUP theory, *EMBEDDINGS (Mathematics), *MATHEMATICS
Abstract

In this paper we describe the covers of a left ample semigroup that arise from strict (2,1)-embeddings in left factorizable left ample monoids. [ABSTRACT FROM AUTHOR]

Published
2007
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43. The Generalized Prefix Expansion of a Weakly Left Ample Semigroup.

Author

Gomes, Gracinda M. S.

Subjects
*INVERSE semigroups, *MORPHISMS (Mathematics), *GROUP theory, *CATEGORIES (Mathematics), *SET theory
Abstract

The generalized prefix expansion of inverse semigroups, presented by Lawson, Margolis and Steinberg, is suitably modified to define an expansion for weakly left ample semigroups. We consider the class of FA-morphisms between weakly left ample semigroups and show that this expansion gives rise to a universal FA-morphism onto a weakly left ample semigroup. The methods used are necessarily different from the ones applied in the inverse case. We show how the inverse case can be deduced from this more general situation. [ABSTRACT FROM AUTHOR]

Published
2006
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47. PROPER COVERS OF AMPLE MONOIDS

Author

Fountain, John and Gomes, Gracinda M. S.

Abstract

Proper ample monoids are described by means of a certain category acted upon on both sides by a cancellative monoid. Making use of this characterization, we show that every ample monoid $S$ has a proper ample cover, which can be taken to be finite whenever $S$ is finite.

Published
2006

49. On a Class of Lattice Ordered Inverse Semigroups

Author

Gomes, Gracinda M. S., Giraldes, Emilia, and McAlister, Donald B.

Abstract

It is well known that the free group on a non-empty set can be totally ordered and, further, that each compatible latttice ordering on a free group is a total ordering. On the other hand, Saito^ has shown that no non-trivial free inverse semigroup can be totally ordered. In this note we show, however, that every free inverse monoid admits compatible lattice orderings which are closely related to the total orderings on free groups.These orderings are natural in the sense that the imposed partial ordering on the idempotents coincides with the natural partial ordering. For this to happen in a lattice ordered inverse semigroup, the idempotents must form a distributive lattice. The method of construction of the lattice orderings on free inverse monoids can be applied to show that naturally lattice ordered inverse semigroups with a given distributive lattice E of idempotents can have arbitrary Green's relation structure. Analogous results hold for naturally ∧-semilatticed inverse semigroups. In this case, there is no restriction on the semilattice E of idempotents.We also show that every compatible lattice ordering on the free monogenic inverse monoid is of the type considered here. This permits us to prove that there are precisely eight distinct compatible lattice orderings on this semigroup. They belong to two families, each of which contains four members, of conjuguate lattice orderings.

Published
2000
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50. Braids and factorizable inverse monoids

Author

Araujo, Isabel M, Branco, Mario J J, Fernandes, Vitor H, Gomes, Gracinda M S, Easdown, D, East, J, FitzGerald, DG, Araujo, Isabel M, Branco, Mario J J, Fernandes, Vitor H, Gomes, Gracinda M S, Easdown, D, East, J, and FitzGerald, DG

Abstract

What is the untangling effect on a braid if one is allowed to snip a string, or if two specified strings are allowed to pass through each other, or even allowed to merge and part as newly reconstituted strings? To calculate the effects, one works in an appropriate factorizable inverse monoid, some aspects of a general theory of which are discussed in this paper. The coset monoid of a group arises, and turns out to have a universal property within a certain class of factorizable inverse monoids. This theory is dual to the classical construction of fundamental inverse semigroups from semilattices. In our braid examples, we will focus mainly on the ``merge and part'' alternative, and introduce a monoid which is a natural preimage of the largest factorizable inverse submonoid of the dual symmetric inverse monoid on a finite set, and prove that it embeds in the coset monoid of the braid group.

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