81 research outputs found
On separability finiteness conditions in semigroups
Funding: The first author is grateful to EPSRC for financial support. The second author is grateful to the School of Mathematics and Statistics of the University of St Andrews for financial support.Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by SchĂŒtzenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H -classes, we investigate whether it has one of these properties if and only if all its SchĂŒtzenberger groups have the property.Publisher PDFPeer reviewe
Languages associated with saturated formations of groups
In a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.The authors are supported by Proyecto MTM2010-19938-C03-01 from MICINN (Spain). The first author acknowledges support from MEC. The second author is supported by the project ANR 2010 BLAN 0202 02 FREC. The third author was supported by the Grant PAID-02-09 from Universitat PolitĂšcnica de ValĂšncia
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
The Covering Problem
An important endeavor in computer science is to understand the expressive
power of logical formalisms over discrete structures, such as words. Naturally,
"understanding" is not a mathematical notion. This investigation requires
therefore a concrete objective to capture this understanding. In the
literature, the standard choice for this objective is the membership problem,
whose aim is to find a procedure deciding whether an input regular language can
be defined in the logic under investigation. This approach was cemented as the
right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on
first-order logic and has been in use since then. However, membership questions
are hard: for several important fragments, researchers have failed in this
endeavor despite decades of investigation. In view of recent results on one of
the most famous open questions, namely the quantifier alternation hierarchy of
first-order logic, an explanation may be that membership is too restrictive as
a setting. These new results were indeed obtained by considering more general
problems than membership, taking advantage of the increased flexibility of the
enriched mathematical setting. This opens a promising research avenue and
efforts have been devoted at identifying and solving such problems for natural
fragments. Until now however, these problems have been ad hoc, most fragments
relying on a specific one. A unique new problem replacing membership as the
right one is still missing. The main contribution of this paper is a suitable
candidate to play this role: the Covering Problem. We motivate this problem
with 3 arguments. First, it admits an elementary set theoretic formulation,
similar to membership. Second, we are able to reexplain or generalize all known
results with this problem. Third, we develop a mathematical framework and a
methodology tailored to the investigation of this problem
A generic characterization of generalized unary temporal logic and two-variable first-order logic
We investigate an operator on classes of languages. For each class , it
outputs a new class associated with a variant of two-variable
first-order logic equipped with a signature built from . For , we get the variant equipped with the linear
order. For , we get the variant
, which also includes the successor. If consists of all Boolean
combinations of languages where is a letter, we get the variant
, which also includes ``between relations''. We prove a generic
algebraic characterization of the classes . It smoothly and
elegantly generalizes the known ones for all aforementioned cases. Moreover, it
implies that if has decidable separation (plus mild properties), then
has a decidable membership problem.
We actually work with an equivalent definition of \fodc in terms of unary
temporal logic. For each class , we consider a variant of unary
temporal logic whose future/past modalities depend on and such that . Finally, we also characterize and , the
pure-future and pure-past restrictions of . These characterizations as
well imply that if \Cs is a class with decidable separation, then and
have decidable membership
Todd-Coxeter methods for inverse monoids
Let P be the inverse monoid presentation (X|U) for the inverse monoid M, let Ï be the set of generators for a right congruence on M and let u Đ M. Using the work of J. Stephen [15], the current work demonstrates a coset enumeration technique for the R-class Rᔀ similar to the coset enumeration algorithm developed by J. A. Todd and H. S. M. Coxeter for groups. Furthermore it is demonstrated how to test whether Rᔀ = Rᔄ, for u, v Đ M and so a technique for enumerating inverse monoids is described. This technique is generalised to enumerate the H-classes of M. The algorithms have been implemented in GAP 3.4.4 [25], and have been used to analyse some examples given in Chapter 6. The thesis concludes by a related discussion of normal forms and automaticity of free inverse semigroups
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