Abelian Kernels of Monoids of Order-preserving Maps and of Some of Its Extensions

Delgado, Manuel; Fernandes, Vítor H.

doi:10.1007/s00233-003-0014-z

Cited by 14 publications

(17 citation statements)

References 19 publications

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“…Furthermore, when n is an even integer, since we have precisely that K Ab (PODI n ) = K Ab (POI n ) [24], as a corollary of Theorem 3.2, we can also state: Corollary 3.3. If n is an even integer then Ab (PODI n ) = n − 1.…”

Section: On Some Inverse Monoidsmentioning

confidence: 78%

“…The abelian kernels of the monoids PODI n and PORI n have been recently determined by the authors in [24] and we have not yet a complete solution for their abelian kernel lengths.…”

Section: On Some Inverse Monoidsmentioning

confidence: 93%

“…In particular, the abelian kernels of POI n and of POPI n were determined in [22] and the abelian kernels of PODI n , PORI n and I n in [24]. In [23] it is shown, in particular, that a monoid M whose idempotents commute is solvable exactly when all subgroups of M are solvable.…”

Section: On Some Transformation Monoidsmentioning

confidence: 99%

“…Since K Ab (PODI n ) ⊆ POI n and K Ab (PORI n ) ⊆ POPI n [24] (note that these inclusions were deduced by applying Theorem 1.2) and considering the equalities of Theorem 3.2 (observe that POI n is a submonoid of PODI n and PORI n ), we have the following bounds for the abelian kernel lengths of PODI n and PORI n :…”

Section: On Some Inverse Monoidsmentioning

confidence: 99%

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Abelian Kernels, Solvable Monoids and the Abelian Kernel Length of a Finite Monoid

Delgado

Fernandes

2004

Semigroups and Languages

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Section: On Some Inverse Monoidsmentioning

confidence: 78%

“…The abelian kernels of the monoids PODI n and PORI n have been recently determined by the authors in [24] and we have not yet a complete solution for their abelian kernel lengths.…”

Section: On Some Inverse Monoidsmentioning

confidence: 93%

Section: On Some Transformation Monoidsmentioning

confidence: 99%

Section: On Some Inverse Monoidsmentioning

confidence: 99%

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Abelian Kernels, Solvable Monoids and the Abelian Kernel Length of a Finite Monoid

Delgado

Fernandes

2004

Semigroups and Languages

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“…Together with Delgado the author [9,10] have computed the abelian kernels of the monoids POI n , POPI n , PODI n and PORI n . More recently, the same authors together with Cordeiro determined all relative abelian kernels of these four monoids [7].…”

Section: Introductionmentioning

confidence: 99%

On Divisors of Pseudovarieties Generated by Some Classes of Full Transformation Semigroups

Fernandes

2008

Algebra Colloq.

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Classical Finite Transformation Semigroups

Ganyushkin¹,

Mazorchuk²

2009

Algebra and Applications

210

199

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Algebra and Applications aims to publish well written and carefully refereed monographs with up-to-date information about progress in all fields of algebra, its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C*algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.Printed on acid-free paper Springer Science+Business Media springer.com PrefaceSemigroup theory is a relatively young part of mathematics. As a separate direction of algebra with its own objects, formulations of problems, and methods of investigations, semigroup theory was formed about 60 years ago.One of the main motivations for the existence of some mathematical theories are interesting and natural examples. For semigroup theory the obvious candidates for such examples are transformation semigroups. Various transformations of different sets appear everywhere in mathematics all the time. As the usual composition of transformations is associative, each set of transformations, closed with respect to the composition, forms a semigroup.Among all transformation semigroups one can distinguish three classical series of semigroups: the full symmetric semigroup T (M ) of all transformations of the set M ; the symmetric inverse semigroup IS(M ) of all partial (that is, not necessarily everywhere defined) injective transformations of M ; and, finally, the semigroup PT (M ) of all partial transformations of M . If M = {1, 2, . . . , n}, then the above semigroups are usually denoted by T n...

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Abelian Kernels of Monoids of Order-preserving Maps and of Some of Its Extensions

Abstract: We compute the abelian kernels of some monoids of partial transformations. More precisely, the cases of the symmetric inverse monoid, monoids of order-preserving or order-reversing transformations, so as monoids of orientation-preserving or orientationreversing transformations are treated.

Cited by 14 publications

References 19 publications

Abelian Kernels, Solvable Monoids and the Abelian Kernel Length of a Finite Monoid

Abelian Kernels, Solvable Monoids and the Abelian Kernel Length of a Finite Monoid

On Divisors of Pseudovarieties Generated by Some Classes of Full Transformation Semigroups

Classical Finite Transformation Semigroups

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