Classical Finite Transformation Semigroups

Abstract

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...

Maximal subsemigroups containing a particular semigroup

“…The maximal subsemigroups of T (X) containing the symmetric group Sym(X) of all bijective mappings on an infinite set X are already known. They were determined by G. P. Gavrilov (X is countable) and by M. Pinsker (any infinite set X) characterizing maximal clones ( [3], [6], [9]). …”

“…the cardinality of im α, is denoted by rank(α) Moreover, we put [1] for α, β ∈ T (X). For more background in the theory of transformation semigroups see [3] and [8].…”

Two generalisations of the symmetric inverse semigroups

“…Beside pure combinatorial interest in this semigroup, it plays an important role for the class of all inverse semigroups similar to that played by the symmetric group S X for the class of all groups. For some facts about semigroup and combinatorial properties of I X we refer the reader to [5].…”

Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

“…In Section 3 we show how the latter model can be used to construct combinatorial Gelfand models for semigroup algebras of those finite semigroups, for which each trace of a regular D-class is an inverse semigroup in which maximal subgroups are direct sums of symmetric groups. Examples of such semigroups include the symmetric inverse semigroup (see [GM3,2.5]), the dual symmetric inverse semigroup (see [FL]), and the maximal factorizable subsemigroup in the dual symmetric inverse semigroup (see [FL]). Another, rather surprising, natural example is the factor power of the symmetric group (see [GM1]), which, in particular, is not even regular.…”