On the semigroups of partial one-to-one order-decreasing finite transformations

Umar, Abdullahi

doi:10.1017/s0308210500025737

Cited by 25 publications

(23 citation statements)

References 9 publications

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“…Every finite inverse semigroup S is embeddable in I n , the analogue of Cayley's theorem for finite groups. Thus, just as the study of symmetric, alternating and dihedral groups has made a significant contribution to group theory, so has the study of various subsemigroups of I n , see for example [3,5,6,10,14,19,20].…”

Section: Introductionmentioning

confidence: 99%

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Combinatorial Results for Certain Semigroups of Order-Decreasing Partial Isometries of a Finite Chain

Al-Kharousi¹,

Kehinde²,

Umar³

2018

Jour. Alg. Num. Theory Adv. Appl.

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Section: Introductionmentioning

confidence: 99%

“…Remark 1.2 DDP n = DP n ∩ I − n and ODDP n = ODP n ∩ I − n , where I − n is the semigroup of partial one-to-one order-decreasing transformations of X n [19].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Results for Certain Semigroups of Order-Decreasing Partial Isometries of a Finite Chain

Al-Kharousi¹,

Kehinde²,

Umar³

2018

Jour. Alg. Num. Theory Adv. Appl.

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“…Also (iα −1 )αξ = iα m and iα m−1 , iα m+l−1 / ∈ dom(αξ ). Thus, by (11) and (12), From the proof of this lemma we can immediately deduce Corollary 4.6 Let be an orbit of α ∈ SI n and ξ = i j ∈ Q E 1 ∩ J n−1 with i, j ∈ . Then…”

Section: Lemma 45 If Is An Orbit Of α ∈ Si N and I J ∈ Then May Bmentioning

confidence: 83%

“…Despite the appearance of the books of Lipscomb [9], and Ganyushkin and Mazorchuk [1] as well as a handful of papers (for example, [2,3,11]), the study of I n is still in its infancy relative to that of T n . The semigroup SI n = I n \S n , of all strictly partial one-to-one maps on X n (being an inverse semigroup) is not generated by its idempotent elements.…”

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confidence: 99%

“…In [4], it was shown that, when n is even, SI n is generated by its nilpotent elements (elements α satisfying α m = 0, for some positive integer m) and that the nilpotent elements only generate a proper subsemigroup of SI n when n is odd. In this paper, we borrow the concept of quasi-idempotent elements (elements α satisfying α 4 = α 2 ) introduced by Umar [11] and show that the semigroup SI n is generated by a set of quasi-idempotent elements regardless of whether n is even or odd. We then find local and global depths for SI n .…”

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confidence: 99%

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Products of quasi-idempotents in finite symmetric inverse semigroups

Garba

Imam

2015

Semigroup Forum

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Let X n = {1, 2, . . . , n} and let S n and I n be the symmetric group and symmetric inverse semigroup on X n respectively. In this paper, we show that the semigroup SI n = I n \S n , of all strictly partial one-to-one maps on X n , is generated by quasi-idempotent elements (non-idempotent elements α satisfying α 4 = α 2 ). Also, we give the least upper bound for the minimum length factorisation of each α ∈ SI n into a product of quasi-idempotents.

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Classical Finite Transformation Semigroups

Ganyushkin¹,

Mazorchuk²

2009

Algebra and Applications

209

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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On the semigroups of partial one-to-one order-decreasing finite transformations

Cited by 25 publications

References 9 publications

Combinatorial Results for Certain Semigroups of Order-Decreasing Partial Isometries of a Finite Chain

Combinatorial Results for Certain Semigroups of Order-Decreasing Partial Isometries of a Finite Chain

Products of quasi-idempotents in finite symmetric inverse semigroups

Classical Finite Transformation Semigroups

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