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Stojan M. Bogdanovi´c, PhD, full professor Faculty of Economics, University of Niˇs Miroslav D. Ciri´c´ , PhD, full professor Faculty of Sciences and Mathematics, University of Niˇs Zarkoˇ Lj. Popovi´c, PhD, associate professor Faculty of Economics, University of Niˇs SEMILATTICE DECOMPOSITIONS OF SEMIGROUPS first edition, 2011 Published by: Faculty of Economics, University of Niˇs For the publisher: Evica Petrovi´c,PhD, Dean Editor-in-Chief: Ljiljana Stankovi´c,PhD Reviewers: Petar Proti´c,PhD Predrag Stanimirovi´c,PhD Jelena Ignjatovi´c,PhD Proofreading: Marta Dimitrijevi´c,MA Printed by: M Kops Centar, Niˇs Circulation: 100 ISBN 978-86-6139-032-6 ⃝c 2011, Faculty of Economics, University of Niˇs All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invited, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publisher and the authors. This monograph was supported by Ministry of Education and Science, Republic of Serbia, Grant No. 174013. Stojan M. Bogdanovi´c Miroslav D. Ciri´c´ Zarkoˇ Lj. Popovi´c SEMILATTICE DECOMPOSITIONS OF SEMIGROUPS Niˇs,2011 Preface General decomposition problems hold a central place in the general struc- ture theory of semigroups, as they look for different ways to break a semi- group into parts, with as simple a structure as possible, in order to ex- amine these parts in detail, as well as the relationships between the parts within the whole semigroup. The main problem is to determine whether the greatest decomposition of a given type exists, the decomposition having the finest components, and to give a characterization and construction of this greatest decomposition. Another important issue is whether a given type of decomposition is atomic, in the sense that the components of the greatest decomposition of the given type cannot further be broken down by decom- position of the same type. In semigroup theory only five types of atomic decompositions are known so far. The atomicity of semilattice decomposi- tions was proved by Tamura [Osaka Math. J. 8 (1956) 243{261], of ordinal decompositions by Lyapin [Semigroups, Fizmatgiz, Moscow, 1960], of the so-called U-decompositions by Shevrin [Dokl. Akad. Nauk SSSR 138 (1961) 796{798], of orthogonal decompositions by Bogdanovi´cand Ciri´c[Israel´ J. Math 90 (1995) 423{428], whereas the atomicity of subdirect decompositions follows from a more general result of universal algebra proved by Birkhoff [Bull. AMS 50 (1944) 764{768]. Semilattice decompositions of semigroups were first defined and studied by A. H. Clifford [Annals of Math. 42 (1941) 1037{1049]. Later T. Tamura and N. Kimura [Kodai Math. Sem. Rep. 4 (1954) 109{112] proved the existence of the greatest semilattice decompo- sition of an arbitrary semigroup, and as we have already noted, while T. Tamura [Osaka Math. J. 8 (1956) 243{261] proved the atomicity of semilat- tice decompositions. The theory of the greatest semilattice decompositions of semigroups has been developed from the middle of the 1950s to the middle i ii PREFACE of the 1970s by T. Tamura, M. S. Putcha, M. Petrich, and others. For a long time after that there were no new results in this area. In the mid of 1990s, the authors of this book initiated the further development of this theory by introducing completely new ideas and methodology. The purpose of this book is to give an overview of the main results on semilattice decompositions of semigroups which appeared in the last 15 years, as well as to connect them with earlier results. The structure of the book is as follows. The first three chapters of the book provide an introduction to the basic concepts of semigroup theory, var- ious types of regularity and the concepts of simple, 0-simple, Archimedean and 0-Archimedean semigroups. Chapter 4 develops the general theory of the greatest semilattice decompositions of semigroups, using the methodol- ogy that was built by the authors. This methodology is based on the compu- tation of the principal radicals of a semigroup, which is an iterative process that, in general, may consist of infinitely many iterations. For this reason, later this chapter discusses the various cases where the greatest semilattice decompositions can be achieved by methods that involve only finitely many iterations. The first effective construction of the smallest semilattice congruence on a semigroup, provided by T. Tamura [Semigroup Forum 4 (1972) 255{ 261], was based on the arrow relation !, which was defined as a natural generalization of the division relation. Namely, two elements a and b of a semigroup are said to be in the relation !, written as a ! b, if the element b divides some power of the element a. If each pair of elements of a semigroup is in that relation, then this semigroup is said to satisfy the famous Archimedean property, which Archimedes proved for natural numbers, and such a semigroup is called an Archimedean semigroup. In the above mentioned paper, T. Tamura proved that the smallest semilattice congruence on a semigroup can be constructed as the symmetric opening of the transitive closure of the arrow relation, whereas M. S. Putcha [Trans. Amer. Math. Soc. 189 (1974), 93{106] showed that these two operations can be permuted, i.e., the smallest semilattice congruence can be computed as the transitive closure of the symmetric opening of the arrow relation. In Chapter 4 the authors discuss various situations where the transitive closure of the arrow relation can be computed in a finite number of steps, and in Chapter 5 they consider the situation when the arrow relation is transitive. PREFACE iii Semigroups with the latter property are actually semigroups that can be represented as a semilattice of Archimedean semigroups. Chapter 5 also deals with various special types of semilattices of Archimedean semigroups. A particular case of Archimedean semigroups are semigroups in which each element divides a fixed power of any other element, and such semigroups are called k-Archimedean. The semilattices of k-Archimedean semigroups and many of their special cases are studied in Chapter 6. A very important special case of semilattices of Archimedean semigroups are semilattices of completely Archimedean semigroups, or equivalently, se- milattices of nil-extensions of completely simple semigroups. At a scientific conference held back in 1977, L. N. Shevrin announced that a semigroup can be decomposed into a semilattice of completely Archimedean semigroups if and only if each of its elements has a regular power, and each of its regular elements is completely regular (i.e., belongs to a subgroup of this semigroup). However, this result along with other related results was published with proof 17 years later [Mat. Sbornik 185 (8) (1994) 129{160, 185 (9) (1994) 153{176]. In the meantime, other authors have studied these decompositions building their own methodology, for example J. L. Galbiati and M. L. Veronesi [Rend. Ist. Lomb. Cl. Sc. (A) 116 (1982) 180{189; Riv. Mat. Univ. Parma (4) 10 (1984) 319{329], and others. The first author of this book began his research in this area in 1985, and later the other two authors joined him. In a series of papers, the authors of this book built their own methodology, which not only led to the same results announced by L. N. Shevrin, but also provided some significant improvements. A complete theory of the decompositions of a semigroup into a semilattice of completely Archimedean semigroups was presented for the first time in the book by the first two authors [Semigroups, Prosveta, Niˇs,1993]. Chapter 7 of this book outlines not only these results, but also many results obtained later. The authors wish to express their gratitude to Professors Petar Proti´c, Predrag Stanimirovi´cand Jelena Ignjatovi´c,for their valuable comments and suggestions. Niˇs,On Saint Petka, 2011 Authors iv PREFACE Contents Preface i Contents v 1 Introduction 1 1.1 The Definition of a Semigroup . 1 1.2 Semigroups of Relations and Mappings . 7 1.3 Congruences and Homomorphisms . 12 1.4 Maximal Subgroups and Monogenic Semigroups . 17 1.5 Ordered Sets and Lattices . 20 1.6 Ideals . 30 1.7 Ideal and Retractive Extensions of Semigroups . 39 1.8 Green's Relations . 46 2 Regularity on Semigroups 53 2.1 π-regular Semigroups . 54 2.2 Completely π-regular Semigroups . 59 2.3 The Union of Groups . 64 2.4 π-inverse Semigroups . 67 2.5 Quasi-regular Semigroups . 74 2.6 Idempotent Generated Semigroups . 77 2.7 Left Regular Semigroups . 80 v vi CONTENTS 3 (0)-Archimedean Semigroups 85 3.1 Completely 0-simple Semigroups . 86 3.2 0-Archimedean Semigroups . 96 3.3 Archimedean Semigroups . 104 3.4 Semigroups in Which Proper Ideals are Archimedean . 111 4 The Greatest Semilattice Decompositions of Semigroups 119 4.1 Principal Radicals and Semilattice Decompositions of Semigroups . 121 4.2 Semilattices of σn-simple Semigroups . 132 4.3 Semilattices of λ-simple Semigroups . 139 4.4 Chains of σ-simple Semigroups . 146 4.5 Semilattices of σbn-simple Semigroups . 149 4.6 Semilattices of λb-simple Semigroups . 153 4.7 The Radicals of Green's J -relation . 155 5 Semilattices of Archimedean Semigroups 159 5.1 The General Case . 160 5.2 Semilattices of Hereditary Archimedean Semigroups . 175 5.3 Semilattices of Weakly Left Archimedean Semigroups . 179 5.4 Semilattices of Left Completely Archimedean Semigroups . 186 5.5 Bands of Left Archimedean Semigroups . 188 6 Semilattices of k-Archimedean Semigroups 205 6.1 k-Archimedean Semigroups . 207 6.2 Bands of Jk-simple Semigroups . 211 6.3 Bands of Lk-simple Semigroups . 217 6.4 Bands of Hk-simple Semigroups .
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