Special Classes of Semigroups

266 Pages·2001·10.322 MB·English

by  Attila Nagy (auth.)

Special Classes ofSemigroups
Advances in Mathematics
VOLUME I
Series Editor:
J.Szep,Budapest University 0/Economics,Hungary
AdvisoryBoard:
G.Erjaee, Shira: University, Iran
W.Fouche,University 0/South Africa, South Africa
P.Grillet,Tulane University, U.S.A.
HJ.Hoehnke, Germany
F.Szidarovszky, University 0/Arizona, U.S.A.
P.Zecca,Universitädi Firenze, Italy
Special Classes
of Semigroups
by
ATTILANAGY
DepartmentofAlgebra,
Institute ofMathematics,
BudapestUniversityofTechnologyand
Economics,Hungary
Springer-Science+Business Media, B.V.
AC.I.P.Cataloguerecord forthis bookisavailable fromtheLibraryofCongress.
Printed onacid-freepaper
AllRightsReserved
ISBN978-1-4419-4853-3 ISBN978-1-4757-3316-7(eBook)
DOI 10.1007/978-1-4757-3316-7
©2001SpringerScience+BusinessMediaDordrecht
Originallypublished byKluwerAcademic Publishers in2001.
Softcoverreprintofthehardcover 1stedition2001
Nopartofthematerial protected bythiscopyright notice maybereproducedor
utilized inanyform orbyanymeans,electronic ormechanical,
including photocopying,recording orbyanyinformation storageand
retrievalsystem, without written permission from thecopyrightowner
Contents
Preface vii
1 Preliminaries 1
2 Putcha semigroups 35
3 Commutative semigroups 43
4 Weakly commutative semigroups 59
5 ~,I:--, 1l-commutative semigroups 69
6 Conditionally commutative semigroups 77
7 'R.C-commutative semigroups 93
8 Quasi commutative semigroups 109
9 Medial semigroups 119
10 Right commutative semigroups 137
11 Externally commutative semigroups 175
12 E-m semigroups, exponential semigroups 183
13 WE-m semigroups 199
14 Weakly exponential semigroups 215
15 (m,n)-commutative semigroups 223
16 n(2)-permutable semigroups 247
Bibliography 259
Index 267
v
Preface
Semigroups are generalizations ofgroups and rings. A group is a semigroup in
which the operation is invertible; a ring is a multiplieative semigroup in wich
the operation together with an additive operation satisfies eertain eonditions.
In the beginning of the development of semigroup theory investigations were
strongly motivated by this fact. Semigroups in which every element has an in
versewereinfoeus, and the results ofring theory wereadapted for semigroups.
In algebra, eongruenees play a eentral role. In this respect, there is a differ
enee between semigroups and groups or rings. The eongruenees ofa group are
uniquely determined by its normal subgroups, and there isa bijeetion between
the eongruenees and the ideals ofa ring. In semigroup theory the situation is
more eomplieated. Although an ideal of a semigroup definesa special congru
enee, there are no subsemigroups which uniquely determine the eongruenees of
semigroups. This problem envolves many diffieulties. Thus semigroup theory
has developed special methods and new semigroup classes have eome into the
center ofinterest.
In semigroup theory there are eertain kinds of band deeompositions which
are veryuseful inthestudyofthe struetureofsemigroups. Thereisa numberof
special semigroup classes in which these deeompositions ean be used very suc
eessfully,beeause the semigroups belonging to them are deeomposable into spe
cial bands of left arehimedean or right arehimedean, archimedean semigroups.
The strueture ofthese different types ofarchimedean semigroups is thorougWy
studied in these semigroup classes. In this book, wefoeusour attention on such
classes of semigroups. Some of them are partially diseussed in earlier books,
but in the last thirthy years new semigroup classes have appeared and a fairly
large body ofmaterial has been published on them. In this book we provide a
systematie review on this subject.
In the first ehapter ofthe book wepresent notions and results ofsemigroup
theory needed in the sequel. This chapter also contains theorems and lern
mas (with proof) whieh are used throughout the book. The other ehapters are
devoted to special semigroup classes. These are Putcha semigroups, commuta
tive semigroups, weakly eommutative semigroups,'R.-eommutativesemigroups,
.L:-eommutativesemigroups, 1l-eommutativesemigroups, eonditionally commu
tativesemigroups,R.C-eommutativesemigroups,quasicommutativesemigroups,
medialsemigroups,right eommutativesemigroups,externallyeommutativesemi
groups, Ern semigroups, exponential semigroups, WE-m semigroups, weakly
vii
viii PREFACE
exponential semigroups, (m,n)-commutative semigroups and n(2)-permutable
semigroups. In any ofthese semigroup classes we deal with different kinds of
band decompositions, describe the structure of simple semigroups and that of
archimedean semigroups, characterize regular semigroups, inverse semigroups,
study the embedding of semigroups into groups and into semigroups which
are unions of groups, construct least left (right) separative and weakly sep
arative congruences, determine subdirect irreducible semigroups and describe
semigroups whose lattice ofcongruences is a chain with respect to inc1usion.
In this book we also present theorems stated and proved in other books.
Othertheorems, lemmas and corollaries are fuHy proved. In general, wepresent
the original proofs, but in a number ofcases wegivea new and shorter one.
Finally, I wouldliketo express my heartythanks to Professor Jeno Szepfor
hisassistanceineveryphase ofwritingthisbook. Iwouldalsoliketo thank Mrs.
Eva Nemetn for helping mein preparingthe carnera-redyversion ofthe LaTeX
file. I further acknowledge the encouragement and support ofthe publisher in
producingthe book.
This work was supported by the Hungarian NFSR grant NoT029525.
Budapest, 2000.
Attila Nagy
Chapter 1
Preliminaries
In this chapter we present those basic notions and results ofsemigroup theory
which are used in this book. This chapter contains further theorems and lem
mas, There are several assertions corresponding to different semigroup classes
examined in this book whose proofs are similar to each other and based on
common ideas. The common parts ofthese proofs are formulated as theorems
and lemmas,and they are presented and proved in this chapter.
Semigroups
Definition 1.1 Let S be a nonempty set. Ey a binary operation on S we mean
afunetion */rom S x S into S. The image in S of the elemenis (a,b) E S x S
* *
is denoted by a b. Frequently, we write ab for a b,
Definition 1.2 A binary operation on aset S is said tobeassociativeif a(bc) =
(ab)c is satisfied for all a,b,c E S. If ab= ba holds for every a,bE S then we
say that tlie operation is commutative.
Definition 1.3 A set together with an associative binary operation is called
a semigroup. A semigroup having only one element is said to be trivial. A
semigroup is said to be acommutativesemigroup if theoperationis commutative.
Subsemigroups
Definition 1.4 A nonempty subsei A ofasemigroup S is calledasubsemigroup
ofS if A is closed under the operation, that is, ab E A for every a,bE A.
Definition 1.5 A subset X of a semigroup S is called a set of generators of
S (or S is generated by X) if, for every element sES, there are elements
Xl,•••,Xn E X such that s = Xl " 'Xn' In such a case, we write S = (X). A
semigroup is said to befinitely generated if it has afiniteset of generators. We
1
2 CHAPTER 1. PRELIMINARIES
say that a semigroup is a cyclic semigroup if it is generated by a single elemeni,
An element a ofa semigroup S is called periodic ifihe cyclic subsemigroup (a)
of S generated by a is finite. A semigroup is called a periodic semigroup if its
every element is periodic.
Definition 1.6 We say that a semigroup S has the permutation property P
n
if, for every sequence (Xl,•••,Xn) of elements of S, there is a non-identity
permutation(J' ofthesei{I, 2,...,n} such that XlX2 •••Xn = Xu(l)Xu(2) •••Xu(n)'
We say that a semigroup has the permutation property if it has thepermutation
property P for some positive integer n :2: 2.
n
Theorem 1.1 ([84J) A finitely generated semigroup is finite ifand only ifit is
periodic and has the permutation property.
Free semigroups
Definition 1.7 Let X be a non-empty set and let FX denote the sei ofallfinite
sequences ofelements ofX. If(Xl,,'" Xn) and (Yl,'" ,Ym) are elemenis of:FX
then we define their produci by simple juxtaposition:
this product is associative. The semigroup :FX is called the free semigroup over
the sei X. The elemenis of:FX is called words. As (Xl,...,Xn) = (xt}...(Xn)I
the sei X is a sei ofgenerators of:Fx .
Identities
Definition 1.8 An elemeni e of a semigroup S is called a left (right) identity
elemeni ofS ifea = a (ae = a) holds for every a ES. We say that e E S is an
identity element ofa scmigroup S ife is both a left and a right identity elemeni
ofS.
It is easy to see that every semigroup has at most one identity element.
Moreover,ifa semigrouphasa right identity element anda left identityelement
then it contains an identity element.
Definition 1.9 A semigroup containing an identity element is called a monoid.
If S is a semigroup then let SI denote the semigroup S U{I} arising from
S by the adjunetion of an identity element 1 unless S already has an identity
element, in which case SI = S.