Nabilah Hani Abu-Ghazalh PhD thesis

155 Pages·2013·0.93 MB·English

FINITENESS CONDITIONS FOR UNIONS OF
SEMIGROUPS
Nabilah Hani Abu-Ghazalh
A Thesis Submitted for the Degree of PhD
at the
University of St Andrews
2013
Full metadata for this item is available in
Research@StAndrews:FullText
at:
http://research-repository.st-andrews.ac.uk/
Please use this identifier to cite or link to this item:
http://hdl.handle.net/10023/3687
This item is protected by original copyright
This item is licensed under a
Creative Commons License
FINITENESS CONDITIONS FOR UNIONS OF
SEMIGROUPS
Nabilah Hani Abu-Ghazalh
Thesis submitted to the University of St Andrews
for the degree of Doctor of Philosophy
March, 2013

CONTENTS
DECLARATION 6
ACKNOWLEDGMENTS 8
ABSTRACT 10
PREFACE 11
1 SEMIGROUPTHEORYPRELIMINARIES 15
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Basicsandmonogenicsemigroups . . . . . . . . . . . . . . . . . . . . 15
1.3 Relations,congruencesandhomomorphisms . . . . . . . . . . . . . . 16
1.4 IdealsandGreenrelations . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Classesandconstructions . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Freesemigroupsandpresentations . . . . . . . . . . . . . . . . . . . . 23
1.7 Residualfiniteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
I DISJOINT UNIONS OF A SMALL NUMBER OF COPIES
OF A (SEMI)GROUP 26
2 GENERALDISJOINTUNIONSOFTWOSEMIGROUPS 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Syntacticcongruencecondition . . . . . . . . . . . . . . . . . . . . . . 29
2.3 (Left,Right)idealsconditions . . . . . . . . . . . . . . . . . . . . . . . 31
3 CLASSIFICATIONOFDISJOINTUNIONSOFTWOANDTHREECOPIES
OFAGROUP 35
1
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Unionsoftwocopiesofagroup . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Unionsofthreecopiesofagroup . . . . . . . . . . . . . . . . . . . . . 36
3.4 Finitenessconditionsfordisjointunionsofgroups . . . . . . . . . . . 41
4 CLASSIFICATION OF DISJOINT UNIONS OF TWO COPIES OF THE
FREEMONOGENICSEMIGROUP 42
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Classificationofsemigroupswhicharedisjointunionsoftwocopies
ofthefreemonogenicsemigroup . . . . . . . . . . . . . . . . . . . . . 43
4.3 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Comparisonbetweendisjointunionsoftwocopiesofthefreemono-
genicsemigroupandtwocopiesoftheinfinitecyclicgroup . . . . . . 49
5 CLASSIFICATIONOFDISJOINTUNIONSOFTHREECOPIESOFTHE
FREEMONOGENICSEMIGROUP 51
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Classificationofsemigroupswhicharedisjointunionsofthreecopies
ofthefreemonogenicsemigroup . . . . . . . . . . . . . . . . . . . . . 55
5.3 Semigroupsofthesametype . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Comparisonbetweendisjointunionsofthreecopiesofthefreemono-
genicsemigroupandthreecopiesoftheinfinitecyclicgroup . . . . . 77
5.5 Disjointunionofaninfinitecyclicgroupandafreemonogenicsemi-
group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 CLASSIFICATION OF DISJOINT UNIONS OF TWO COPIES OF THE
FREESEMIGROUPOFRANKTWO 89
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Sixbalancedsemigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3 Preliminary,technicalresults . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Classificationofbalancedsemigroups . . . . . . . . . . . . . . . . . . 105
6.5 Residualfiniteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
II DISJOINT UNIONS OF ANY NUMBER OF COPIES OF
2
THE FREE MONOGENIC SEMIGROUP 113
7 RECTANGULAR BANDS OF FINITELY MANY COPIES OF THE FREE
MONOGENICSEMIGROUP 114
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2 Finitepresentability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3 Specialexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.4 Residualfiniteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8 FINITENESSCONDITIONSFORDISJOINTUNIONSOFFINITELYMANY
COPIESOFTHEFREEMONOGENICSEMIGROUP 122
8.1 Preliminaries: multiplicationandarithmeticprogressions . . . . . . . 122
8.2 Finitepresentability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.3 Residualfiniteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.4 Hopficity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.5 Commutativesemigroups . . . . . . . . . . . . . . . . . . . . . . . . . 134
9 DECIDABILITYFORDISJOINTUNIONSOFFINITELYMANYCOPIES
OFTHEFREEMONOGENICSEMIGROUP 137
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.2 Wordproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9.3 Subsemigroupmembershipproblem . . . . . . . . . . . . . . . . . . . 141
10 CONCLUDINGREMARKSANDCONJECTURES 147
Bibliography 150
3
LIST OF TABLES
5.1 The nine types of semigroups which are disjoint unions of three
copies of the free monogenic semigroup (up to isomorphism and
anti-isomorphism) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Generalforbiddentypes . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Forbiddentypesinthefamily (a,a,∗,∗,∗,∗) . . . . . . . . . . . . . . 57
5.4 Forbiddentypesinthefamily (b,a,∗,∗,∗,∗) . . . . . . . . . . . . . . 59
5.5 Themultiplicationof xj and y where x, y ∈ {a,b,c} inthepresenta-
tion (cid:104)a,b,c|ab = ai,ba = ai,ac = c2,ca = a2,bc = ci,cb = ai(cid:105) . . . . . . 67
5.6 Themultiplicationof xi and y where x, y ∈ {a,b,c} inthepresenta-
tion (cid:104)a,b,c|ab = b2,ba = a2,ac = c2,ca = b2,bc = c2,cb = a2(cid:105) . . . . . 69
5.7 Themultiplicationof xi and y where x, y ∈ {a,b,c} inthepresenta-
tion (cid:104)a,b,c|ab = b2,ba = a2,ac = c2,ca = a2,bc = c2,cb = a2(cid:105) . . . . . 71
5.8 Themultiplicationof xj and y where x, y ∈ {a,b,c} inthepresenta-
tion (cid:104)a,b,c|ab = b2,ba = a2,ac = bi,ca = ai,bc = ai,cb = bi(cid:105) . . . . . 73
6.1 Themultiplicationofxi andyj wherex, y ∈ {a,b,c,d}inthepresen-
tation (cid:104)a,b,c,d | ac = c2, ca = a2, bc = dc, cb = ab, ad = cd, da =
ba, bd = d2, db = b2 (cid:105) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Themultiplicationofxi andyj wherex, y ∈ {a,b,c,d}inthepresen-
tation (cid:104)a,b,c,d | ac = d2, ca = b2, bc = cd, cb = ba, ad = dc, da =
ab, bd = c2, db = a2 (cid:105) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.1 Themultiplicationofarectangularband . . . . . . . . . . . . . . . . . 114
k+1
(cid:70)
7.2 Therectangularbandsemigroup S . . . . . . . . . . . . . . . . . . 115
i
i=1
k+1 h k+1
(cid:70) (cid:70) (cid:70)
7.3 Asplitofthesemigroup S to S and S . . . . . . . . . . . 115
i i i
i=1 i=1 i=h+1
4
k+1
(cid:70)
7.4 Theonerowrectangularbandsemigroup S . . . . . . . . . . . . 116
i
i=1
k+1 k
(cid:70) (cid:70)
7.5 Asplitoftheonerowsemigroup S to S and S . . . . . . . 116
i i k+1
i=1 i=1
7.6 The multiplication on the subsemigroup S of the semigroup free
P
4
productoftwotrivialsemigroups . . . . . . . . . . . . . . . . . . . . . 117
7.7 Rectangularbandoffourcopiesofthefreemonogenicsemigroup . . 119
5
DECLARATION
I, Nabilah Hani Abu-Ghazalh, hereby certify that this thesis, which is approxi-
mately26.000wordsinlength,hasbeenwrittenbyme,thatitistherecordofwork
carried out by me and that it has not been submitted in any previous application
for a higher degree. I was admitted as a research student in January 2009 and as
a candidate for the degree of Doctor of Philosophy in September 2009; the higher
study for which this is a record was carried out in the University of St Andrews
between2009and2012.
Signature: Name: NabilahAbu-Ghazalh Date
I, Nik Rusˇkuc, hereby certify that the candidate has fulfilled the conditions of the
Resolution and Regulations appropriate for the degree of Doctor of Philosophy in
theUniversityofStAndrewsandthatthecandidateisqualifiedtosubmitthisthe-
sisinapplicationforthatdegree.
Signature: Name: NikRusˇkuc Date
In submitting this thesis to the University of St Andrews we understand that we
are giving permission for it to be made available for use in accordance with the
regulations of the University Library for the time being in force, subject to any
copyright vested in the work not being affected thereby. We also understand that
the title and the abstract will be published, and that a copy of the work may be
madeandsuppliedtoanybonafidelibraryorresearchworker,thatmythesiswill
beelectronicallyaccessibleforpersonalorresearchuseunlessexemptbyawardof
an embargo as requested below, and that the library has the right to migrate my
thesis into new electronic forms as required to ensure continued access to the the-
6
sis. Wehaveobtainedanythird-partycopyrightpermissionsthatmayberequired
in order to allow such access and migration, or have requested the appropriate
embargo below. The following is an agreed request by candidate and supervisor
regardingtheelectronicpublicationofthisthesis:
Access to Printed copy and electronic publication of thesis through the University
ofStAndrews.
Signature: Name: NabilahAbu-Ghazalh Date
Signature: Name: NikRusˇkuc Date
7