Elements of compact semigroups:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Columbus, Ohio
Merrill
1966
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 384 S. |
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| 245 | 1 | 0 | |a Elements of compact semigroups |c Karl Heinrich Hofmann ; Paul S. Mostert |
| 264 | 1 | |a Columbus, Ohio |b Merrill |c 1966 | |
| 300 | |a XIII, 384 S. | ||
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| adam_text | Table of Contents
INTRODUCTION. 1
Chapter A. PRELIMINARIES. 11
Section 1. The minimal ideal. 11
Review of nets. Definition of a topological semigroup. Sub
semigroups, ideals, left and right ideals. The closure of a subgroup
in a compact semigroup. The Rees product. Paragroups, sandwich
functions, completely simple compact semigroups. Rectangular
semigroups. The existence of minimal ideals. Each compact semi¬
group contains an idempotent. Elementary properties of monothetic
semigroups. The swelling lemma. The partial order of idempotents.
Primitive idempotents. The minimal ideal is a paragroup.
Section 2. Homomorphisms and congruences. 17
Normality, normalizer, centralizer, center. The group of units.
If xy is a unit, then x and y are units. Homomorphisms, mono
morphisms, surmorphisms, endomorphisms, automorphisms, iso¬
morphisms. Inner automorphisms; Clifford Miller endomorphisms.
The minimal ideal maps onto the minimal ideal. Congruences.
Quotient semigroups. The kernel congruence. The monotone light
factorisation. The second isomorphy theorem for semigroups. The
congruence relation generated by a subset of 5 x S. Congruence
relations reducing the minimal ideal. The function £. The natural
retraction onto a group in the minimal ideal. Surmorphisms of
vii
viii TABLE OF CONTENTS
semigroups onto groups. The Rees quotient. The semigroups H,
H*» H* The definition of a one parameter semigroup. Two
special congruences. Factoring the group of units. Split extensions.
Section 3. Ideals and quasi orders. 27
The filterbasis of complements of proper ideals. The maximal
ideal contained in a neighborhood of the minimal ideal. Connec¬
tivity of ideals. The standard quasi orders, Green s relations.
Elementary properties of Green s relations. Homomorphisms and
Green s relations. Idempotents and Green s relations. Green s
relations and congruences. Totally ordered ^ decomposition space.
Section 4. The Schutzenberger group and regular ^ classes. 34
Actions of semigroups. The Schutzenberger group. An J^ class
in the normalizer of the group of units. Decomposition of regular
^ classes. Fibering regular ^ classes. Green s classes and minimal
idempotents above.
Section 5. Rees products. 40
Various lemmas about a particular kind of Rees product with
totally ordered ^ class space. Summary.
Section 6. Clifford semigroups. 43
Definition of Clifford semigroups, semilattices. The ^ relation
is a congruence in Clifford semigroups. The space of cross sections
in a Clifford semigroup. Clifford semigroups which have 2 ^ classes.
Clifford semigroups with totally ordered ^ class space. A certain
operation on the space of cross sections.
Section 7. The semigroup of compact sets. 47
The definition of the semigroup of compact sets. Certain sub
semigroups of this semigroup.
Section 8. Projectire limits. 48
The definition of a projective limit. Representation of compact
semigroups as projective limits. Metric semigroups. Every compact
semigroup is a projective limit of metric ones with subinvariant
metrics. Certain groups of isometrics. Every totally disconnected
compact semigroup is a projective limit of finite ones.
Section 9. Cohomology properties of compact semigroups. 52
The cohomology is carried by the minimal ideal. The space of
idempotents in the minimal ideal is weakly contractible. The coho¬
mology is carried by any maximal group of the minimal ideal.
A lemma about finite dimensional semigroups. A few technical
lemmas.
TABLE OF CONTENTS ix
Section 10. The First Fundamental Theorem of Compact Semigroups. 56
Section 11. Historical Notes. 58
Chapter B. BASIC THEORY 61
Section 0. The centralizer of abelian groups of units. 61
THEOREM I. Exercises: Appropriate subgroups of automorphism
groups. Problems. Historical comments. 62
Section 1. Monothetic and solenoidal semigroups. 66
General definition. Surmorphisms of monothetic and solenoidal
semigroups. The universal compact monothetic resp., solenoidal,
semigroup. THEOREM II. Exercises: Arcwise connectivity of
solenoidal semigroups; finite cyclic semigroups; characterizations of
monothetic groups in terms of semigroups; bicyclic semigroups; the
universal compact semigroup generated by two idempotents; a result
about inverse semigroups. Historical comments. Terminology.
Section 2. Cylindrical semigroups. 83
Factoring the group of units gradually. The category of cylindri¬
cal semigroups. Universal cylindrical semigroups. Cylindrical semi¬
groups with a one parameter semigroup going straight to the
minimal ideal. Exercises: Compact semigroups which are locally
embeddable in a Lie group; Cylindrical semigroups which are em
beddable in a manifold. Historical comments.
Section 3. The existence of one parameter semigroups. 97
Semigroup nuclei. Subnuclei, homomorphism of nuclei. Rays,
perpendicular rays. Units of nuclei. Characterization of perpen¬
dicular rays. The group C. The set /QO, 1[)*. Continuity of rays.
Summary of preceding results. Producing discontinuous rays. Defi¬
nition of the function associated with a net. The existence of a ray
under special conditions. Technical lemmas. Lie group nuclei.
L nuclei. Summary of results about L nuclei. A principal lemma.
The existence of a ray. A compact automorphism group enters the
picture. One parameter semigroups fixed under automorphisms.
Appropriate subgroups of automorphism groups. Isotropy modulo
a compact normal subgroup. THEOREM III. Exercises: An exist¬
ence theorem for one parameter semigroups in locally compact groups;
another existence theorem; a conclusion for abelian semigroups;
I semigroups; one parameter semigroups of sets; existence of arcs
in the presence of one parameter semigroups of sets. Problems.
Historical comments.
X TABLE OF CONTENTS
Section 4. Connected abelian subsemigroups. 128
The set of compact abelian subsemigroups. U connectivity.
A chaining lemma. A characterization of connectivity. THEOREM
IV. Exercises: Koch s arcs; Koch s arcs in inverse semigroups;
connectivity of a compact semigroup and the connectivity of its minimal
ideal; Problem. Historical comments.
Section 5. The Hormos and irreducibility. 139
Chainable collections. The generalized hormos, the hormos.
Irreducible semigroups. An irreducible semigroup is a hormos. The
group of units in an irreducible semigroup is singleton. Characteri¬
zation of an irreducible hormos. The universal irreducible semigroup
and its properties. THEOREM V. The Second Fundamental
Theorem of Compact Semigroups. Exercises: Various results involving
irreducible semigroups; I semigroups in semigroups with totally dis¬
connected groups. Problems. Historical comments.
Section 6. The peripheral position of the group of units. 162
Sufficient conditions that the group of units is open. Peripher
ality. A theorem about compact semigroups whose identity com¬
ponent consists of units. Exercises: Semigroups with identity on a
compact manifold; semigroups with identity on totally ordered con¬
nected spaces; peripherality of left and right units; other concepts of
peripherality; the (n, G) rim; marginal points; all units are marginal;
the wedge theorem; weak cut points; no unit is a weak cut point;
finite dimensional homogeneous compact connected semigroups with
identity are groups; semigroups on limit manifolds are groups.
Problems. Historical comments.
Section 7. Peripherality and the location of one parameter
semigroups. 174
THEOREM VI. Exercises: One parameter semigroups if the
group of units is normal; characterization of a hormos; the centralizer
of the group of units; various theorems about the location of a hormos;
the dimension of S and S ^f; dim Sl3t = 1; a surmorphic image of
a hormos is a hormos; subsemigroups meeting all 3f classes. Problems.
Historical comments.
Chapter C. FURTHER DEVELOPMENTS. 186
Section 1. Totally ordered S1 class decompositions. 186
The role of cylindrical semigroups. The topmost ^ class is a
subsemigroup. The action of totally ordered semilattices on spaces.
Linking homomorphisms. Linked cross sections. The existence of
TABLE OF CONTENTS xi
linked homomorphisms and cross sections. Two compatible actions
of semilattices. Admissible spaces. The uniqueness of cross sections.
Admissible semigroups. Admissible Clifford semigroups.
THEOREM VII. Exercises: Partial solutions to Problem 1 of
Section 0 in Chapter B; idempotent compact connected semigroups
with totally ordered Zt class space. Problems. Historical comments.
Section 2. Semigroups which are totally ordered modulo a group
of units. 200
The orbit decomposition defines a congruence. Semigroups on
an interval. THEOREM VIII. Exercises: Corollaries to the theorem;
locally compact semigroups with dense open subgroup and compact
complement; locally compact semigroups with identity which are
embeddable in an n manifold and have an n — l dimensional compact
group of units; semigroups on compact manifolds with regular bounda¬
ry; semigroups with high dimensional Jt classes embedded in mani¬
folds; n dimensional compact connected semigroups with identity
and (n — ) dimensional group of units; (n — 2) dimensional groups
of units; the skin of a compact semigroup—another concept of
peripherality; various results involving skins; a theorem about semi¬
groups on a cell; semigroups embedded in R . Historical comments.
Section 3. Fibering over the minimal ideal. 221
The minimal ideal a translate of an ^ class. The fibering
theorem. Historical comments.
Section 4. One dimensional semigroups. 224
Hereditary unicoherence. ^ classes are totally disconnected.
Endpoints and units. Monotone surmorphisms. Summary of pre¬
ceding results. More general minimal ideals. The theorem about
one dimensional semigroups. Exercises: dim S/Jtf = 1 and JP a
congruence; endpoints in the center; endpoints idempotent and com¬
muting; miscellaneous facts about one dimensionality. Historical
comments.
Chapter D. EXAMPLES. 236
Simple building blocks. Subsemigroups. Direct products. Quo¬
tients. Cylindrical semigroups. Raising dimension. The Rees prod¬
uct. Splitting extensions—some important counter examples. Lots
of semigroups on the 2 cell. The cone technique. A semigroup
with zero without the fixed point property. Epimorphisms which are
not surmorphisms. Sticker techniques. The trunk of a semigroup,
universal semigroups having a given trunk. Making big minimal
ideals. More stickers. Chaining semigroups. Examples of hormoi.
xii TABLE OF CONTENTS
The catena. Examples of idempotent semigroups with totally
ordered S? class space. The technique of Koch and McAuley. The
Cantorian Swastika. Semigroups of transformations. Equicontinuity.
Linear compact semigroups. Affine semigroups. Measure semi¬
groups. Taylor s semigroups. Semigroups in physics.
Appendix I. GROUPS. 281
Section 1. Locally compact abelian groups. 281
Review of duality. Adjoint morphism. Character group. Duality
theorem. Exactness of the duality functor. Connectivity. Weight.
Monothetic and solenoidal compact groups. Universal monothetic
and solenoidal groups. Supplementing a closed connected subgroup.
Some special technical results about long diagrams.
Section 2. Miscellaneous theorems about compact groups. 295
The counter image of the center under an epimorphism of
compact groups. A conclusion for one parameter groups. Supple¬
menting a closed connected normal subgroup. The center of non
connected compact groups. Contracting a compact group. Charac¬
terizing a Lie group cohomologically. Compact groups embedded
in fl^ . Epimorphisms of compact abelian groups and cohomology.
One dimensional quotient spaces of compact connected groups.
The centralizer of a connected abelian subgroup.
Section 3. Some facts about Lie groups. 304
The Campbell Hausdorff formula. Some facts concerning the
local structure of a Lie group.
Section 4. Acyclic coset spaces of compact groups. 306
The Borel theorem.
Appendix II. TRANSFORMATION GROUPS. 311
Section 1. The local cross section theorems. 311
Local G spaces. Local cross sections to local orbits. The local
Tietze Gleason extension theorem for local actions. The existence
of local cross sections to the local orbits. Local isotropy. The local
cross section theorem in the presence of isotropy. Compact trans¬
formation groups with totally disconnected or contractible orbit
spaces. The local cross section theorem for locally compact groups.
TABLE OF CONTENTS xiii
Section 2. (n — l) dimensional orbits on manifolds. 319
Section 3. Compact groups acting on acyclic compact spaces. 321
Universal and classifying spaces. Conner s first lemma. The
existence of fixed points. The acyclicity of the fixed point set.
Conner s second lemma. The Borel Conner theorem. Generaliza¬
tions. An application of the Smith Swan fixed point theory. Com¬
pact groups acting on totally ordered connected spaces.
Appendix III. TOPOLOGICAL RESULTS. 335
Section 1. Miscellaneous results of topology. 335
The generalized homotopy theorem. Hereditarily unicoherent
and decomposable spaces. Uniform convergence and set conver¬
gence. Some facts about set convergence. Maximal and minimal
elements on quasi ordered spaces.
Section 2. Peripherality in topological spaces. 339
(*) invariant categories of connected spaces. Peripheral and
intrinsic points. Transformation groups and peripherality. A result
about subsets of euclidean spaces.
Section 3. Admissibility of finite dimensional spaces. 345
Bibliography 353
List of Symbols 365
Index 375
|
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| spelling | Hofmann, Karl H. 1932- Verfasser (DE-588)115780734 aut Elements of compact semigroups Karl Heinrich Hofmann ; Paul S. Mostert Columbus, Ohio Merrill 1966 XIII, 384 S. txt rdacontent n rdamedia nc rdacarrier Semigroups Kompakte Halbgruppe (DE-588)4164841-9 gnd rswk-swf Topologische Halbgruppe (DE-588)4200462-7 gnd rswk-swf Topologische Halbgruppe (DE-588)4200462-7 s DE-604 Kompakte Halbgruppe (DE-588)4164841-9 s Mostert, Paul Stallings Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001753038&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hofmann, Karl H. 1932- Mostert, Paul Stallings Elements of compact semigroups Semigroups Kompakte Halbgruppe (DE-588)4164841-9 gnd Topologische Halbgruppe (DE-588)4200462-7 gnd |
| subject_GND | (DE-588)4164841-9 (DE-588)4200462-7 |
| title | Elements of compact semigroups |
| title_auth | Elements of compact semigroups |
| title_exact_search | Elements of compact semigroups |
| title_full | Elements of compact semigroups Karl Heinrich Hofmann ; Paul S. Mostert |
| title_fullStr | Elements of compact semigroups Karl Heinrich Hofmann ; Paul S. Mostert |
| title_full_unstemmed | Elements of compact semigroups Karl Heinrich Hofmann ; Paul S. Mostert |
| title_short | Elements of compact semigroups |
| title_sort | elements of compact semigroups |
| topic | Semigroups Kompakte Halbgruppe (DE-588)4164841-9 gnd Topologische Halbgruppe (DE-588)4200462-7 gnd |
| topic_facet | Semigroups Kompakte Halbgruppe Topologische Halbgruppe |
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