Lattices and Ordered Sets:
"...A thorough introduction to the subject of ordered sets and lattices, with an emphasis on the latter. Topic coverage includes: modular, semimodular and distributive lattices, boolean algebras, representation of distributive lattices, algebraic lattices, congruence relations on lattices, free...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2008
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "...A thorough introduction to the subject of ordered sets and lattices, with an emphasis on the latter. Topic coverage includes: modular, semimodular and distributive lattices, boolean algebras, representation of distributive lattices, algebraic lattices, congruence relations on lattices, free lattices, fixed-point theorems, duality theory and more..." |
| Physical Description: | XIV, 305 S. graph. Darst. |
| ISBN: | 9780387789002 0387789006 |
Staff View
MARC
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| 245 | 1 | 0 | |a Lattices and Ordered Sets |c Steven Roman |
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| 300 | |a XIV, 305 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 520 | 3 | |a "...A thorough introduction to the subject of ordered sets and lattices, with an emphasis on the latter. Topic coverage includes: modular, semimodular and distributive lattices, boolean algebras, representation of distributive lattices, algebraic lattices, congruence relations on lattices, free lattices, fixed-point theorems, duality theory and more..." | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
Contents
Part I: Basic Theory
1
Partially Ordered Sets,
1
Basic Definitions,
1
Duality,
10
Monotone Maps,
13
Down-Sets and the Down Map,
13
Height and Graded Posets,
15
Chain Conditions,
16
Chain Conditions and Finiteness,
17
Dilworth s Theorem,
18
Symmetric and Transitive Closures,
20
Compatible Total Orders,
21
The
Poset
of Partial Orders,
23
Exercises,
23
2
Well-Ordered Sets,
27
Well-Ordered Sets,
27
Ordinal Numbers,
31
Transfinite
Induction,
37
Cardinal Numbers,
37
Ordinal and Cardinal Arithmetic,
42
Complete Posets,
43
Cofinality,
46
Exercises,
47
3
Lattices,
49
Closure and Inheritance,
49
Semilattices,
51
Arbitrary Meets Equivalent to Arbitrary Joins,
52
Lattices,
53
Meet-Structures and Closure Operators,
55
Properties of Lattices,
60
Join-Irreducible and Meet-Irreducible Elements,
64
Completeness,
65
Sublattices,
68
Denseness,
70
Lattice Homomorphisms,
71
The B-Down Map,
73
Ideals and Filters,
74
Prime and Maximal Ideals,
77
Lattice Representations,
79
Special Types of Lattices,
80
The Dedekind—MacNeille Completion,
85
Exercises,
89
Modular and Distributive Lattices,
95
Quadrilaterals,
95
The Definitions,
95
Examples,
96
Characterizations,
98
Modularity and Semimodularity,
104
Partition Lattices and Representations,
110
Distributive Lattices,
121
Irredundant
Join-Irreducible Representations,
123
Exercises,
124
Boolean Algebras,
129
Boolean Lattices,
129
Boolean Algebras,
130
Boolean Rings,
131
Boolean Homomorphisms,
134
Characterizing Boolean Lattices,
136
Complete and Infinite Distributivity,
138
Exercises,
142
The Representation of Distributive Lattices,
145
The Representation of Distributive Lattices with
DCC,
145
The Representation of Atomic Boolean Algebras,
146
The Representation of Arbitrary Distributive Lattices,
147
Summary,
149
Exercises,
150
Algebraic Lattices,
153
Motivation,
153
Algebraic Lattices,
155
ГНЈ
-Structures,
156
Algebraic Closure Operators,
158
The Main Correspondence,
159
Subalgebra
Lattices,
160
Congruence Lattices,
162
Meet-Representations,
163
Exercises,
166
8
Prime and Maximal Ideals; Separation Theorems,
169
Separation Theorems,
169
Exercises,
176
9
Congruence Relations on Lattices,
179
Congruence Relations on Lattices,
180
The Lattice of Congruence Relations,
185
Commuting Congruences and Joins,
187
Quotient Lattices and Kernels,
189
Congruence Relations and Lattice Homomorphisms,
191
Standard Ideals and Standard Congruence Relations,
195
Exercises,
202
Part II: Topics
10
Duality for Distributive Lattices: The Priestley Topology,
209
The Duality Between Finite Distributive Lattices and Finite Posets,
215
Totally Order-Separated Spaces,
218
The Priestley Prime Ideal Space,
219
The Priestley Duality,
222
The Case of Boolean Algebras,
229
Applications,
230
Exercises,
235
11
Free Lattices,
239
Lattice Identities,
239
Free and Relatively Free Lattices,
240
Constructing a Relatively Free Lattice,
243
Characterizing Equational Classes of Lattices,
245
The Word Problem for Free Lattices,
247
Canonical Forms,
250
The Free Lattice on Three Generators Is Infinite,
255
Exercises,
259
12
Fixed-Point Theorems,
263
Fixed Point Terminology,
264
Fixed-Point Theorems: Complete Lattices,
265
Fixed-Point Theorems: Complete Posets,
269
Exercises,
274
Al A
Bit of Topology,
277
Topological Spaces,
277
Subspaces,
277
Bases and
Subbases,
277
Connectedness and Separation,
278
Compactness,
278
Continuity,
280
The Product Topology,
280
A2 A Bit of Category Theory,
283
Categories,
283
Functors,
285
Natural Transformations,
287
References,
293
Index of Symbols,
297
Index,
299
|
| adam_txt |
Contents
Preface
Contents
Part I: Basic Theory
1
Partially Ordered Sets,
1
Basic Definitions,
1
Duality,
10
Monotone Maps,
13
Down-Sets and the Down Map,
13
Height and Graded Posets,
15
Chain Conditions,
16
Chain Conditions and Finiteness,
17
Dilworth's Theorem,
18
Symmetric and Transitive Closures,
20
Compatible Total Orders,
21
The
Poset
of Partial Orders,
23
Exercises,
23
2
Well-Ordered Sets,
27
Well-Ordered Sets,
27
Ordinal Numbers,
31
Transfinite
Induction,
37
Cardinal Numbers,
37
Ordinal and Cardinal Arithmetic,
42
Complete Posets,
43
Cofinality,
46
Exercises,
47
3
Lattices,
49
Closure and Inheritance,
49
Semilattices,
51
Arbitrary Meets Equivalent to Arbitrary Joins,
52
Lattices,
53
Meet-Structures and Closure Operators,
55
Properties of Lattices,
60
Join-Irreducible and Meet-Irreducible Elements,
64
Completeness,
65
Sublattices,
68
Denseness,
70
Lattice Homomorphisms,
71
The B-Down Map,
73
Ideals and Filters,
74
Prime and Maximal Ideals,
77
Lattice Representations,
79
Special Types of Lattices,
80
The Dedekind—MacNeille Completion,
85
Exercises,
89
Modular and Distributive Lattices,
95
Quadrilaterals,
95
The Definitions,
95
Examples,
96
Characterizations,
98
Modularity and Semimodularity,
104
Partition Lattices and Representations,
110
Distributive Lattices,
121
Irredundant
Join-Irreducible Representations,
123
Exercises,
124
Boolean Algebras,
129
Boolean Lattices,
129
Boolean Algebras,
130
Boolean Rings,
131
Boolean Homomorphisms,
134
Characterizing Boolean Lattices,
136
Complete and Infinite Distributivity,
138
Exercises,
142
The Representation of Distributive Lattices,
145
The Representation of Distributive Lattices with
DCC,
145
The Representation of Atomic Boolean Algebras,
146
The Representation of Arbitrary Distributive Lattices,
147
Summary,
149
Exercises,
150
Algebraic Lattices,
153
Motivation,
153
Algebraic Lattices,
155
ГНЈ
-Structures,
156
Algebraic Closure Operators,
158
The Main Correspondence,
159
Subalgebra
Lattices,
160
Congruence Lattices,
162
Meet-Representations,
163
Exercises,
166
8
Prime and Maximal Ideals; Separation Theorems,
169
Separation Theorems,
169
Exercises,
176
9
Congruence Relations on Lattices,
179
Congruence Relations on Lattices,
180
The Lattice of Congruence Relations,
185
Commuting Congruences and Joins,
187
Quotient Lattices and Kernels,
189
Congruence Relations and Lattice Homomorphisms,
191
Standard Ideals and Standard Congruence Relations,
195
Exercises,
202
Part II: Topics
10
Duality for Distributive Lattices: The Priestley Topology,
209
The Duality Between Finite Distributive Lattices and Finite Posets,
215
Totally Order-Separated Spaces,
218
The Priestley Prime Ideal Space,
219
The Priestley Duality,
222
The Case of Boolean Algebras,
229
Applications,
230
Exercises,
235
11
Free Lattices,
239
Lattice Identities,
239
Free and Relatively Free Lattices,
240
Constructing a Relatively Free Lattice,
243
Characterizing Equational Classes of Lattices,
245
The Word Problem for Free Lattices,
247
Canonical Forms,
250
The Free Lattice on Three Generators Is Infinite,
255
Exercises,
259
12
Fixed-Point Theorems,
263
Fixed Point Terminology,
264
Fixed-Point Theorems: Complete Lattices,
265
Fixed-Point Theorems: Complete Posets,
269
Exercises,
274
Al A
Bit of Topology,
277
Topological Spaces,
277
Subspaces,
277
Bases and
Subbases,
277
Connectedness and Separation,
278
Compactness,
278
Continuity,
280
The Product Topology,
280
A2 A Bit of Category Theory,
283
Categories,
283
Functors,
285
Natural Transformations,
287
References,
293
Index of Symbols,
297
Index,
299 |
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| id | DE-604.BV035172939 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:55:01Z |
| indexdate | 2024-07-09T21:26:39Z |
| institution | BVB |
| isbn | 9780387789002 0387789006 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016979853 |
| oclc_num | 261123137 |
| open_access_boolean | |
| owner | DE-739 DE-20 DE-11 DE-188 DE-706 |
| owner_facet | DE-739 DE-20 DE-11 DE-188 DE-706 |
| physical | XIV, 305 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| spelling | Roman, Steven Verfasser aut Lattices and Ordered Sets Steven Roman New York, NY Springer 2008 XIV, 305 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier "...A thorough introduction to the subject of ordered sets and lattices, with an emphasis on the latter. Topic coverage includes: modular, semimodular and distributive lattices, boolean algebras, representation of distributive lattices, algebraic lattices, congruence relations on lattices, free lattices, fixed-point theorems, duality theory and more..." Lattice theory Ordered sets Gitter Mathematik (DE-588)4157375-4 gnd rswk-swf Geordnete Menge (DE-588)4156748-1 gnd rswk-swf Gitter Mathematik (DE-588)4157375-4 s Geordnete Menge (DE-588)4156748-1 s DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016979853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Roman, Steven Lattices and Ordered Sets Lattice theory Ordered sets Gitter Mathematik (DE-588)4157375-4 gnd Geordnete Menge (DE-588)4156748-1 gnd |
| subject_GND | (DE-588)4157375-4 (DE-588)4156748-1 |
| title | Lattices and Ordered Sets |
| title_auth | Lattices and Ordered Sets |
| title_exact_search | Lattices and Ordered Sets |
| title_exact_search_txtP | Lattices and Ordered Sets |
| title_full | Lattices and Ordered Sets Steven Roman |
| title_fullStr | Lattices and Ordered Sets Steven Roman |
| title_full_unstemmed | Lattices and Ordered Sets Steven Roman |
| title_short | Lattices and Ordered Sets |
| title_sort | lattices and ordered sets |
| topic | Lattice theory Ordered sets Gitter Mathematik (DE-588)4157375-4 gnd Geordnete Menge (DE-588)4156748-1 gnd |
| topic_facet | Lattice theory Ordered sets Gitter Mathematik Geordnete Menge |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016979853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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