Ring of sets 1_5

Ring of sets 1From Wikipedia, the free encyclopedia

Contents

1 Birkhos representation theorem 11.1 Understanding the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The partial order of join-irreducibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Birkhos theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Rings of sets and preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Borel set 72.1 Generating the Borel algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Standard Borel spaces and Kuratowski theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Non-Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Alternative non-equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Cyclic order 113.1 Finite cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 The ternary relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Rolling and cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.3 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.4 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Monotone functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.1 Functions on nite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Further constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

i

ii CONTENTS

3.4.1 Unrolling and covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 Products and retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6.2 Modied axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.7 Symmetries and model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 Notes on usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Delta-ring 254.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Distributive lattice 265.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Characteristic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.6 Free distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Dynkin system 326.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Empty set 347.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.2.1 Operations on the empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.3.1 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.3.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.4 Questioned existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

CONTENTS iii

7.4.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.4.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Family of sets 408.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9 Field of sets 429.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 42

9.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.1.2 Separative and compact elds of sets: towards Stone duality . . . . . . . . . . . . . . . . . 42

9.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2.2 Topological elds of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2.3 Preorder elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.2.4 Complex algebras and elds of sets on relational structures . . . . . . . . . . . . . . . . . . 44

9.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Intersection (set theory) 4610.1 Basic denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.1.1 Intersecting and disjoint sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.2 Arbitrary intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.3 Nullary intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11 Lebesgue measure 5211.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

iv CONTENTS

11.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.4 Null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.5 Construction of the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.6 Relation to other measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

12 Mathematics 5712.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

12.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12.2 Denitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.2.1 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 6412.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

12.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7312.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

13 Measure (mathematics) 7513.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

13.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.3.2 Measures of innite unions of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . 7713.3.3 Measures of innite intersections of measurable sets . . . . . . . . . . . . . . . . . . . . . 77

13.4 Sigma-nite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7813.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7813.7 Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7813.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7813.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7913.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7913.11Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8013.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

CONTENTS v

14 Pi system 8314.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8314.2 Relationship to -Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

14.2.1 The - Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8414.3 -Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

14.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8514.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8614.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8614.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

15 Ring of sets 8715.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8715.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

16 Semiring 8916.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8916.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

16.2.1 In general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9016.2.2 Specic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

16.3 Semiring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.5 Complete and continuous semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9116.6 Star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

16.6.1 Complete star semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9216.7 Further generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.8 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.9 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.12Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.13Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

17 Set (mathematics) 9617.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

17.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

17.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

vi CONTENTS

17.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10017.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

17.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

17.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10517.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10517.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10617.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10617.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10717.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10717.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10717.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

18 Sigma-algebra 10818.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

18.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10818.1.2 Limits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10918.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

18.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11118.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11118.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

18.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11218.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11218.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

18.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11218.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 11218.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11218.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11318.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11318.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11318.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 11418.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 114

18.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11418.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11518.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

CONTENTS vii

19 Sigma-ring 11619.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.3 Similar concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11719.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

20 Subset 11820.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11920.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11920.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11920.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12020.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12020.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12020.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

21 Unit interval 12221.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

21.1.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12221.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12321.3 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12321.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12321.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

22 Upper set 12422.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12522.2 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12522.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12522.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12522.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 126

22.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12622.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13022.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Chapter 1

Birkhos representation theorem

This is about lattice theory. For other similarly named results, see Birkhos theorem (disambiguation).

In mathematics, Birkhos representation theorem for distributive lattices states that the elements of any nitedistributive lattice can be represented as nite sets, in such a way that the lattice operations correspond to unions andintersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributivelattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between nite topological spacesand preorders. It is named after Garrett Birkho, who published a proof of it in 1937.[1]

The name Birkhos representation theorem has also been applied to two other results of Birkho, one from 1935on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called elds of sets, closely related to the rings of sets used by Birkho to represent distributive lattices), and BirkhosHSP theorem representing algebras as products of irreducible algebras. Birkhos representation theorem has alsobeen called the fundamental theorem for nite distributive lattices.[2]

1.1 Understanding the theoremMany lattices can be dened in such a way that the elements of the lattice are represented by sets, the join operationof the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. Forinstance, the Boolean lattice dened from the family of all subsets of a nite set has this property. More generally anynite topological space has a lattice of sets as its family of open sets. Because set unions and intersections obey thedistributive law, any lattice dened in this way is a distributive lattice. Birkhos theorem states that in fact all nitedistributive lattices can be obtained this way, and later generalizations of Birkhos theorem state a similar thing forinnite distributive lattices.

1.2 ExamplesConsider the divisors of some composite number, such as (in the gure) 120, partially ordered by divisibility. Anytwo divisors of 120, such as 12 and 20, have a unique greatest common factor 12 20 = 4, the largest number thatdivides both of them, and a unique least common multiple 12 20 = 60; both of these numbers are also divisors of120. These two operations and satisfy the distributive law, in either of two equivalent forms: (x y) z = (x z) (y z) and (x y) z = (x z) (y z), for all x, y, and z. Therefore, the divisors form a nite distributive lattice.One may associate each divisor with the set of prime powers that divide it: thus, 12 is associated with the set {2,3,4},while 20 is associated with the set {2,4,5}. Then 12 20 = 4 is associated with the set {2,3,4} {2,4,5} = {2,4},while 12 20 = 60 is associated with the set {2,3,4} {2,4,5} = {2,3,4,5}, so the join and meet operations of thelattice correspond to union and intersection of sets.The prime powers 2, 3, 4, 5, and 8 appearing as elements in these sets may themselves be partially ordered bydivisibility; in this smaller partial order, 2 4 8 and there are no order relations between other pairs. The 16 setsthat are associated with divisors of 120 are the lower sets of this smaller partial order, subsets of elements such thatif x y and y belongs to the subset, then x must also belong to the subset. From any lower set L, one can recover the

1

2 CHAPTER 1. BIRKHOFFS REPRESENTATION THEOREM

{2} {3} {5}

{2,4} {2,3} {2,5} {3,5}

{2,4,8} {2,3,4} {2,4,5} {2,3,5}

{2,3,4,8} {2,4,5,8} {2,3,4,5}

{2,3,4,5,8}

1

2 3 5

4 6 10 15

8 12 20 30

24 40 60

120

The distributive lattice of divisors of 120, and its representation as sets of prime powers.

associated divisor by computing the least common multiple of the prime powers in L. Thus, the partial order on theve prime powers 2, 3, 4, 5, and 8 carries enough information to recover the entire original 16-element divisibilitylattice.Birkhos theorem states that this relation between the operations and of the lattice of divisors and the operations and of the associated sets of prime powers is not coincidental, and not dependent on the specic properties ofprime numbers and divisibility: the elements of any nite distributive lattice may be associated with lower sets of apartial order in the same way.As another example, the application of Birkhos theorem to the family of subsets of an n-element set, partiallyordered by inclusion, produces the free distributive lattice with n generators. The number of elements in this latticeis given by the Dedekind numbers.

1.3 The partial order of join-irreduciblesIn a lattice, an element x is join-irreducible if x is not the join of a nite set of other elements. Equivalently, x isjoin-irreducible if it is neither the bottom element of the lattice (the join of zero elements) nor the join of any twosmaller elements. For instance, in the lattice of divisors of 120, there is no pair of elements whose join is 4, so 4 isjoin-irreducible. An element x is join-prime if, whenever x y z, either x y or x z. In the same lattice, 4 isjoin-prime: whenever lcm(y,z) is divisible by 4, at least one of y and z must itself be divisible by 4.In any lattice, a join-prime element must be join-irreducible. Equivalently, an element that is not join-irreducible isnot join-prime. For, if an element x is not join-irreducible, there exist smaller y and z such that x = y z. But then x y z, and x is not less than or equal to either y or z, showing that it is not join-prime.There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but ina distributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and that x y z.This inequality is equivalent to the statement that x = x (y z), and by the distributive law x = (x y) (x z).But since x is join-irreducible, at least one of the two terms in this join must be x itself, showing that either x = x y(equivalently x y) or x = x z (equivalently x z).The lattice ordering on the subset of join-irreducible elements forms a partial order; Birkhos theorem states thatthe lattice itself can be recovered from the lower sets of this partial order.

1.4 Birkhos theoremIn any partial order, the lower sets form a lattice in which the lattices partial ordering is given by set inclusion, thejoin operation corresponds to set union, and the meet operation corresponds to set intersection, because unions andintersections preserve the property of being a lower set. Because set unions and intersections obey the distributive

1.5. RINGS OF SETS AND PREORDERS 3

law, this is a distributive lattice. Birkhos theorem states that any nite distributive lattice can be constructed in thisway.

Theorem. Any nite distributive lattice L is isomorphic to the lattice of lower sets of the partial orderof the join-irreducible elements of L.

That is, there is a one-to-one order-preserving correspondence between elements of L and lower sets of the partialorder. The lower set corresponding to an element x of L is simply the set of join-irreducible elements of L that areless than or equal to x, and the element of L corresponding to a lower set S of join-irreducible elements is the join ofS.If one starts with a lower set S of join-irreducible elements, lets x be the join of S, and constructs lower set T of thejoin-irreducible elements less than or equal to x, then S = T. For, every element of S clearly belongs to T, and anyjoin-irreducible element less than or equal to x must (by join-primality) be less than or equal to one of the membersof S, and therefore must (by the assumption that S is a lower set) belong to S itself. Conversely, if one starts with anelement x of L, lets S be the join-irreducible elements less than or equal to x, and constructs y as the join of S, thenx = y. For, as a join of elements less than or equal to x, y can be no greater than x itself, but if x is join-irreduciblethen x belongs to S while if x is the join of two or more join-irreducible items then they must again belong to S, so y x. Therefore, the correspondence is one-to-one and the theorem is proved.

1.5 Rings of sets and preordersBirkho (1937) dened a ring of sets to be a family of sets that is closed under the operations of set unions and setintersections; later, motivated by applications in mathematical psychology, Doignon & Falmagne (1999) called thesame structure a quasi-ordinal knowledge space. If the sets in a ring of sets are ordered by inclusion, they form adistributive lattice. The elements of the sets may be given a preorder in which x y whenever some set in the ringcontains x but not y. The ring of sets itself is then the family of lower sets of this preorder, and any preorder givesrise to a ring of sets in this way.

1.6 FunctorialityBirkhos theorem, as stated above, is a correspondence between individual partial orders and distributive lattices.However, it can also be extended to a correspondence between order-preserving functions of partial orders andbounded homomorphisms of the corresponding distributive lattices. The direction of these maps is reversed in thiscorrespondence.Let 2 denote the partial order on the two-element set {0, 1}, with the order relation 0 < 1, and (following Stanley)let J(P) denote the distributive lattice of lower sets of a nite partial order P. Then the elements of J(P) correspondone-for-one to the order-preserving functions from P to 2.[2] For, if is such a function, 1(0) forms a lower set,and conversely if L is a lower set one may dene an order-preserving function L that maps L to 0 and that mapsthe remaining elements of P to 1. If g is any order-preserving function from Q to P, one may dene a function g*from J(P) to J(Q) that uses the composition of functions to map any element L of J(P) to L g. This compositefunction maps Q to 2 and therefore corresponds to an element g*(L) = (L g)1(0) of J(Q). Further, for any x andy in J(P), g*(x y) = g*(x) g*(y) (an element of Q is mapped by g to the lower set x y if and only if belongsboth to the set of elements mapped to x and the set of elements mapped to y) and symmetrically g*(x y) = g*(x) g*(y). Additionally, the bottom element of J(P) (the function that maps all elements of P to 0) is mapped by g* tothe bottom element of J(Q), and the top element of J(P) is mapped by g* to the top element of J(Q). That is, g* is ahomomorphism of bounded lattices.However, the elements of P themselves correspond one-for-one with bounded lattice homomorphisms from J(P) to 2.For, if x is any element of P, one may dene a bounded lattice homomorphism jx that maps all lower sets containingx to 1 and all other lower sets to 0. And, for any lattice homomorphism from J(P) to 2, the elements of J(P) thatare mapped to 1 must have a unique minimal element x (the meet of all elements mapped to 1), which must bejoin-irreducible (it cannot be the join of any set of elements mapped to 0), so every lattice homomorphism has theform jx for some x. Again, from any bounded lattice homomorphism h from J(P) to J(Q) one may use composition offunctions to dene an order-preserving map h* from Q to P. It may be veried that g** = g for any order-preservingmap g from Q to P and that and h** = h for any bounded lattice homomorphism h from J(P) to J(Q).


In category theoretic terminology, J is a contravariant hom-functor J = Hom(,2) that denes a duality of categoriesbetween, on the one hand, the category of nite partial orders and order-preserving maps, and on the other hand thecategory of nite distributive lattices and bounded lattice homomorphisms.

1.7 GeneralizationsIn an innite distributive lattice, it may not be the case that the lower sets of the join-irreducible elements are inone-to-one correspondence with lattice elements. Indeed, there may be no join-irreducibles at all. This happens, forinstance, in the lattice of all natural numbers, ordered with the reverse of the usual divisibility ordering (so x ywhen y divides x): any number x can be expressed as the join of numbers xp and xq where p and q are distinct primenumbers. However, elements in innite distributive lattices may still be represented as sets via Stones representationtheorem for distributive lattices, a form of Stone duality in which each lattice element corresponds to a compact openset in a certain topological space. This generalized representation theorem can be expressed as a category-theoreticduality between distributive lattices and coherent spaces (sometimes called spectral spaces), topological spaces inwhich the compact open sets are closed under intersection and form a base for the topology.[3] Hilary Priestley showedthat Stones representation theorem could be interpreted as an extension of the idea of representing lattice elementsby lower sets of a partial order, using Nachbins idea of ordered topological spaces. Stone spaces with an additionalpartial order linked with the topology via Priestley separation axiom can also be used to represent bounded distributivelattices. Such spaces are known as Priestley spaces. Further, certain bitopological spaces, namely pairwise Stonespaces, generalize Stones original approach by utilizing two topologies on a set to represent an abstract distributvelattice. Thus, Birkhos representation theorem extends to the case of innite (bounded) distributive lattices in atleast three dierent ways, summed up in duality theory for distributive lattices.Birkhos representation theorem may also be generalized to nite structures other than distributive lattices. In adistributive lattice, the self-dual median operation[4]

m(x; y; z) = (x _ y) ^ (x _ z) ^ (y _ z) = (x ^ y) _ (x ^ z) _ (y ^ z)

gives rise to a median algebra, and the covering relation of the lattice forms a median graph. Finite median algebrasand median graphs have a dual structure as the set of solutions of a 2-satisability instance; Barthlemy & Constantin(1993) formulate this structure equivalently as the family of initial stable sets in a mixed graph.[5] For a distributivelattice, the corresponding mixed graph has no undirected edges, and the initial stable sets are just the lower sets ofthe transitive closure of the graph. Equivalently, for a distributive lattice, the implication graph of the 2-satisabilityinstance can be partitioned into two connected components, one on the positive variables of the instance and theother on the negative variables; the transitive closure of the positive component is the underlying partial order of thedistributive lattice.Another result analogous to Birkhos representation theorem, but applying to a broader class of lattices, is thetheorem of Edelman (1980) that any nite join-distributive lattice may be represented as an antimatroid, a familyof sets closed under unions but in which closure under intersections has been replaced by the property that eachnonempty set has a removable element.

1.8 Notes

[1] Birkho (1937).

[2] (Stanley 1997).

[3] Johnstone (1982).

[4] Birkho & Kiss (1947).

[5] A minor dierence between the 2-SAT and initial stable set formulations is that the latter presupposes the choice of a xedbase point from the median graph that corresponds to the empty initial stable set.

1.9. REFERENCES 5

1.9 References Barthlemy, J.-P.; Constantin, J. (1993), Median graphs, parallelism and posets, Discrete Mathematics 111

(13): 4963, doi:10.1016/0012-365X(93)90140-O. Birkho, Garrett (1937), Rings of sets, Duke Mathematical Journal 3 (3): 443454, doi:10.1215/S0012-

7094-37-00334-X.

Birkho, Garrett; Kiss, S. A. (1947), A ternary operation in distributive lattices, Bulletin of the AmericanMathematical Society 53 (1): 749752, doi:10.1090/S0002-9904-1947-08864-9, MR 0021540.

Doignon, J.-P.; Falmagne, J.-Cl. (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2. Edelman, Paul H. (1980), Meet-distributive lattices and the anti-exchange closure, Algebra Universalis 10

(1): 290299, doi:10.1007/BF02482912. Johnstone, Peter (1982), II.3 Coherent locales, Stone Spaces, Cambridge University Press, pp. 6269, ISBN

978-0-521-33779-3. Priestley, H. A. (1970), Representation of distributive lattices by means of ordered Stone spaces, Bulletin ofthe London Mathematical Society 2 (2): 186190, doi:10.1112/blms/2.2.186.

Priestley, H. A. (1972), Ordered topological spaces and the representation of distributive lattices, Proceedingsof the London Mathematical Society 24 (3): 507530, doi:10.1112/plms/s3-24.3.507.

Stanley, R. P. (1997), Enumerative Combinatorics, Volume I, Cambridge Studies in Advanced Mathematics 49,Cambridge University Press, pp. 104112.


Distributive example lattice, with join-irreducible elements a,...,g (shadowed nodes). The lower set a node corresponds to by Birkhosisomorphism is shown in light blue.

Chapter 2

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, fromclosed sets) through the operations of countable union, countable intersection, and relative complement. Borel setsare named after mile Borel.For a topological space X, the collection of all Borel sets on X forms a -algebra, known as the Borel algebra orBorel -algebra. The Borel algebra on X is the smallest -algebra containing all open sets (or, equivalently, all closedsets).Borel sets are important in measure theory, since any measure dened on the open sets of a space, or on the closedsets of a space, must also be dened on all Borel sets of that space. Any measure dened on the Borel sets is called aBorel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.In some contexts, Borel sets are dened to be generated by the compact sets of the topological space, rather thanthe open sets. The two denitions are equivalent for many well-behaved spaces, including all Hausdor -compactspaces, but can be dierent in more pathological spaces.

2.1 Generating the Borel algebraIn the case X is a metric space, the Borel algebra in the rst sense may be described generatively as follows.For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let

T be all countable unions of elements of T T be all countable intersections of elements of T T = (T):

Now dene by transnite induction a sequence Gm, where m is an ordinal number, in the following manner:

For the base case of the denition, let G0 be the collection of open subsets of X. If i is not a limit ordinal, then i has an immediately preceding ordinal i 1. Let

Gi = [Gi1]:

If i is a limit ordinal, set

Gi =[j

8 CHAPTER 2. BOREL SET

G 7! G:to the rst uncountable ordinal.To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets.In particular, complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountablelimit ordinal, Gm is closed under countable unions.Note that for each Borel set B, there is some countable ordinal B such that B can be obtained by iterating theoperation over B. However, as B varies over all Borel sets, B will vary over all the countable ordinals, and thus therst ordinal at which all the Borel sets are obtained is 1, the rst uncountable ordinal.

2.1.1 ExampleAn important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It isthe algebra on which the Borel measure is dened. Given a real random variable dened on a probability space, itsprobability distribution is by denition also a measure on the Borel algebra.The Borel algebra on the reals is the smallest -algebra on R which contains all the intervals.In the construction by transnite induction, it can be shown that, in each step, the number of sets is, at most, thepower of the continuum. So, the total number of Borel sets is less than or equal to

@1 2@0 = 2@0 :

2.2 Standard Borel spaces and Kuratowski theoremsLet X be a topological space. The Borel space associated to X is the pair (X,B), where B is the -algebra of Borelsets of X.Mackey dened a Borel space somewhat dierently, writing that it is a set together with a distinguished -eld ofsubsets called its Borel sets. [1] However, modern usage is to call the distinguished sub-algebra measurable sets andsuch spaces measurable spaces. The reason for this distinction is that the Borel sets are the -algebra generated byopen sets (of a topological space), whereas Mackeys denition refers to a set equipped with an arbitrary -algebra.There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.[2]

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. Afunction f : X ! Y is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, f1(B) is ameasurable set in X.Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which denes thetopology of X and which makes X a complete separable metric space. Then X as a Borel space is isomorphic to oneof (1) R, (2) Z or (3) a nite space. (This result is reminiscent of Maharams theorem.)Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic.A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized upto isomorphism by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injectivemaps dened on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.See analytic set.Every probability measure on a standard Borel space turns it into a standard probability space.

2.3 Non-Borel setsAn example of a subset of the reals which is non-Borel, due to Lusin[4] (see Sect. 62, pages 7678), is describedbelow. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.

2.4. ALTERNATIVE NON-EQUIVALENT DEFINITIONS 9

Every irrational number has a unique representation by a continued fraction

x = a0 +1

a1 +1

a2 +1

a3 +1

. . .where a0 is some integer and all the other numbers ak are positive integers. Let A be the set of all irrationalnumbers that correspond to sequences (a0; a1; : : : ) with the following property: there exists an innite subsequence(ak0 ; ak1 ; : : : ) such that each element is a divisor of the next element. This set A is not Borel. In fact, it is analytic,and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris,especially Exercise (27.2) on page 209, Denition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.Another non-Borel set is an inverse image f1[0] of an innite parity function f : f0; 1g! ! f0; 1g . However, thisis a proof of existence (via the axiom of choice), not an explicit example.

2.4 Alternative non-equivalent denitionsAccording to Halmos (Halmos 1950, page 219), a subset of a locally compact Hausdor topological space is calleda Borel set if it belongs to the smallest ring containing all compact sets.Norberg and Vervaat [5] redene the Borel algebra of a topological space X as the algebra generated by its opensubsets and its compact saturated subsets. This denition is well-suited for applications in the case where X is notHausdor. It coincides with the usual denition if X is second countable or if every compact saturated subset isclosed (which is the case in particular if X is Hausdor).

2.5 See also Baire set Cylindrical -algebra Polish space Descriptive set theory Borel hierarchy

2.6 References William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent expo-

sition of Polish topology)

Richard Dudley, Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989

Halmos, Paul R. (1950). Measure theory. D. van Nostrand Co. See especially Sect. 51 Borel sets and Bairesets.

Halsey Royden, Real Analysis, Prentice Hall, 1988

Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol.156)

10 CHAPTER 2. BOREL SET

2.7 Notes[1] Mackey, G.W. (1966), Ergodic Theory and Virtual Groups, Math. Annalen. (Springer-Verlag) 166 (3): 187207,

doi:10.1007/BF01361167, ISSN 0025-5831, (subscription required (help))

[2] Jochen Wengenroth (mathoverflow.net/users/21051), Is every sigma-algebra the Borel algebra of a topology?, http://mathoverflow.net/questions/87888 (version: 2012-02-09)

[3] Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7

[4] Lusin, Nicolas (1927), Sur les ensembles analytiques, Fundamenta Mathematicae (Institute of mathematics, Polishacademy of sciences) 10: 195.

[5] Tommy Norberg and Wim Vervaat, Capacities on non-Hausdor spaces, in: Probability and Lattices, in: CWI Tract, vol.110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150

2.8 External links Hazewinkel, Michiel, ed. (2001), Borel set, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

Formal denition of Borel Sets in the Mizar system, and the list of theorems that have been formally provedabout it.

Weisstein, Eric W., Borel Set, MathWorld.

Chapter 3

Cyclic order

In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory,a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is more clockwise thanwest. Instead, a cyclic order is dened as a ternary relation [a, b, c], meaning after a, one reaches b before c. Forexample, [June, October, February]. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive,

11

12 CHAPTER 3. CYCLIC ORDER

and total. Dropping the total requirement results in a partial cyclic order.A set with a cyclic order is called a cyclically ordered set or simply a cycle.[nb] Some familiar cycles are discrete,having only a nite number of elements: there are seven days of the week, four cardinal directions, twelve notes inthe chromatic scale, and three plays in rock-paper-scissors. In a nite cycle, each element has a next element anda previous element. There are also continuously variable cycles with innitely many elements, such as the orientedunit circle in the plane.Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line. Any linear ordercan be bent into a circle, and any cyclic order can be cut at a point, resulting in a line. These operations, along with therelated constructions of intervals and covering maps, mean that questions about cyclic orders can often be transformedinto questions about linear orders. Cycles have more symmetries than linear orders, and they often naturally occur asresidues of linear structures, as in the nite cyclic groups or the real projective line.

3.1 Finite cycles

V1 V2

V3

V4

V5

A 5-element cycle

A cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Eachelement x in X has a next element and a previous element, and taking either successors or predecessors cycles

3.2. DEFINITIONS 13

exactly once through the elements as x(1), x(2), ..., x(n).There are a few equivalent ways to state this denition. A cyclic order on X is the same as a permutation that makes allof X into a single cycle. A cycle with n elements is also a Zn-torsor: a set with a free transitive action by a nite cyclicgroup.[1] Another formulation is to make X into the standard directed cycle graph on n vertices, by some matchingof elements to vertices.It can be instinctive to use cyclic orders for symmetric functions, for example as in

xy + yz + zx

where writing the nal monomial as xz would distract from the pattern.A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements g andh of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and y1with y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rulesthat allow one to remove or add adjacent y and y1.A cyclic order on a set X can be determined by a linear order on X, but not in a unique way. Choosing a linear orderis equivalent to choosing a rst element, so there are exactly n linear orders that induce a given cyclic order. Sincethere are n! possible linear orders, there are (n 1)! possible cyclic orders.

3.2 DenitionsAn innite set can also be ordered cyclically. Important examples of innite cycles include the unit circle, S1, andthe rational numbers, Q. The basic idea is the same: we arrange elements of the set around a circle. However, inthe innite case we cannot rely upon an immediate successor relation, because points may not have successors. Forexample, given a point on the unit circle, there is no next point. Nor can we rely upon a binary relation to determinewhich of two points comes rst. Traveling clockwise on a circle, neither east or west comes rst, but each followsthe other.Instead, we use a ternary relation denoting that elements a, b, c occur after each other (not necessarily immediately)as we go around the circle. For example, in clockwise order, [east, south, west]. By currying the arguments of theternary relation [a, b, c], one can think of a cyclic order as a one-parameter family of binary order relations, calledcuts, or as a two-parameter family of subsets of K, called intervals.

3.2.1 The ternary relationThe general denition is as follows: a cyclic order on a set X is a relation C X3, written [a, b, c], that satises thefollowing axioms:[nb]

1. Cyclicity: If [a, b, c] then [b, c, a]2. Asymmetry: If [a, b, c] then not [c, b, a]3. Transitivity: If [a, b, c] and [a, c, d] then [a, b, d]4. Totality: If a, b, and c are distinct, then either [a, b, c] or [c, b, a]

The axioms are named by analogy with the asymmetry, transitivity, and totality axioms for a binary relation, whichtogether dene a strict linear order. Edward Huntington (1916, 1924) considered other possible lists of axioms,including one list that was meant to emphasize the similarity between a cyclic order and a betweenness relation. Aternary relation that satises the rst three axioms, but not necessarily the axiom of totality, is a partial cyclic order.

3.2.2 Rolling and cutsGiven a linear order < on a set X, the cyclic order on X induced by < is dened as follows:[2]

[a, b, c] if and only if a < b < c or b < c < a or c < a < b


Two linear orders induce the same cyclic order if they can be transformed into each other by a cyclic rearrangement,as in cutting a deck of cards.[3] One may dene a cyclic order relation as a ternary relation that is induced by a strictlinear order as above.[4]

Cutting a single point out of a cyclic order leaves a linear order behind. More precisely, given a cyclically orderedset (K, [ ]), each element a K denes a natural linear order

3.4. FURTHER CONSTRUCTIONS 15

3.3.1 Functions on nite setsA cyclic order on a nite set X can be determined by an injection into the unit circle, X S1. There are many possiblefunctions that induce the same cyclic orderin fact, innitely many. In order to quantify this redundancy, it takesa more complex combinatorial object than a simple number. Examining the conguration space of all such mapsleads to the denition of an (n 1)-dimensional polytope known as a cyclohedron. Cyclohedra were rst applied tothe study of knot invariants;[11] they have more recently been applied to the experimental detection of periodicallyexpressed genes in the study of biological clocks.[12]

The category of homomorphisms of the standard nite cycles is called the cyclic category; it may be used to constructAlain Connes' cyclic homology.One may dene a degree of a function between cycles, analogous to the degree of a continuous mapping. For example,the natural map from the circle of fths to the chromatic circle is a map of degree 7. One may also dene a rotationnumber.

3.3.2 Completion A cut with both a least element and a greatest element is called a jump. For example, every cut of a nite cycleZn is a jump. A cycle with no jumps is called dense.[13][14]

A cut with neither a least element nor a greatest element is called a gap. For example, the rational numbers Qhave a gap at every irrational number. They also have a gap at innity, i.e. the usual ordering. A cycle with nogaps is called complete.[15][14]

A cut with exactly one endpoint is called a principal or Dedekind cut. For example, every cut of the circle S1 isa principal cut. A cycle where every cut is principal, being both dense and complete, is called continuous.[16][14]

The set of all cuts is cyclically ordered by the following relation: [


[

3.5. TOPOLOGY 17

c = a b and z < x

a = b = c and [x, y, z]

The lexicographic product K L globally looks like K and locally looks like L; it can be thought of as K copies of L.This construction is sometimes used to characterize cyclically ordered groups.[21]

One can also glue together dierent linearly ordered sets to form a circularly ordered set. For example, given twolinearly ordered sets L1 and L2, one may form a circle by joining them together at positive and negative innity. Acircular order on the disjoint union L1 L2 {, } is dened by < L1 < < L2 < , where the inducedordering on L1 is the opposite of its original ordering. For example, the set of all longitudes is circularly ordered byjoining together all points west and all points east, along with the prime meridian and the 180th meridian. Kuhlmann,Marshall & Osiak (2011) use this construction while characterizing the spaces of orderings and real places of doubleformal Laurent series over a real closed eld.[22]

3.5 Topology

The open intervals form a base for a natural topology, the cyclic order topology. The open sets in this topology areexactly those sets which are open in every compatible linear order.[23] To illustrate the dierence, in the set [0, 1),the subset [0, 1/2) is a neighborhood of 0 in the linear order but not in the cyclic order.Interesting examples of cyclically ordered spaces include the conformal boundary of a simply connected Lorentzsurface[24] and the leaf space of a lifted essential lamination of certain 3-manifolds.[25] Discrete dynamical systemson cyclically ordered spaces have also been studied.[26]

The interval topology forgets the original orientation of the cyclic order. This orientation can be restored by enrich-ing the intervals with their induced linear orders; then one has a set covered with an atlas of linear orders that arecompatible where they overlap. In other words, a cyclically ordered set can be thought of as a locally linearly orderedspace: an object like a manifold, but with order relations instead of coordinate charts. This viewpoint makes it easierto be precise about such concepts as covering maps. The generalization to a locally partially ordered space is studiedin Roll (1993); see also Directed topology.

3.6 Related structures


3.6.1 Groups

Main article: Cyclically ordered group

A cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplicationboth preserve the cyclic order. Cyclically ordered groups were rst studied in depth by Ladislav Rieger in 1947.[27]They are a generalization of cyclic groups: the innite cyclic group Z and the nite cyclic groups Z/n. Since a linearorder induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rationalnumbers Q, the real numbers R, and so on. Some of the most important cyclically ordered groups fall into neitherprevious category: the circle group T and its subgroups, such as the subgroup of rational points.Every cyclically ordered group can be expressed as a quotient L / Z, where L is a linearly ordered group and Z is acyclic conal subgroup of L. Every cyclically ordered group can also be expressed as a subgroup of a product T L,where L is a linearly ordered group. If a cyclically ordered group is Archimedean or compact, it can be embedded inT itself.[28]

3.6.2 Modied axioms

A partial cyclic order is a ternary relation that generalizes a (total) cyclic order in the same way that a partial ordergeneralizes a total order. It is cyclic, asymmetric, and transitive, but it need not be total. An order variety is a partialcyclic order that satises an additional spreading axiom. Replacing the asymmetry axiom with a complementaryversion results in the denition of a co-cyclic order. Appropriately total co-cyclic orders are related to cyclic ordersin the same way that is related to

3.9. NOTES ON USAGE 19

3.9 Notes on usage^cyclic order The relation may be called a cyclic order (Huntington 1916, p. 630), a circular order (Huntington1916, p. 630), a cyclic ordering (Kok 1973, p. 6), or a circular ordering (Mosher 1996, p. 109). Some authors callsuch an ordering a total cyclic order (Isli & Cohn 1998, p. 643), a complete cyclic order (Novk 1982, p. 462), alinear cyclic order (Novk 1984, p. 323), or an l-cyclic order or -cyclic order (ernk 2001, p. 32), to distinguishfrom the broader class of partial cyclic orders, which they call simply cyclic orders. Finally, some authors may takecyclic order to mean an unoriented quaternary separation relation (Bowditch 1998, p. 155).^cycle A set with a cyclic order may be called a cycle (Novk 1982, p. 462) or a circle (Giraudet & Holland 2002,p. 1). The above variations also appear in adjective form: cyclically ordered set (cyklicky uspodan mnoiny, ech1936, p. 23), circularly ordered set, total cyclically ordered set, complete cyclically ordered set, linearly cyclicallyordered set, l-cyclically ordered set, -cyclically ordered set. All authors agree that a cycle is totally ordered.^ternary relation There are a few dierent symbols in use for a cyclic relation. Huntington (1916, p. 630) usesconcatenation: ABC. ech (1936, p. 23) and (Novk 1982, p. 462) use ordered triples and the set membershipsymbol: (a, b, c) C. Megiddo (1976, p. 274) uses concatenation and set membership: abc C, understanding abcas a cyclically ordered triple. The literature on groups, such as wierczkowski (1959a, p. 162) and ernk & Jakubk(1987, p. 157), tend to use square brackets: [a, b, c]. Giraudet & Holland (2002, p. 1) use round parentheses: (a, b,c), reserving square brackets for a betweenness relation. Campero-Arena & Truss (2009, p. 1) use a function-stylenotation: R(a, b, c). Rieger (1947), cited after Pecinov 2008, p. 82) uses a less-than symbol as a delimiter: < x,y, z


[14] Novk & Novotn 1987, p. 409410.

[15] Novk 1984, pp. 325, 331.

[16] Novk 1984, p. 333.

[17] Novk 1984, p. 330.

[18] Roll 1993, p. 469; Freudenthal & Bauer 1974, p. 10

[19] Freudenthal 1973, p. 475; Freudenthal & Bauer 1974, p. 10

[20] wierczkowski 1959a, p. 161.

[21] wierczkowski 1959a.

[22] Kuhlmann, Marshall & Osiak 2011, p. 8.

[23] Viro et al. 2008, p. 44.

[24] Weinstein 1996, pp. 8081.

[25] Calegari & Duneld 2003, pp. 1213.

[26] Bass et al. 1996, p. 19.

[27] Pecinov-Kozkov 2005, p. 194.

[28] wierczkowski 1959a, pp. 161162.

[29] Knuth 1992, p. 4.

[30] Huntington 1935.

[31] Macpherson 2011.

Bibliography

Bass, Hyman; Otero-Espinar, Maria Victoria; Rockmore, Daniel; Tresser, Charles (1996), Cyclic renormallzat-lon and automorphism groups of rooted trees, Lecture Notes in Mathematics 1621, Springer, doi:10.1007/BFb0096321,ISBN 978-3-540-60595-9

Bowditch, Brian H. (September 1998), Cut points and canonical splittings of hyperbolic groups (PDF), ActaMathematica 180 (2): 145186, doi:10.1007/BF02392898, retrieved 25 April 2011

Bowditch, Brian H. (November 2004), Planar groups and the Seifert conjecture, Journal fr die reine undangewandte Mathematik 576: 1162, doi:10.1515/crll.2004.084, retrieved 31 May 2011

Brown, Kenneth S. (February 1987), Finiteness properties of groups (PDF), Journal of Pure and AppliedAlgebra 44 (13): 4575, doi:10.1016/0022-4049(87)90015-6, retrieved 21 May 2011

Calegari, Danny (13 December 2004), Circular groups, planar groups, and the Euler class (PDF), Geometry& Topology Monographs 7: 431491, arXiv:math/0403311, doi:10.2140/gtm.2004.7.431, retrieved 30 April2011

Calegari, Danny; Duneld, Nathan M. (April 2003), Laminations and groups of homeomorphisms of thecircle, Inventiones Mathematicae 152 (1): 149204, arXiv:math/0203192, doi:10.1007/s00222-002-0271-6

Campero-Arena, G.; Truss, John K. (April 2009), 1-transitive cyclic orderings (PDF), Journal of Combina-torial Theory, Series A 116 (3): 581594, doi:10.1016/j.jcta.2008.08.006, retrieved 25 April 2011

ech, Eduard (1936), Bodov mnoiny (in Czech), Prague: Jednota eskoslovenskch matematik a fysik,hdl:10338.dmlcz/400435, retrieved 9 May 2011

ernk, tefan (2001), Cantor extension of a half linearly cyclically ordered group (PDF), DiscussionesMathematicae - General Algebra and Applications 21 (1): 3146, doi:10.7151/dmgaa.1025, retrieved 22 May2011

3.10. REFERENCES 21

ernk, tefan; Jakubk, Jn (1987), Completion of a cyclically ordered group (PDF), Czechoslovak Mathe-matical Journal 37 (1): 157174, MR 875137, Zbl 0624.06021, hdl:10338.dmlcz/102144, retrieved 25 April2011

erny, Ilja (1978), Cuts in simple connected regions and the cyclic ordering of the system of all boundaryelements (PDF), asopis pro pstovn matematiky 103 (3): 259281, hdl:10338.dmlcz/117983, retrieved 11May 2011

Courcelle, Bruno (21 August 2003), 2.3 Circular order, in Berwanger, Dietmar; Grdel, Erich, Problems inFinite Model Theory (PDF), p. 12, retrieved 15 May 2011

H. S. M. Coxeter (1949) The Real Projective Plane, chapter 3: Order and continuity. Evans, David M.; Macpherson, Dugald; Ivanov, Alexandre A. (1997), Finite Covers, in Evans, David M.,Model theory of groups and automorphism groups: Blaubeuren, August 1995, London Mathematical SocietyLecture Note Series 244, Cambridge University Press, pp. 172, ISBN 0-521-58955-X, retrieved 5 May 2011

Freudenthal, Hans (1973), Mathematics as an educational task, D. Reidel, ISBN 90-277-0235-7 Freudenthal, Hans; Bauer, A. (1974), GeometryA Phenomenological Discussion, in Behnke, Heinrich;

Gould, S. H., Fundamentals of mathematics 2, MIT Press, pp. 328, ISBN 0-262-02069-6

Freudenthal, Hans (1983), Didactical phenomenology of mathematical structures, D. Reidel, ISBN 90-277-1535-1

Giraudet, Michle; Holland, W. Charles (September 2002), Ohkuma Structures (PDF), Order 19 (3): 223237, doi:10.1023/A:1021249901409, retrieved 28 April 2011

Huntington, Edward V. (1 November 1916), A Set of Independent Postulates for Cyclic Order (PDF), Pro-ceedings of the National Academy of Sciences of theUnited States of America 2 (11): 630631, doi:10.1073/pnas.2.11.630,retrieved 8 May 2011

Huntington, Edward V. (15 February 1924), Sets of Completely Independent Postulates for Cyclic Order(PDF), Proceedings of the National Academy of Sciences of the United States of America 10 (2): 7478,doi:10.1073/pnas.10.2.74, retrieved 8 May 2011

Huntington, Edward V. (July 1935), Inter-Relations Among the Four Principal Types of Order (PDF),Transactions of the American Mathematical Society 38 (1): 19, doi:10.1090/S0002-9947-1935-1501800-1,retrieved 8 May 2011

Isli, Amar; Cohn, Anthony G. (1998), An algebra for cyclic ordering of 2D orientations, AAAI '98/IAAI'98 Proceedings of the fteenth national/tenth conference on Articial intelligence/Innovative applications ofarticial intelligence (PDF), ISBN 0-262-51098-7, retrieved 23 May 2011

Knuth, Donald E. (1992), Axioms and Hulls, Lecture Notes in Computer Science 606, Heidelberg: Springer-Verlag, pp. ix+109, doi:10.1007/3-540-55611-7, ISBN 3-540-55611-7, retrieved 5 May 2011

Kok, H. (1973), Connected orderable spaces, Amsterdam: Mathematisch Centrum, ISBN 90-6196-088-6 Kuhlmann, Salma; Marshall, Murray; Osiak, Katarzyna (1 June 2011), Cyclic 2-structures and spaces of or-

derings of power series elds in two variables (PDF), Journal of Algebra 335 (1): 3648, doi:10.1016/j.jalgebra.2011.02.026,retrieved 11 May 2011

Kulpeshov, Beibut Sh. (December 2006), On 0-categorical weakly circularly minimal structures, Mathe-matical Logic Quarterly 52 (6): 555574, doi:10.1002/malq.200610014

Kulpeshov, Beibut Sh. (March 2009), Denable functions in the 0-categorical weakly circularly mini-mal structures, Siberian Mathematical Journal 50 (2): 282301, doi:10.1007/s11202-009-0034-3 Transla-tion of Kulpeshov (2009), " 0- ", Sibirski Matematicheski Zhurnal 50 (2): 356379, retrieved 24 May 2011

Kulpeshov, Beibut Sh.; Macpherson, H. Dugald (July 2005), Minimality conditions on circularly orderedstructures, Mathematical Logic Quarterly 51 (4): 377399, doi:10.1002/malq.200410040, MR 2150368


Macpherson, H. Dugald (2011), A survey of homogeneous structures (PDF),DiscreteMathematics, doi:10.1016/j.disc.2011.01.024,retrieved 28 April 2011

McMullen, Curtis T. (2009), Ribbon R-trees and holomorphic dynamics on the unit disk (PDF), Journal ofTopology 2 (1): 2376, doi:10.1112/jtopol/jtn032, retrieved 15 May 2011

Megiddo, Nimrod (March 1976), Partial and complete cyclic orders (PDF), Bulletin of the American Math-ematical Society 82 (2): 274276, doi:10.1090/S0002-9904-1976-14020-7, retrieved 30 April 2011

Morton, James; Pachter, Lior; Shiu, Anne; Sturmfels, Bernd (January 2007), The Cyclohedron Test for Find-ing Periodic Genes in Time Course Expression Studies, Statistical Applications in Genetics and MolecularBiology 6 (1), arXiv:q-bio/0702049, doi:10.2202/1544-6115.1286

Mosher, Lee (1996), A users guide to the mapping class group: once-punctured surfaces, in Baumslag,Gilbert, Geometric and computational perspectives on innite groups, DIMACS 25, AMS Bookstore, pp. 101174, arXiv:math/9409209, ISBN 0-8218-0449-9

Novk, Vtzslav (1982), Cyclically ordered sets (PDF), Czechoslovak Mathematical Journal 32 (3): 460473, hdl:10338.dmlcz/101821, retrieved 30 April 2011

Novk, Vtzslav (1984), Cuts in cyclically ordered sets (PDF), Czechoslovak Mathematical Journal 34 (2):322333, hdl:10338.dmlcz/101955, retrieved 30 April 2011

Novk, Vtzslav; Novotn, Miroslav (1987), On completion of cyclically ordered sets (PDF), CzechoslovakMathematical Journal 37 (3): 407414, hdl:10338.dmlcz/102168, retrieved 25 April 2011

Pecinov-Kozkov, Elika (2005), Ladislav Svante Rieger and His Algebraic Work, in Safrankova, Jana,WDS 2005 - Proceedings of Contributed Papers, Part I, Prague: Matfyzpress, pp. 190197, ISBN 80-86732-59-2, retrieved 25 April 2011

Pecinov, Elika (2008), Ladislav Svante Rieger (19161963), Djiny matematiky (in Czech) 36, Prague: Mat-fyzpress, ISBN 978-80-7378-047-0, hdl:10338.dmlcz/400757, retrieved 9 May 2011

Rieger, L. S. (1947), " uspodanch a cyklicky uspodanch grupch II (On ordered and cyclically orderedgroups II)", Vstnk Krlovsk esk spolecnosti nauk, Tda mathematicko-prodovdn (Journal of the RoyalCzech Society of Sciences, Mathematics and natural history) (in Czech) (1): 133

Roll, J. Blair (1993), Locally partially ordered groups (PDF), Czechoslovak Mathematical Journal 43 (3):467481, hdl:10338.dmlcz/128411, retrieved 30 April 2011

Stashe, Jim (1997), From operads to 'physically' inspired theories, in Loday, Jean-Louis; Stashe, JamesD.; Voronov, Alexander A., Operads: Proceedings of Reneassance Conferences, Contemporary Mathematics202, AMS Bookstore, pp. 5382, ISBN 978-0-8218-0513-8, retrieved 1 May 2011

wierczkowski, S. (1959a), On cyclically ordered groups (PDF), Fundamenta Mathematicae 47: 161166,retrieved 2 May 2011

Tararin, Valeri Mikhailovich (2001), On Automorphism Groups of Cyclically Ordered Sets, Siberian Mathe-matical Journal 42 (1): 190204, doi:10.1023/A:1004866131580. Translation of Tamarin (2001), , Sibirskii Matematicheskii Zhurnal (in Russian) 42(1): 212230, retrieved 30 April 2011

Tararin, Valeri Mikhailovich (2002), On c-3-Transitive Automorphism Groups of Cyclically Ordered Sets,Mathematical Notes 71 (1): 110117, doi:10.1023/A:1013934509265. Translation of Tamarin (2002), " c-3- ", Matematicheskie Zametki71 (1): 122129, doi:10.4213/mzm333, retrieved 22 May 2011

Truss, John K. (2009), On the automorphism group of the countable dense circular order (PDF), FundamentaMathematicae 204 (2): 97111, doi:10.4064/fm204-2-1, retrieved 25 April 2011

Viro, Oleg; Ivanov, Oleg; Netsvetaev, Nikita; Kharlamov, Viatcheslav (2008), 8. Cyclic Orders, Elementarytopology: problem textbook (PDF) (1st English ed.), AMS Bookstore, pp. 4244, ISBN 978-0-8218-4506-6,retrieved 25 April 2011

Weinstein, Tilla (July 1996), An introduction to Lorentz surfaces, De Gruyter Expositions in Mathematics 22,Walter de Gruyter, ISBN 978-3-11-014333-1

3.11. FURTHER READING 23

3.11 Further reading Bhattacharjee, Meenaxi; Macpherson, Dugald; Mller, Rgnvaldur G.; Neumann, Peter M. (1998), Notes onInnite PermutationGroups, Lecture Notes in Mathematics 1698, Springer, pp. 108109, doi:10.1007/BFb0092550

Bodirsky, Manuel; Pinsker, Michael (to appear), Reducts of Ramsey Structures, Model Theoretic Methods inFinite Combinatorics, Contemporary Mathematics, AMS, arXiv:1105.6073 Check date values in: |date= (help)

Cameron, Peter J. (June 1976), Transitivity of permutation groups on unordered sets,Mathematische Zeitschrift148 (2): 127139, doi:10.1007/BF01214702

Cameron, Peter J. (June 1977), Cohomological aspects of two-graphs, Mathematische Zeitschrift 157 (2):101119, doi:10.1007/BF01215145

Cameron, Peter J. (1997), The algebra of an age, in Evans, David M., Model theory of groups and automor-phism groups: Blaubeuren, August 1995, London Mathematical Society Lecture Note Series 244, CambridgeUniversity Press, pp. 126133, ISBN 0-521-58955-X, CiteSeerX: 10 .1 .1 .39 .2321

Courcelle, Bruno; Engelfriet, Joost (April 2011), Graph Structure and Monadic Second-Order Logic, a Lan-guage Theoretic Approach (PDF), Cambridge University Press, retrieved 17 May 2011

Droste, M.; Giraudet, M.; Macpherson, D. (March 1995), Periodic Ordered Permutation Groups and CyclicOrderings, Journal of Combinatorial Theory, Series B 63 (2): 310321, doi:10.1006/jctb.1995.1022

Droste, M.; Giraudet, M.; Macpherson, D. (March 1997), Set-Homogeneous Graphs and Embeddings ofTotal Orders, Order 14 (1): 920, doi:10.1023/A:1005880810385, CiteSeerX: 10 .1 .1 .22 .9135

Evans, David M. (17 November 1997), Finite covers with nite kernels, Annals of Pure and Applied Logic88 (23): 109147, doi:10.1016/S0168-0072(97)00018-3, CiteSeerX: 10 .1 .1 .57 .5323

Ivanov, A. A. (January 1999), Finite Covers, Cohomology and Homogeneous Structures, Proceedings of theLondon Mathematical Society 78 (1): 128, doi:10.1112/S002461159900163X

Jakubk, Jn (2006), On monotone permutations of -cyclically ordered sets (PDF), Czechoslovak Mathe-matical Journal 45 (2): 403415, hdl:10338.dmlcz/128075, retrieved 30 April 2011

Kennedy, Christine Cowan (August 1955), On a cyclic ternary relation ... (M.A. Thesis), Tulane University,OCLC 16508645

Knya, Eszter Herendine (2006), A mathematical and didactical analysis of the concept of orientation (PDF),Teaching Mathematics and Computer Science 4 (1): 111130, retrieved 17 May 2011

Knya, Eszter Herendine (2008), Geometrical transformations and the concept of cyclic ordering, in Maj,Boena; Pytlak, Marta; Swoboda, Ewa, Supporting Independent Thinking Through Mathematical Education(PDF), Rzeszw University Press, pp. 102108, ISBN 978-83-7338-420-0, retrieved 17 May 2011

Leloup, Grard (February 2011), Existentially equivalent cyclic ultrametric spaces and cyclically valued groups(PDF), Logic Journal of the IGPL 19 (1): 144173, doi:10.1093/jigpal/jzq024, retrieved 30 April 2011

Marongiu, Gabriele (1985), Some remarks on the 0-categoricity of circular orderings, Unione MatematicaItaliana. Bollettino. B. Serie VI (in Italian) 4 (3): 883900, MR 0831297

McCleary, Stephen; Rubin, Matatyahu (6 October 2005), Locally Moving Groups and the Reconstruction Prob-lem for Chains and Circles, arXiv:math/0510122

Mller, G. (1974), Lineare und zyklische Ordnung, Praxis der Mathematik 16: 261269, MR 0429660 Rubin, M. (1996), Locally moving groups and reconstruction problems, in Holland, W. Charles, Orderedgroups and innite permutation groups, Mathematics and Its Applications 354, Kluwer, pp. 121157, ISBN978-0-7923-3853-6

wierczkowski, S. (1956), On cyclic ordering relations, Bulletin de l'Acadmie Polonaise des Sciences, ClasseIII 4: 585586

wierczkowski, S. (1959b), On cyclically ordered intervals of integers (PDF), Fundamenta Mathematicae47: 167172, retrieved 2 May 2011


Truss, J.K. (July 1992), Generic Automorphisms of Homogeneous Structures, Proceedings of the LondonMathematical Society, s3-65 (1): 121141, doi:10.1112/plms/s3-65.1.121

3.12 External links cyclic order in nLab

Chapter 4

Delta-ring

In mathematics, a nonempty collection of setsR is called a -ring (pronounced delta-ring) if it is closed under union,relative complementation, and countable intersection:

1. A [B 2 R if A;B 2 R2. AB 2 R if A;B 2 R3. T1n=1An 2 R if An 2 R for all n 2 N

If only the rst two properties are satised, then R is a ring but not a -ring. Every -ring is a -ring, but not every-ring is a -ring.-rings can be used instead of -elds in the development of measure theory if one does not wish to allow sets ofinnite measure.

4.1 See also Ring of sets Sigma eld Sigma ring

4.2 References Cortzen, Allan. Delta-Ring. From MathWorldA Wolfram Web Resource, created by Eric W. Weisstein.

http://mathworld.wolfram.com/Delta-Ring.html

25

Chapter 5

Distributive lattice

In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other.The prototypical examples of such structures are collections of sets for which the lattice operations can be given byset union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is up to isomorphism given as such a lattice of sets.

5.1 DenitionAs in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of ordertheory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. Inthe present situation, the algebraic description appears to be more convenient:A lattice (L,,) is distributive if the following additional identity holds for all x, y, and z in L:

x (y z) = (x y) (x z).

Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty nite joins. It is abasic fact of lattice theory that the above condition is equivalent to its dual:[1]

x (y z) = (x y) (x z) for all x, y, and z in L.[2]

In every lattice, dening pq as usual to mean pq=p, the inequation x (y z) (x y) (x z) holds as well asits dual inequation x (y z) (x y) (x z). A lattice is distributive if one of the converse inequations holds,too. More information on the relationship of this condition to other distributivity conditions of order theory can befound in the article on distributivity (order theory).

5.2 MorphismsA morphism of distributive lattices is just a lattice homomorphism as given in the article on lattices, i.e. a functionthat is compatible with the two lattice operations. Because such a morphism of lattices preserves the lattice structure,it will consequently also preserve the distributivity (and thus be a morphism of distributive lattices).

5.3 ExamplesDistributive lattices are ubiquitous but also rather specic structures. As already mentioned the main example fordistributive lattices are lattices of sets, where join and meet are given by the usual set-theoretic operations. Furtherexamples include:

26

5.3. EXAMPLES 27

Youngs lattice

The Lindenbaum algebra of most logics that support conjunction and disjunction is a distributive lattice, i.e.and distributes over or and vice versa.

Every Boolean algebra is a distributive lattice.

Every Heyting algebra is a distributive lattice. Especially this includes all locales and hence all open set latticesof topological spaces. Also note that Heyting algebras can be viewed as Lindenbaum algebras of intuitionisticlogic, which makes them a special case of the above example.

Every totally ordered set is a distributive lattice with max as join and min as meet.

The natural numbers form a distributive lattice (complete as a meet-semilattice) with the greatest commondivisor as meet and the least common multiple as join.

Given a positive integer n, the set of all positive divisors of n forms a distributive lattice, again with the greatestcommon divisor as meet and the least common multiple as join. This is a Boolean algebra if and only if n issquare-free.

A lattice-ordered vector space is a distributive lattice.

Youngs lattice given by the inclusion ordering of Young diagrams representing integer partitions is a distributivelattice.

Early in the development of the lattice theory Charles S. Peirce believed that all lattices are distributive, that is,distributivity follows from the rest of the lattice axioms.[3][4] However, independence proofs were given by Schrder,Voigt,(de) Lroth, Korselt,[5] and Dedekind.[3]

28 CHAPTER 5. DISTRIBUTIVE LATTICE

5.4 Characteristic propertiesVarious equivalent formulations to the above denition exist. For example, L is distributive if and only if the followingholds for all elements x, y, z in L:

(x ^ y) _ (y ^ z) _ (z ^ x) = (x _ y) ^ (y _ z) ^ (z _ x).

Similarly, L is distributive if and only if

x ^ z = y ^ z and x _ z = y _ z always imply x=y.

Hasse diagrams of the two prototypical non-distributive lattices The diamond lattice M3 is non-distributive: x (y z) = x 1 = x 0 = 0 0 = (x y) (x z). The pentagon lattice N5 is non-distributive: x (y z) = x 1 = x z = 0 z = (x y) (x z).

Distributive lattice which contains N5 (solid lines, left) and M3 (right) as subset, but not as sublattice, respectively

The simplest non-distributive lattices are M3, the diamond lattice, and N5, the pentagon lattice. A lattice isdistributive if and only if none of its sublattices is isomorphic to M3 or N5; a sublattice is a subset that is closed underthe meet and join operations of the original lattice. Note that this is not the same as being a subset that is a latticeunder the original order (but possibly with dierent join and meet operations). Further characterizations derive fromthe representation theory in the next section.Finally distributivity entails several other pleasant properties. For example, an element of a distributive lattice ismeet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property. By duality, thesame is true for join-prime and join-irreducible elements.[6] If a lattice is distributive, its covering relation forms amedian graph.[7]

Furthermore, every distributive lattice is also modular.

5.5 Representation theoryThe introduction already hinted at the most important characterization for distributive lattices: a lattice is distributiveif and only if it is isomorphic to a lattice of sets (closed under set union and intersection). That set union andintersection are indeed distributive in the above sense is an elementary fact. The other direction is less trivial, inthat it requires the representation theorems stated below. The important insight from this characterization is that theidentities (equations) that hold in all distributive lattices are exactly the ones that hold in all lattices of sets in the abovesense.

5.6. FREE DISTRIBUTIVE LATTICES 29

Birkhos representation theorem for distributive lattices states that every nite distributive lattice is isomorphic tothe lattice of lower sets of the poset of its join-prime (equivalently: join-irreducible) elements. This establishes abijection (up to isomorphism) between the class of all nite posets and the class of all nite distributive lattices.This bijection can be extended to a duality of categories between homomorphisms of nite distributive lattices andmonotone functions of nite posets. Generalizing this result to innite lattices, however, requires adding furtherstructure.Another early representation theorem is now known as Stones representation theorem for distributive lattices (thename honors Marshall Harvey Stone, who rst proved it). It characterizes distributive lattices as the lattices ofcompact open sets of certain topological spaces. This result can be viewed both as a generalization of Stones famousrepresentation theorem for Boolean algebras and as a specialization of the general setting of Stone duality.A further important representation was established by Hilary Priestley in her representation theorem for distributivelattices. In this formulation, a distrib

Date post:	08-Sep-2015
Category:	Documents
Upload:	man
View:	239 times
Download:	11 times

Ring of sets 1_5

Documents