Graph factorization From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Profile ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 Abelian group 5 2.1 Definition ............................................... 5 2.2 Facts ................................................. 5 2.2.1 Notation ........................................... 5 2.2.2 Multiplication table ...................................... 6 2.3 Examples ............................................... 6 2.4 Historical remarks .......................................... 6 2.5 Properties ............................................... 6 2.6 Finite abelian groups ......................................... 7 2.6.1 Classification ......................................... 7 2.6.2 Automorphisms ....................................... 7 2.7 Infinite abelian groups ........................................ 8 2.7.1 Torsion groups ........................................ 9 2.7.2 Torsion-free and mixed groups ................................ 9 2.7.3 Invariants and classification .................................. 9 2.7.4 Additive groups of rings ................................... 9 2.8 Relation to other mathematical topics ................................. 10 2.9 A note on the typography ....................................... 10 2.10 See also ................................................ 10 2.11 Notes ................................................. 11 2.12 References .............................................. 11 2.13 External links ............................................. 11 3 Associative algebra 12 3.1 Formal definition ........................................... 12 3.1.1 From R-modules ....................................... 12 i ii CONTENTS 3.1.2 From rings .......................................... 13 3.2 Algebra homomorphisms ....................................... 13 3.3 Examples ............................................... 13 3.4 Constructions ............................................. 14 3.5 Associativity and the multiplication mapping ............................. 15 3.6 Coalgebras ............................................... 15 3.7 Representations ............................................ 15 3.7.1 Motivation for a Hopf algebra ................................. 16 3.7.2 Motivation for a Lie algebra .................................. 16 3.8 See also ................................................ 17 3.9 References ............................................... 17 4 Automata theory 18 4.1 Automata ............................................... 19 4.1.1 Informal description ..................................... 19 4.1.2 Formal definition ....................................... 19 4.2 Variant definitions of automata .................................... 19 4.3 Classes of automata .......................................... 21 4.3.1 Discrete, continuous, and hybrid automata .......................... 21 4.4 Hierarchy in terms of powers ..................................... 21 4.5 Applications .............................................. 21 4.6 Automata simulators .......................................... 21 4.7 Connection to category theory ..................................... 22 4.8 References ............................................... 22 4.9 Further reading ............................................ 22 4.10 External links ............................................. 23 5 Bijection 24 5.1 Definition ............................................... 25 5.2 Examples ............................................... 25 5.2.1 Batting line-up of a baseball team ............................... 25 5.2.2 Seats and students of a classroom ............................... 25 5.3 More mathematical examples and some non-examples ........................ 26 5.4 Inverses ................................................ 26 5.5 Composition .............................................. 26 5.6 Bijections and cardinality ....................................... 26 5.7 Properties ............................................... 27 5.8 Bijections and category theory ..................................... 27 5.9 Generalization to partial functions ................................... 28 5.10 Contrast with ............................................. 28 5.11 See also ................................................ 28 5.12 Notes ................................................. 28 CONTENTS iii 5.13 References ............................................... 29 5.14 External links ............................................. 29 6 Bipartite graph 30 6.1 Examples ............................................... 31 6.2 Properties ............................................... 32 6.2.1 Characterization ........................................ 32 6.2.2 König’s theorem and perfect graphs .............................. 32 6.2.3 Degree ............................................ 32 6.2.4 Relation to hypergraphs and directed graphs ......................... 33 6.3 Algorithms ............................................... 33 6.3.1 Testing bipartiteness ..................................... 33 6.3.2 Odd cycle transversal ..................................... 34 6.3.3 Matching ........................................... 35 6.4 Additional applications ........................................ 35 6.5 See also ................................................ 35 6.6 References .............................................. 36 6.7 External links ............................................. 37 7 Category (mathematics) 38 7.1 Definition ............................................... 39 7.2 History ................................................. 39 7.3 Small and large categories ....................................... 40 7.4 Examples ............................................... 40 7.5 Construction of new categories .................................... 40 7.5.1 Dual category ......................................... 40 7.5.2 Product categories ...................................... 41 7.6 Types of morphisms .......................................... 41 7.7 Types of categories .......................................... 42 7.8 See also ................................................ 42 7.9 Notes ................................................. 42 7.10 References ............................................... 42 8 Class function (algebra) 44 8.1 Characters ............................................... 44 8.2 Inner products ............................................. 44 8.3 See also ................................................ 44 8.4 References .............................................. 44 9 Complete bipartite graph 45 9.1 Definition ............................................... 45 9.2 Examples ............................................... 45 9.3 Properties ............................................... 46 iv CONTENTS 9.4 See also ................................................ 47 9.5 References ............................................... 47 10 Complete graph 48 10.1 Properties ............................................... 48 10.2 Geometry and topology ........................................ 48 10.3 Examples ............................................... 49 10.4 See also ................................................ 49 10.5 References ............................................... 49 10.6 External links ............................................. 49 11 Complete metric space 50 11.1 Examples ............................................... 50 11.2 Some theorems ............................................ 51 11.3 Completion .............................................. 51 11.4 Topologically complete spaces ..................................... 52 11.5 Alternatives and generalizations .................................... 52 11.6 See also ................................................ 52 11.7 Notes ................................................. 53 11.8 References ............................................... 53 12 Conjugacy class 54 12.1 Definition ............................................... 54 12.2 Examples ............................................... 54 12.3 Properties ............................................... 55 12.4 Conjugacy class equation ....................................... 55 12.4.1 Example ............................................ 56 12.5 Conjugacy of subgroups and general subsets ............................. 56 12.6 Conjugacy as group action ....................................... 56 12.7 Geometric interpretation ....................................... 56 12.8 See also ................................................ 57 12.9 References ............................................... 57 13 Connected space 58 13.1 Formal definition ........................................... 59 13.1.1 Connected components .................................... 59 13.1.2 Disconnected spaces ..................................... 59 13.2 Examples ............................................... 59 13.3 Path connectedness .......................................... 60 13.4 Arc connectedness .......................................... 61 13.5 Local connectedness ......................................... 61 13.6 Set operations ............................................