Appendix:Glossary of probability andstatistics

From en.wiktionary.org

Contents

1 Appendix:Glossary of abstract algebra 11.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.9 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Appendix:Glossary of cryptography 32.1 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Appendix:Glossary of ecology 53.1 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.6 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.7 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Appendix:Glossary of game theory 74.1 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.4 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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4.6 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.7 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.8 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.9 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.10 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.11 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.12 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.13 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.14 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.15 Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Appendix:Glossary of grammar 115.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.6 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.8 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.9 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.10 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Appendix:Glossary of graph theory 136.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.10 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.11 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.12 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.13 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.14 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.15 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.16 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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6.17 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.18 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.19 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.20 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.21 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.22 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Appendix:Glossary of group theory 267.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.5 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.8 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.9 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.10 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Appendix:Glossary of linear algebra 288.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.3 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.5 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.6 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.7 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.8 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9 Appendix:Glossary of logic 309.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.4 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.5 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.6 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.7 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.8 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.9 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Appendix:Glossary of order theory 32

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10.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.5 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.6 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.7 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.8 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.9 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.10L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.11M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.12O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.13P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.14Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.15R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.16S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.17T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.18U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.19W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.20Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11 Appendix:Glossary of philosophical isms 3911.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.10J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.11K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.12L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.13M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.14N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.15O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.16P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.17R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.18S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.19T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

CONTENTS v

11.20U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.21V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.22Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.23External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12 Appendix:Glossary of philosophy 6212.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.2 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.3 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.4 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.5 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.6 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.7 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.8 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13 Appendix:Glossary of probability and statistics 6413.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.8 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.9 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.10K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.11L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.12M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.13N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.14O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.15P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.16R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.17S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.18See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.19Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 69

13.19.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.19.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.19.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 1

Appendix:Glossary of abstract algebra

This is a glossary of abstract algebraTable of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

1.1 Aassociative Of an operator *, such that, for any operands a,b,c, (a * b) * c = a * (b * c).

1.2 Ccommutative Of an operator *, such that, for any operands a,b, a * b = b * a.

1.3 Ddistributive Of an operation * with respect to the operation o, such that a * (b o c) = (a * b) o (a * c).

1.4 Feld A set having two operations called addition and multiplication under both of which all the elements of the set

are commutative and associative; for which multiplication distributes over addition; and for both of which thereexist an identity element and an inverse element.

1.5 Ggroup A set with an associative binary operation, under which there exists an identity element, and such that each

element has an inverse.

1.6 Iideal A subring closed under multiplication by its containing ring.

identity element Amember of a structure which, when applied to any other element via a binary operation inducesan identity mapping.

1

2 CHAPTER 1. APPENDIX:GLOSSARY OF ABSTRACT ALGEBRA

1.7 Mmonoid A set which is closed under an associative binary operation, and which contains an element which is an

identity for the operation.

1.8 Rring An algebraic structure which is a group under addition and a monoid under multiplication.

1.9 Ssemigroup Any set for which there is a binary operation that is both closed and associative.

semiring An algebraic structure similar to a ring, but without the requirement that each element must have anadditive inverse.

1.10 See also Appendix:English arities and adicities Appendix:English polynomial degrees Appendix:Glossary of group theory

Chapter 2

Appendix:Glossary of cryptography

This is a glossary of cryptography.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

2.1 Ccipher A cryptographic system using an algorithm that converts letters or sequences of bits into ciphertext.

ciphertext Encoded text; text that is unreadable.

cleartext The unencrypted form of an encrypted text; plaintext.

code A cryptographic system using a codebook that converts words or phrases into codewords.

codebook A book, table, database, or other object that stores the mapping between plaintext words or phrases andtheir equivalents in a code.

codeword A string representing an encoded piece of text.

cryptanalysis The science of analyzing and breaking of codes and ciphers.

cryptology The practice of analysing encoded messages, in order to decode them.

cryptography The discipline concerned with communication security (eg, condentiality of messages, integrity ofmessages, sender authentication, non-repudiation of messages, and many other related issues), regardless ofthe used medium such as pencil and paper or computers.

2.2 Kkey A piece of information (e.g. a passphrase) used to encode or decode a message or messages.

keyspace The notional space that contains all possible keys.

2.3 Pplain text Un-encrypted text, text that is readable.

private key The unpublished key in a cryptographic system that uses two keys.

public key The public one of the two keys used in asymmetric cryptography.

3

4 CHAPTER 2. APPENDIX:GLOSSARY OF CRYPTOGRAPHY

2.4 Ssubstitution cipher A method of encryption by which units of plaintext are substituted with ciphertext according

to a regular system; the units may be single letters (the most common), pairs of letters, triplets of letters,mixtures of the above, and so forth. The receiver deciphers the text by performing an inverse substitution.

Chapter 3

Appendix:Glossary of ecology

This is a glossary of ecology.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

3.1 Ccommensal An organism partaking in a commensal relationship.

commensalism A sharing of the same environment by two organisms without specic harm or benet to either.

3.2 Eecosystem A system formed by an ecological community and its environment that functions as a unit.

3.3 Hhost A cell or organism which harbors another organism or biological entity, usually a parasite.

3.4 Mmutualism Any interaction between two species that benets both; typically involves the exchange of substances or

services.

3.5 Nniche A function within an ecological system to which an organism is especially suited.

3.6 Pparasite A (generally undesirable) living organism that exists by stealing the resources needed by another living

organism.

parasitism An interaction between two organisms, in which one organism (the parasite) benets and the other (thehost) is harmed

5

6 CHAPTER 3. APPENDIX:GLOSSARY OF ECOLOGY

3.7 Ssymbiote An organism in a partnership with another such that each prots from their being together; a symbiont.

symbiosis A close, prolonged association between two or more dierent organisms of dierent species that normallybenets both members.

Chapter 4

Appendix:Glossary of game theory

This is a glossary of game theorythe branch of mathematics in which games are studied: that is, models describinghuman behaviour.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

4.1 Notational conventions

real numbers R .

The set of players N .

strategy space =Qi2N i . Where:player is strategy space i is the space of all possible ways in which player i can play the game.

strategy for player i i is an element of .

complements i an element of i =Qj2N;j 6=i

j , is a tuple of strategies for all players other than i.

outcome space is in most textbooks identical to -

payos RN , describing how much gain (money, pleasure, etc.) the players are allocated by the end of the game.

4.2 A

acceptable game A game form such that for every possible preference proles, the game has pure Nash equilibria,all of which are pareto ecient.

allocation of goods A function : ! RN . The allocation is a cardinal approach for determining the good (e.g.money) the players are granted under the dierent outcomes of the game.

4.3 B

best reply The best reply to a given complement i is a strategy i that maximizes player i's payment. Formally,we want:8 i 2 i ( i; i) ( i; i) .

7

8 CHAPTER 4. APPENDIX:GLOSSARY OF GAME THEORY

4.4 Ccoalition Any subset of the set of players: S N .

Condorcet winner An outcome such that all non-dummy players prefer it to all other outcomes, given a preference on the outcome space.

cooperative game A game in which players are allowed form coalitions (and to enforce coalitionary discipline). Acooperative game is given by stating a value for every coalition: : 2P(N) ! RIt is always assumed that the empty coalition gains nil. Solution concepts for cooperative games usually assumethat the players are forming the grand coalition N , whose value (N) is then divided among the players togive an allocation.

coordination game A normal form game in which players have the same sets of strategies and their payos arehigher if they chose the same strategies than they are if they choose dierent strategies.

4.5 Ddictator A player is a strong dictator if he can guarantee any outcome regardless of the other players. m 2 N is a

weak dictator if he can guarantee any outcome, but his strategies for doing so might depend on the complementstrategy vector. Naturally, every strong dictator is a weak dictator. Formally:m is a Strong dictator if:8a 2 A; 9 n 2 n s:t: 8 n 2 n : ( n; n) = am is aWeak dictator if:8a 2 A; 8 n 2 n 9 n 2 n s:t: ( n; n) = a

Another way to put it is:a weak dictator is -eective for every possible outcome.A strong dictator is -eective for every possible outcome.See Eectiveness. Antonym: dummy.

dominated outcome Given a preference on the outcome space, we say that an outcome a is dominated by outcomeb (hence, b is the dominant strategy) if it is preferred by all players. If, in addition, some player strictly prefersb over a, then we say that a is strictly dominated. Formally:8j 2 N j(a) j(b) for domination, and9i 2 N s:t: i(a) < i(b) for strict domination.An outcome a is (strictly) dominated if it is (strictly) dominated by some other outcome.An outcome a is dominated for a coalition S if all players in S prefer some other outcome to a. See alsoCondorcet winner.

dominated strategy A strategy is (strongly) dominated by strategy i if for any complement strategies tuple i, player i benets by playing i . Formally speaking:8 i 2 i ( i; i) ( i; i) and9 i 2 i s:t: ( i; i) < ( i; i) .A strategy is (strictly) dominated if it is (strictly) dominated by some other strategy.

dummy A player i is a dummy if he has no eect on the outcome of the game. I.e. if the outcome of the game isinsensitive to player i's strategy. Antonyms: say, veto, dictator.

4.6 Eeectiveness A coalition (or a single player) S is eective for a if it can force a to be the outcome of the game. S

is -eective if the members of S have strategies s.t. no matter what the complement of S does, the outcomewill be a.

4.7. F 9

S is -eective if for any strategies of the complement of S, the members of S can answer with strategies thatensure outcome a.

extensive form game A tree, where, at each node of the tree, a dierent player has the choice of choosing an edge.

4.7 Fnite game A game with nitely many players, each of which has a nite set of strategies.

4.8 Mmixed strategy For player i, a probability distribution P on i . It is understood that player i chooses a strategy

randomly according to P.

mixed Nash equilibrium Same as Pure Nash Equilibrium, dened on the space of mixed strategies. Every nitegame has Mixed Nash Equilibria.

4.9 NNash equilibrium The set of choices of players strategies for which no player can benet by changing his or her

strategy while the other players keep theirs unchanged.

normal form game A function: i ! RN , given the tuple of strategies chosen by the players, one is given anallocation of payments (given as real numbers).A further generalization can be achieved by splitting the game into a composition of two functions: i ! the outcome function of the game (some authors call this function the game form), and : ! RN theallocation of payos (or preferences) to players, for each outcome of the game.

4.10 Ooutcome function A function that assigns to each combination of chosen strategies an outcome from an outcome

space. It appears in one of the formalizations of a normal form game; see normal form game.

outcome space A set of all the possible outcomes of a gameoutcomes to which a real-valued award is assignedfor each player. Conventionally labeled as . It appears in one of the formalizations of a normal form game;see normal form game.

4.11 PPareto eciency The property of being Pareto ecient.

Pareto ecient An outcome a of game form is (strongly) Pareto ecient if it is undominated under all preferenceproles.

Pareto optimal Of an allocation of goods to individual, such that it is Pareto ecient.

Pareto optimal Of a strategy of a player, such that there is no alternative strategy outperforming it agaist at leastone opponents strategy while not underperforming it against any opponents strategy.

Pareto optimality The property of being Pareto optimal.

10 CHAPTER 4. APPENDIX:GLOSSARY OF GAME THEORY

payo function Informally, a mathematical function describing the real-valued award given to a single player atthe outcome of a game. Formally, a function F : ! R , where is the strategy space. Accordingly, tocompletely specify a game, the payo function has to be specied for each player in the player set P= {1, 2,..., m}.

preference prole A real-valued function that species for each outcome of the game and each of N players to whatdegree he prefers the outcome; that is, : ! RN , where is the outcome space. See allocation of goods.This is the ordinal approach at describing the outcome of the game.

pure Nash equilibrium point An element = i2N of the strategy space of a game is a pure Nash equilibriumpoint if no player i can benet by deviating from his strategy i , given that the other players are playing in . Formally:8i 2 N 8 i 2 i ( ; i) ( ) .No equilibrium point is dominated.

4.12 Ssay A player i has a Say if he is not a Dummy, i.e. if there is some tuple of complement strategies s.t. (_i) is not

a constant function. Antonym: dummy.

simple game A simplied form of a cooperative game, where the possible gain is assumed to be eiter '0' or '1'. Asimple game is couple (N,W), whereW is the list of winning coalitions, capable of gaining the loot ('1'), andN is the set of players.

4.13 Vvalue A value of a game is a rationally expected outcome. There are more than a few denitions of value, describing

dierent meathods of obtaining a solution to the game.

veto A veto denotes the ability (or right) of some player to prevent a specic alternative from being the outcome ofthe game. A player who has that ability is called a veto player. Antonym: dummy.

4.14 Wweakly acceptable game A game that has pure Nash equilibria some of which are pareto ecient.

4.15 Zzero-sum game A normal form game for which, for each combination of strategies chosen by each player, the sum

of the players payos is zero.

4.16 External links Glossary of game theory on Wikipedia.Wikipedia

Chapter 5

Appendix:Glossary of grammar

This is a glossary of grammar.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

5.1 Aaspect Grammatical quality of a verb which determines the relationship of the speaker to the internal temporal ow

of the event the verb describes; whether the speaker views the event from outside as a whole, or from withinas it is unfolding.

auxiliary verb A verb that accompanies the main verb in a clause in order to make distinctions in tense, mood,voice or aspect.

5.2 Ccausative An expression of an agent causing or forcing a patient to perform an action (or to be in a certain condition).

clause A word or group of words ordinarily consisting of a subject and a predicate; in some languages and typesof clauses, the subject may not appear explicitly; one clause may be coordinated with or embedded in anotherwithin a single sentence.

complement Aword or group of words that completes a grammatical construction in the predicate and that describesor is identied with the subject or object.

complement Any word or group of words used to complete a grammatical construction, typically in the predicate,including adverbials, innitives, and sometimes objects.

copula Aword used to link the subject of a sentence with a predicate (usually a subject complement or an adverbial);it serves to unite (or associate) the subject with the predicate. (e.g. be).

5.3 Ddangling modier A word or clause that modies another word or clause ambiguously, possibly causing confusion

with regard to the speakers intended meaning.

5.4 Ffrequentative Serving to express the frequent repetition of an action.

11

12 CHAPTER 5. APPENDIX:GLOSSARY OF GRAMMAR

5.5 Iiterative Expressive of an action that is repeated with frequency.

5.6 Mmodier A word, phrase, or clause that limits or qualies the sense of another word or phrase.

mood A verb form that depends on how its containing clause relates to the speakers or writers wish, intent, orassertion about reality.

5.7 Oobject The noun phrase which is an internal complement of a verb phrase or a prepositional phrase. In a verb phrase

with a transitive action verb, it is typically the receiver of the action.

5.8 Pphrasal verb A phrase consisting of a verb and either or both a preposition or adverb, that has idiomatic meaning.

preposition Amember of a closed class of non-inecting words typically employed to connect a noun or a pronoun,in an adjectival or adverbial sense, with some other word; a particle used with a noun or pronoun (in Englishalways in the objective case) to make a phrase limiting some other word; so called because it is usuallyplaced before the word with which it is phrased; as, a bridge of iron; he comes from town; it is good forfood; he escaped by running. Prepositions are a heterogeneous class of words in some languages, with fuzzyboundaries that tend to overlap with other categories (like adverbs, adjectives, and conjunctions).

prepositional phrase A phrase that has both a preposition and its object or complement; may be used as an adjunctor a modier.

5.9 Qqualier A word or phrase, such as an adjective or adverb, that describes or characterizes another word or phrase,

such as a noun or verb; a modier; that adds or subtracts attributes to another.

5.10 Ssubject In a clause: the word or word group (usually a noun phrase) that is dealt with. In active clauses with verbs

denoting an action, the subject and the actor are usually the same.

5.11 See also Appendix:Grammatical cases

Chapter 6

Appendix:Glossary of graph theory

This is a glossary of graph theorya mathematical theory of graphs consisting of vertices and edges that connectvertices.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

6.1 A

123

546

An illustration of a graph

acyclic Of graph, such one that it does not contain any directed cycle. A nite, acyclic digraph with no isolatedvertices necessarily contain at least one source and at least one sink. See also directed acyclic graph (DAG forshort) for more.

acyclic graph A graph that contains no cycles.

13

14 CHAPTER 6. APPENDIX:GLOSSARY OF GRAPH THEORY

adjacent Two edges are adjacent if they have a node in common; two nodes are adjacent if they have an edge incommon.

anti-edge An edge that is not there. More formally, for two vertices u and v , fu; vg is an anti-edge in a graphG if (u; v) is not an edge in G . This means that there is either no edge between the two vertices or there isonly an edge (v; u) from v to u if G is directed.

anti-triangle A set of three vertices none of which are connected.

arborescence (also out-tree or branching) An oriented tree in which all vertices are reachable from a single vertex.Likewise, an in-tree is an oriented tree in which a single vertex is reachable from every other one.

adjacency matrix In computers, a nite, directed or undirected graph (with n vertices, say) is often represented byits adjacency matrix: an n-by-nmatrix whose entry in row i and column j gives the number of edges from thei-th to the j-th vertex.

adjacent Two vertices u and v are considered adjacent if an edge exists between them. We denote this by u v.In the above graph, vertices 1 and 2 are adjacent, but vertices 2 and 4 are not. The set of neighbors, called a(open) neighborhood NG(v) for a vertex v in a graph G, consists of all vertices adjacent to v but not includingv. When v is also included, it is called a closed neighborhood, denoted by NG[v]. When stated without anyqualication, a neighborhood is assumed to be open. The subscript G is usually dropped when there is nodanger of confusion. In the example graph, vertex 1 has two neighbors: vertices 2 and 5. For a simple graph,the number of neighbors that a vertex has coincides with its degree.

arc (also directed edge) An ordered pair of endvertices.

articulation point A cut vertex.

6.2 Bbalanced k-partite graph A k-partite graph such that each partite set diers in size by at most 1 with any other.

biclique A complete bipartite graph.

biconnected component A maximal set of edges in which any two members lie on a common simple cycle.

bipartite graph A graph that can be decomposed into two partite sets but not fewer.

biregular graph A graph that has unequal maximum and minimum degrees and every vertex has one of those twodegrees.

block Either a maximally 2-connected subgraph or a bridge with its endvertices.

bond A minimal (but not necessarily minimum), nonempty set of edges whose removal disconnects a graph.

bridge An edge whose removal disconnects a graph. (For example, a tree is made entirely of bridges.)

6.3 Ccenter A vertex with minimum eccentricity.

circuit A circuit of n vertices, denoted by Cn, is usually assumed to be a simple cycle, or a simple circuit, meaningthat every vertex is incident to exactly two edges. In the above graph (1, 5, 2, 1) is a simple cycle.

6.3. C 15

circumference The length of a longest (simple) cycle, or innity if the graph is acyclic.

claw An induced star with 3 edges.

clique In a graph, a set of pairwise adjacent vertices. Since any subgraph induced by a clique is a complete subgraph,the two terms and their notations are usually used interchangeably.

k-clique A clique of order k. In the example graph above, vertices 1, 2 and 5 form a 3-clique, or a triangle.

clique number (G) of a graph G, the order of a largest clique in G.

color, colored, identied Nodes or edges which are considered as individuals. Although they may actually be ren-dered in diagrams in dierent colors, working mathematicians generally pencil in numbers or letters.

k-colorable graph' A graph that can be decomposed into k partite sets.

complement G of a graph G is a graph with the same vertex set as G but with an edge set such that xy is an edge inG if and only if xy is not an edge in G.

complete A graph in which every node is linked to every other node. For a complete digraph, this means one linkin either direction.

complete graph Kn of order n, a simple graph with n vertices in which every vertex is adjacent to every other. Theexample graph is not complete. The complete graph on n vertices is often denoted by Kn. It has n(n1)/2edges (corresponding to all possible choices of pairs of vertices).

complete multipartite graph A graph in which vertices are adjacent if and only if they belong to dierent partitesets.

component A maximally connected subgraph.

connected If some route exists from every node to every other, the graph is connected. Note that some graphs arenot connected. A diagram of an unconnected graph may look like two or more unrelated diagrams, but all thenodes and edges shown are considered as one graph.

connectivity extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency.

connected graph If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is saidto be connected.

k-connected If it is always possible to establish a path from any vertex to every other one even after removing any k- 1 vertices, then the graph is said to be k-connected. Note that a graph is k-connected if and only if it containsk internally disjoint paths between any two vertices. The example graph above is connected (and therefore1-connected), but not 2-connected.

connectivity (G) of a graph G is the minimum number of vertices needed to disconnect G. By convention, Kn hasconnectivity n - 1; and a disconnected graph has connectivity 0.

crossing A pair of intersecting edges.

crossing number The minimum number of crossings that must appear when a graph is drawn on a plane is calledthe crossing number.

cut A partition of the vertices of a graph into two disjoint subsets.

16 CHAPTER 6. APPENDIX:GLOSSARY OF GRAPH THEORY

cut edge A bridge.

cut set A set of vertices whose removal disconnects the graph.

cut vertex A vertex whose removal disconnects a graph. Also known as Cut Point or Articulation Point. See alsocut set and bridge.

cycle In a graph was a closed walk.

6.4 DDAG A directed acyclic graph.

degree The number of edges which connect a node.

degree dG(v) of a vertex v in a graph G is the number of edges incident to v, with loops being counted twice. Avertex of degree 0 is an isolated vertex. A vertex of degree 1 is a leaf. In the example graph vertices 1 and 3have a degree of 2, vertices 2,4 and 5 have a degree of 3 and vertex 6 has a degree of 1. If E is nite, then thetotal sum of vertex degrees is equal to twice the number of edges.

degree sequence A list of degrees of a graph in non-increasing order (e.g. d1 d2 dn). A sequence ofnon-increasing integers is realizable if it is a degree sequence of some graph.

diagram A visible rendering of the abstract concept of a graph.

diameter The maximum eccentricity over all vertices in a graph. It is denoted diam(G) of a graph G.

digraph A directed graph.

directed cycle (also cycle) An oriented simple cycle such that all arcs go the same direction, meaning all verticeshave in- and out-degrees 1.

directed A graph in which each edge symbolizes an ordered, non-transitive relationship between two nodes. Suchedges are rendered with an arrowhead at one end of a line or arc.

directed graph A graph whose edges are directed, possibly represented as ordered pairs of vertices; synonyms:digraph. Alternative models of graph exist; e.g., a graph may be thought as a Boolean binary function over theset of vertices or as a square (0,1)-matrix.

disconnected graph A graph that is not connected.

disconnecting set A set of edges whose removal increases the number of components.

distance dG(u, v) between two (not necessary distinct) vertices u and v in a graph G is the length of a shortest pathbetween them. The subscript G is usually dropped when there is no danger of confusion. When u and v areidentical, their distance is 0. When u and v are unreachable from each other, their distance is dened to beinnity .

dominate A vertex v dominates another vertex u if there is an arc from v to u. A vertex subset S is out-dominatingif every vertex not in S is dominated by some vertex in S; and in-dominating if every vertex in S is dominatedby some vertex not in S.

dominating set Of a graph, a vertex subset whose closed neighborhood include all vertices of the graph. A vertexv dominates another vertex u if there is an edge from v to u. A vertex subset V dominates another vertexsubset U if every vertex in U is adjacent to some vertex in V. The minimum size of a dominating set is thedomination number (G).

6.5. E 17

dual A dual, or planar dual when the context needs to be claried, G* of a plane graph G is a graph whose verticesrepresent the faces, including any outerface, of G and are adjacent in G* if and only if their correspondingfaces are adjacent in G. The dual of a planar graph is always a planar pseudograph (e.g. consider the dual of atriangle). In the familiar case of a 3-connected simple planar graph G (isomorphic to a convex polyhedron P),the dual G* is also a 3-connected simple planar graph (and isomorphic to the dual polyhedron P*).

6.5 Eeccentricity G(v) of a vertex v in a graph G is the maximum distance from v to any other vertex.

edge A set of two elements, drawn as a line connecting two vertices, called endvertices, or endpoints. An edge withendvertices x and y is denoted by xy without any symbol in between, so, do not write xy. The edge set of G isusually denoted by E(G), or E when there is no danger of confusion.

edge, link, arc Relationships represented in a graph. These are always rendered as straight or curved lines. Theterm arc may be misleading.

edge cut The set of all edges having one endvertex in some proper vertex subset S and another endvertex in V(G)\S.Edges of K3 form a disconnecting set but not an edge cut. Any two edges of K3 form a minimal disconnectingset as well as an edge cut. An edge cut is necessarily a disconnecting set; and a minimal disconnecting set ofan nonempty graph is necessarily an edge cut.

edgeless graph (empty graph) A graph possibly with some vertices, but no edges. Or, it is a graph with no verticesand no edges.

k-edge-connected Of a graph, such that if any subgraph formed by removing any k - 1 edges is still connected.The edge connectivity (G) of a graph G is the minimum number of edges needed to disconnect G. Onewell-known result is that (G) (G) (G).

embeddable A graph is embeddable on a surface if its vertices and edges can be arranged on it without any crossing.The genus of a graph is the lowest genus of any surface on which the graph can embed.

embedding G1 = (V1; E1) ofG2 = (V2; E2) is an injection from V2 to V1 such that every edge inE2 correspondsto a path(disjoint from all other such paths) in G1 .

equipartite graph A k-partite graph such that each partite set has the same size.

Eulerian circuit Eulerian cycle.

Eulerian cycle A closed walk which uses each edge precisely once.

Eulerian digraph A digraph with equal in- and out-degrees at every vertex.

Eulerian path A path which passes through every edge (once and only once). If the starting and ending nodes arethe same, it is an Euler cycle or an Euler circuit. If the starting and ending nodes are dierent, it is an Eulertrail.

Eulerian path A graph is a walk that uses each edge precisely once; that is, a trail that uses all the edges. If such atrail exists, the graph is called traversable.

Eulerian trail Eulerian path.

Eulerian tour Eulerian cycle.

even cycle A cycle that has an even length.

18 CHAPTER 6. APPENDIX:GLOSSARY OF GRAPH THEORY

6.6 Fface When a graph is drawn without any crossing, any cycle that surrounds a region without any edge reaching from

the cycle inside to such region forms a face. Two faces on a plane graph are adjacent if they share a commonedge.

factor A spanning subgraph.

k-factor A k-regular spanning subgraph. An 1-factor is a perfect matching. A partition of edges of a graph intok-factors is called a k-factorization. A k-factorable graph is a graph that admits a k-factorization.

forest A vertex-disjoint union of trees; or, equivalently, an acyclic graph.

6.7 Ggirth Of a graph, the length of a shortest (simple) cycle in the graph, or innity if the graph is acyclic.

graph, network An abstraction of relationships among objects. Graphs consist exclusively of nodes and edges. Allcharacteristics of a system are either eliminated or subsumed into these elements.

graph A mathematical structure costiting of two types of elements, namely vertices and edges. Every edge has twoendpoints in the set of vertices, ans is said to connect or join the two endpoints. The set of edges can thusbe dened as a subset of the family of all two-element sets of vertices. Often, however, the set of vertices isconsidered as a set, and there is an incidence relation which maps each edge to the pair of vertices that are itsendpoints.

graph invariant A property of a graph G, usually a number or a polynomial, that depends only on the isomorphismclass of G; examples are genus and graph order.

graph labeling The assignment of unique labels (usually natural numbers) to the edges and vertices of a graph.Graphs with labeled edges or vertices are known as labeled, those without as unlabeled. More specically,graphs with labeled vertices only are vertex-labeled, those with labeled edges only are edge-labeled. (Thisusage is to distinguish between graphs with identiable vertex or edge sets on the one hand, and isomorphismtypes or classes of graphs on the other.)

6.8 HHamiltonian cycle A spanning cycle.

Hamiltonian graph A graph that contains a Hamiltonian cycle.

Hamiltonian path A (simple) path that contains every vertex.

Hamiltonian path A path which passes through every node once and only once. If the starting and ending nodesare adjacent, it is a Hamiltonian cycle.

Hamiltonian path A spanning path.

Hamiltonian connected graph Agraph that, given any pair of (distinct) vertices, contains a Hamiltonian path usingthem as endvertices.

H-free graf A graph that does not contain H as an induced subgraph is said to be H-free.

6.9. I 19

head The rst vertex in the arc AKA directed edge. See also tail.

homomorphic Likewise, a graph G is said to be homomorphic to a graph H if there is a mapping, called ahomomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then their correspondingvertices are adjacent in H.

hyperedge An edge that is allowed to take on any number of vertices, possibly more than 2. A graph that allowsany hyperedge is called a hypergraph. A simple graph can be considered a special case of the hypergraph,namely the 2-uniform hypergraph. However, when stated without any qualication, an edge is always assumedto consist of at most 2 vertices, and a graph is never confused with a hypergraph.

6.9 Iin degree The number of edges entering a vertex. Is denoted as d-(v). The degree d(v) of a vertex v is equal to

the sum of its out- and in- degrees. When the context is clear, the subscript can be dropped. Maximum andminimum out degrees are denoted by +() and +(); and maximum and minimum in degrees, -() and -().

in-neighborhood (predecessor set) N-(v) of a vertex v is the set of heads of arc going into v.

incident An edgee connects two vertices; these two vertices are said to be incident to that edge, or, equivalently,that edge incident to those two vertices. All degree-related concepts have to do with adjacency or incidence.

independence number (G) of a graph G is the size of a largest independent set of G.

independent In graph theory, the word independent usually carries the connotation of pairwise disjoint or mutuallynonadjacent. In this sense, independence is a form of immediate nonadjacency.

independent set A set of isolated vertices. Since the graph induced by any independent set is an empty graph, thetwo terms are usually used interchangeably. In the example above, vertices 1, 3, and 6 form an independentset; and 3, 5, and 6 form another one.

induced subgraph A subgraph H of a graph G is said to be induced if, for any pair of vertices x and y of H, xy isan edge of H if and only if xy is an edge of G. In other words, H is an induced subgraph of G if it has the mostedges that appear in G over the same vertex set. If H is chosen based on a vertex subset S of V(G), then H canbe written as G[S] and is said to be induced by S.

innite A graph is innite if it has innitely many vertices or edges or both; otherwise the graph is nite. An innitegraph where every vertex has nite degree is called locally nite. When stated without any qualication, agraph is usually assumed to be nite.

initial vertex A head.

internally disjoint Two paths are internally disjoint (some people call it independent) if they do not have anyvertex in common, except the rst and last ones.

isolated vertex A vertex not incident to any edges.

isomorphic Two graphs G and H are said to be isomorphic, denoted by G ~ H, if there is a one-to-one correspon-dence, called an isomorphism, between the vertices of the graph such that two vertices are adjacent in G ifand only if their corresponding vertices are adjacent in H.

isthmus A bridge.

20 CHAPTER 6. APPENDIX:GLOSSARY OF GRAPH THEORY

6.10 Kk-ary tree A k-ary tree is a rooted tree in which every internal vertex has k children. An 1-ary tree is just a path.

A 2-ary tree is also called a binary tree.

kernel An independent out-dominating set. A digraph is kernel perfect if every induced sub-digraph has a kernel.

knot In a directed graph, a collection of vertices and edges with the property that every vertex in the knot hasoutgoing edges, and all outgoing edges from vertices in the knot have other vertices in the knot as destinations.Thus it is impossible to leave the knot while following the directions of the edges.

6.11 Llength of a cycle The number of its edges. Cn has length n. A cycle, unlike a path, is not allowed to have length 0.

length of a path or walk The number of edges that the path uses. Pn has length n - 1. Some people count multipleedges multiple times. In the example graph, (1, 2, 5, 1, 2, 3) is a path with length 5, and (5, 2, 1) is a simplepath of length 2.

link has two distinct endvertices.

digon A cycle of length 2. In the example graph, (1, 2, 3, 4, 5, 1) is a cycle of length 5.

loop An edge whose endvertices are the same vertex.

loop, cycle A path which ends at the node where it began.

6.12 Mmatching number (G) of a graph G is a the size of a largestmatching, or pairwise vertex disjoint edges, of G.

maximum degree (G) of a graph G, the largest degree over all vertices; theminimum degree (G), the smallest.

minor G2 = (V2; E2) of G1 = (V1; E1) is an injection from V2 to V1 such that every edge in E2 corresponds to apath(disjoint from all other such paths) inG1 such that every vertex in V1 is in one or more paths, or is part ofthe injection from V1 to V2 . This can alternatively be phrased in terms of contractions, which are operationswhich collapse a path and all vertices on it into a single edge (see edge contraction).

multiple A set of arcs are multiple, or parallel, if they share the same head and the same tail. A pair of arcs areanti-parallel if ones head/tail is the others tail/head.

multiple edge An edge such that there is another edge with the same endvertices; antonyms: simple edge. Themultiplicity of an edge is the number of multiple edges sharing the same endvertices; the multiplicity of agraph, the maximum multiplicity of its edges.

multigraph A graph that has multiple edges, but no loops.

multipartite graph A graph that can be decomposed into an unspecic number of partite sets but not fewer.

6.13 Nnetwork A weighted graph, possibly directed or undirected, possibly containing special vertices (nodes), such as

source or sink.

null graph The graph with no vertices and no edges. Or, it is a graph with no edges and any number n of vertices,in which case it may be called the null graph on n vertices. (There is no consistency at all in the literature.)

6.14. O 21

6.14 Oodd cycle A cycle that has odd length.

order The order of a graph is the number of its vertices, i.e. |V(G)|.

orientation An assignment of directions to the edges of an undirected or partially directed graph. When statedwithout any qualication, it is usually assumed that all undirected edges are replaced by a directed one in anorientation. Also, the underlying graph is usually assumed to be undirected and simple.

oriented graph A graph that contains only arcs. When stated without any qualication, a graph is almost alwaysassumed to be undirected. Also, a digraph is usually assumed to contain no undirected edges.

out degree The number of edges leaving a vertex. Is denoted d+(v), given a digraph . See also in degree.

out-neighborhood (successor set) N+(v) of a vertex v is the set of tails of arcs going from v.

outer face Furthermore, since we can establish a sense of inside and outside on a plane, we can identify anoutermost region that contains the entire graph if the graph does not cover the entire plane. Such outermostregion is called an outer face.

outerplanar graph A graph that can be drawn in the planar fashion such that its vertices are all adjacent to the outerface. See also outerplane graph.

outerplane graph A graph that is drawn in the planar fashion such that its vertices are all adjacent to the outer face.See also outerplanar graph.

6.15 Ppancyclic graph A graph that contains cycles of every possible length (from 3 to the order of the graph).

k-partite graph A graph that can be decomposed into k partite sets but not fewer.

partite set A graph can be decomposed into independent sets in the sense that the entire vertex set of the graph canbe partitioned into pairwise disjoint independent subsets. Such independent subsets are called partite sets, orsimply parts.

path A directed path, or just a path when the context is clear, is an oriented simple path such that all arcs go thesame direction, meaning all internal vertices have in- and out-degrees 1.

path A route that does not pass any edge more than once. If the path does not pass any node more than once, it isa simple path.

path In a graph was what is now usually known as an open walk. Nowadays, when stated without any qualication,a path of n vertices, denoted by Pn (but some write Pn), is usually dened to be a simple path or a simpletrail in the old sense, meaning that every vertex is incident to at most two edges.

perfect matching A spanning matching.

peripheral vertex A vertex with maximum eccentricity.

planar graph A graph that can be drawn on the (Euclidean) plane without any crossing; a graph of genus 0.

plane graph A graph that is drawn on the (Euclidean) plane without any crossing.

22 CHAPTER 6. APPENDIX:GLOSSARY OF GRAPH THEORY

point, node, vertex Objects (things) represented in a graph. These are almost always rendered as round dots.

k-th power Gk of a graph G is a supergraph formed by adding an edge between all pairs of vertices of G withdistance at most k. A second power of a graph is also called a square.

pseudograph A graph that contains both multiple edges and loops (the literature is highly inconsistent).

6.16 Rradius The minimum eccentricity over all vertices in a graph. It is denoted rad(G) of a graph G. When there are

two components in G, diam(G) and rad(G) dened to be innity .

reachable Of a vertex v from a given vertex u in a directed graph, such that there is a directed path that starts fromu and ends at v.

regular A graph in which each node has the same degree.

regular graph A graph in which every vertex has the same degree.

k-regular graph A graph in which every vertex has degree k. A 0-regular graph is an independent set. A 1-regulargraph is a matching. A 2-regular graph is a vertex disjoint union of cycles. A 3-regular graph is said to becubic, or trivalent.

route A sequence of edges and nodes from one node to another. Any given edge or node might be used more thanonce.

6.17 Sseparating set A cut set.

simple A digraph is called simple if it has no loops and at most one arc between any pair of vertices. When statedwithout any qualication, a digraph is usually assumed to be simple.

simple graph A graph that has no multiple edges or loops. When stated without any qualication, a graph is almostalways assumed to be simple. See also multigraph.

sink A vertex of a network with out-degree of zero; see also source.

size of a graph The number of its edges, i.e. |E(G)|.

source A vertex of a network with in-degree of zero; see also sink.

k-spanner A spanning subgraph in which every two vertices are at most k times as far apart on S than on G. Thenumber k is the dilation. k-spanner is used for studying geometric network optimization.

spanning subgraph A subgraph H of a graph G such that it has the same vertex set as G. We say H spans G.

spanning tree A spanning subgraph that is a tree. Every graph has a spanning forest. But only a connected graphhas a spanning tree.

star A special kind of tree called star is K,k; see also claw.

6.18. T 23

strongly connected A digraph is strongly connected if every vertex is reachable from every other following thedirections of the arcs. On the contrary, a diagraph is weakly connected if its underlying undirected graph isconnected. A weakly connected graph can be thought of as a digraph in which every vertex is reachable fromevery other but not necessarily following the directions of the arcs. A strong orientation is an orientation thatproduces a strongly connected digraph.

strongly connected component Informally, a subgraph where all nodes in the subgraph are reachable by all othernodes in the subgraph.

subtree of the graph G is a subgraph that is a tree.

subgraph Of a graph G is a graph whose vertex and edge sets are subsets of those of G. In the other direction, asupergraph of a graph G is a graph that contains G as a subgraph. We say a graph G contains another graphH if some subgraph of G is H or is isomorphic to H (depending on the needs of the situation).

semiregular A k-partite graph is semiregular if each of its partite set has a uniform degree

spanning matching A matching that covers all vertices of a graph.

stable set An independent set.

staset A stable set.

strongly regular graph A regular graph such that any adjacent vertices have the same number of common neighborsas other adjacent pairs and that any nonadjacent vertices have the same number of common neighbors as othernonadjacent pairs.

6.18 Ttail The second vertex in the arc AKA directed edge. See also head.

terminal vertex tail.

theta graph The union of three internally disjoint (simple) paths that have the same two distinct endvertices. Atheta0 graph has seven vertices which can be arranged as the vertices of a regular hexagon plus an additionalvertex in the center. The eight edges are the perimeter of the hexagon plus one diameter.

thickness The minimum number of planar graphs needed to cover a graph is the thickness of the graph.

totally disconnected graph A graph is totally disconnected if there is no path connecting any pair of vertices.This is just another name to describe an empty graph or independent set.

tournament A digraph in which each pair of vertices is connected by exactly one arc. In other words, it is anoriented complete graph.

traceable graph A graph that contains a Hamiltonian path.

traceable path A spanning path.

trail A walk in which all the edges are distinct. (A closed trail has been called a tour or circuit, but these areuncommon and the latter is usually reserved for a regular subgraph of degree two.)

tree A connected graph with no loops.

24 CHAPTER 6. APPENDIX:GLOSSARY OF GRAPH THEORY

tree A connected acyclic simple graph. A vertex of degree 1 is called a leaf, or pendant vertex. An edge incident toa leaf is an leaf edge, or pendant edge. (Some people dene a leaf edge as a leaf and then dene a leaf vertexon top of it. These two sets of denitions are often used interchangeably.) A non-leaf vertex is an internalvertex. Sometimes, one vertex of the tree is distinguished, and called the root. A rooted tree is a tree with aroot. Rooted trees are often treated as directed acyclic graphs with the edges pointing away from the root.Trees are commonly used as data structures in computer science (see tree data structure).

triangle C3 is called a triangle.

tripartite graph A graph that can be decomposed into three partite sets but not fewer.

6.19 Uundirected edge An edge that disregards any sense of direction and treats both endvertices interchangeably.

undirected A graph in which each edge symbolizes an unordered, transitive relationship between two nodes. Suchedges are rendered as plain lines or arcs.

unicyclic graph A graph that contains exactly one cycle.

unidentied Nodes or edges which are not considered as individuals. Only the way in which they connect to therest of the graph characterize unidentied nodes and edges.

universal graph In a classK of graphs, a simple graph in which every element in K can be embedded as a subgraph.

unweighted A graph in which all the relationships symbolized by edges are considered equivalent. Such edges arerendered as plain lines or arcs.

6.20 Vvalency A degree.

vertex A basic element of a graph, drawn as a node or a dot. The vertex set of G is usually denoted by V(G), or Vwhen there is no danger of confusion.

vertex cut A cut set.

6.21 Wwalk An alternating sequence of vertices and edges, beginning and ending with a vertex, in which each vertex is

incident to the two edges that precede and follow it in the sequence, and the vertices that precede and followan edge are the endvertices of that edge. A walk is closed if its rst and last vertices are the same, and openif they are dierent.

weighted Weighted edges symbolize relationships between nodes which are considered to have some value, forinstance, distance or lag time. Such edges are usually annotated by a number or letter placed beside the edge.Weighted nodes also have some value; this must be distinguished from identication.

weighted graph A graph that associates a label (weight) with every edge in the graph. Weights are usually realnumbers. Theymay be restricted to rational numbers or integers. Certain algorithms require further restrictionson weights; for instance, the Dijkstra algorithm works properly only for positive weights.

6.22. SEE ALSO 25

weight of a subgraph The sum of the weights of the edges of the subgraph.

Wiener index of a vertex v in a graph G, denoted byWG(v) is the sum of distances between v and all others.

Wiener index of a graph G, denoted byW(G), is the sum of distances over all pairs of vertices.

Wiener polynomial of an undirected graph qd(u,v) over all unordered pairs of vertices u and v.

6.22 See also W:Glossary of graph theory

6.23 References Bollobs, Bla (1998). Modern Graph Theory. New York: Springer-Verlag. ISBN 0-387-98488-7. [Packedwith advanced topics followed by a historical overview at the end of each chapter.]

West, Douglas B. (2001). Introduction to Graph Theory (2ed). Upper Saddle River: Prentice Hall. ISBN0-13-014400-2. [Tons of illustrations, references, and exercises. The most complete introductory guide to thesubject.]

Eric W. Weisstein. Graph. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Graph.html

Zaslavsky, Thomas. Glossary of signed and gain graphs and allied areas. Electronic Journal of Combinatorics,Dynamic Surveys in Combinatorics, # DS 8. http://www.combinatorics.org/Surveys/

Chapter 7

Appendix:Glossary of group theory

This is a glossary of group theory. Throughout the article, we use e to denote the identity element of a group.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

7.1 Aabelian group A group (G,*) is abelian if * is commutative, i.e. g*h=h*g for all g,h G. Likewise, a group is

nonabelian if this relation fails to hold for any pair g,h G.

7.2 Ddirect product direct sum, and semidirect product of groups. These are ways of combining groups to construct

new groups; please refer to the corresponding links for explanation.

7.3 Ffactor group Or quotient group. Given a group G and a normal subgroup N of G, the quotient group is the set G/N

of left cosets {aN : aG} together with the operation aN*bN=abN. The relationship between normal subgroups,homomorphisms, and factor groups is summed up in the fundamental theorem on homomorphisms.

nitely generated group If there exists a nite set S such that = G, then G is said to be nitely generated. If Scan be taken to have just one element, G is a cyclic group of nite order, an innite cyclic group, or possibly agroup {e} with just one element.

free group Given any set A, one can dene a multiplication of words as follows: (abb)*(bca)=abbbca. The freegroup generated by A is the smallest group containing this semigroup.

7.4 Ggeneral linear group Denoted by GL(n, F), is the group of n-by-n invertible matrices, where the elements of the

matrices are taken from a eld F such as the real numbers or the complex numbers.

group A set together with an associative operation which admits an identity element and such that every elementhas an inverse.

group isomorphism Group homomorphisms that have inverse functions. The inverse of an isomorphism, it turnsout, must also be a homomorphism.

26

7.5. I 27

group homomorphism These are functions f : (G,*) (H,) that have the special property thatf(a * b) = f(a) f(b)for any elements a and b of G.

group representation (not to be confused with the presentation of a group). A homomorphism from a group to ageneral linear group. One basically tries to represent a given abstract group as a concrete group of invertiblematrices which is much easier to study.

7.5 Iisomorphic groups Two groups are isomorphic if there exists a group isomorphism mapping from one to the other.

Isomorphic groups can be thought of as essentially the same, only with dierent labels on the individual ele-ments.

7.6 Kkernel Given a group homomorphism , the preimage of the identity in the codomain of the group homomorphism.

Every normal subgroup is the kernel of a group homomorphism and vice versa.

7.7 Nnormal subgroup H is a normal subgroup of G if for all g in G and h in H, g * h * g1 also belongs to H.

7.8 Oorder Of a group (G; ) , the cardinality (i.e. number of elements) of G . (A group with nite order is called a

nite group.)

order Of an element of a group. Suppose x 2 G and there exists a positive integer m such that xm = e , thenthe smallest possiblem is called the order of x . The order of a nite group is divisible by the order of everyelement.

7.9 Pp-group If p is prime, then a p-group is just a group with order pm for some m.

p-subgroup A subgroup which is also p-group. (The study of p-subgroups is the central object of the Sylow theo-rems.)

7.10 Ssimple group Simple groups are those groups with {e} and itself as the only normal subgroups. The name is

misleading as its structure could be extremely complex. An example is the monster group, a group of ordermore than one million. Every nite group is built up from simple groups through the use of group extensions, sothe study and classication of nite simple groups is central to the study of nite groups in general. As a resultof extensive eort over the second half of the 20th century, the nite simple groups have all been classied.

subgroup Given group (G,*), a subset H which remains a group when the operation * is restricted to H. Given a setS of G. We denote by the smallest subgroup of G containing S.

Chapter 8

Appendix:Glossary of linear algebra

This is a glossary of linear algebra.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

8.1 Aane transformation A linear transformation between vector spaces followed by a translation.

8.2 Bbasis In a vector space, a linearly independent set of vectors spanning the whole vector space.

8.3 Ddeterminant The unique scalar function over square matrices which is distributive over matrix multiplication, mul-

tilinear in the rows and columns, and takes the value of 1 for the unit matrix.

diagonal matrix A matrix in which only the entries on the main diagonal are non-zero.

dimension The number of elements of any basis of a vector space.

8.4 Iidentity matrix A diagonal matrix all of the diagonal elements of which are equal to 1.

inverse matrix Of a matrix A, another matrix B such that A multiplied by B and B multiplied by A both equal theidentity matrix.

8.5 Llinear algebra The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems

of linear equations.

linear combination A sum, each of whose summands is an appropriate vector times an appropriate scalar (or ringelement).

28

8.6. M 29

linear equation A polynomial equation of the rst degree (such as x = 2y - 7).

linear transformation A map between vector spaces which respects addition and multiplication.

linearly independent (Of a set of vectors or ring elements) whose nontrivial linear combinations are nonzero.

8.6 Mmatrix A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in

geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory.

8.7 Sspectrum Of a bounded linear operator A, the scalar values such that the operator AI, where I denotes the

identity operator, does not have a bounded inverse.

square matrix A matrix having the same number of rows as columns.

8.8 Vvector A directed quantity, one with both magnitude and direction; an element of a vector space.

vector space A set V, whose elements are called vectors, together with a binary operation + forming a module(V,+), and a set F* of bilinear unary functions f*:VV, each of which corresponds to a "scalar" element f ofa eld F, such that the composition of elements of F* corresponds isomorphically to multiplication of elementsof F, and such that for any vector v, 1*(v) = v.

Chapter 9

Appendix:Glossary of logic

This is a glossary of logic.Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

9.1 Aantecedent The conditional part of a hypothetical proposition

9.2 Cconclusion In a syllogism, the proposition that follows as a necessary consequence of the premises.

consequent The second half of a hypothetical proposition; Q, if the form of the proposition is If P, then Q.

contraposition The statement of the form if not Q then not P, given the statement if P then Q.

9.3 Ddomain of discourse In predicate logic, an indication of the relevant set of entities that are being dealt with by

quantiers.

9.4 Fformula A syntactic expression of a proposition, built up from quantiers, logical connectives, variables, relation

and operation symbols, and, depending on the type of logic, possibly other operators such as modal, temporal,deontic or epistemic ones.

9.5 Iimplication The connective in propositional calculus that, when joining two predicates A and B in that order, has

the meaning if A is true, then B is true.

inference The act or process of inferring; the production of a proposition based on given propositions.

inverse A statement constructed from the negatives of the premise and conclusion of some other statement: ~p ~q is the inverse of p q.

30

9.6. M 31

9.6 Mmaterial implication An implication as dened in classical propositional logic, leading to the truth of paradoxes

of implication such as Q --> (P --> P), to be read as any proposition whatsoever is a sucient condition for atrue proposition.

modus ponens A valid form of argument in which the antecedent of a conditional proposition is armed, therebyentailing the armation of the consequent.

9.7 Ppremise Either of the rst two propositions of a syllogism, from which the conclusion is deduced.

proposition The content of an assertion that may be taken as being true or false and is considered abstractly withoutreference to the linguistic sentence that constitutes the assertion.

9.8 Rreductio ad absurdum The method of proving a statement by assuming the statement is false and, with that as-

sumption, arriving at a blatant contradiction.

9.9 Ssentence A formula with no free variables.

syllogism An inference in which one proposition (the conclusion) follows necessarily from two other propositions,known as the premises.

9.10 See also Category:Logic Appendix:Glossary of fallacies

Chapter 10

Appendix:Glossary of order theory

This is a glossary of some terms used in various branches of mathematics that are related to the elds of order, lattice,and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources mightbe the following overview articles:

completeness properties of partial orders distributivity laws of order theory preservation properties of functions between posets.

In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning isclear from the context, will suce to denote the corresponding relational symbol, even without prior introduction.Furthermore, < will denote the strict order induced by .Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

10.1 AAdjoint See Galois connection.

Alexandrov topology For a preordered set P, any upper setO isAlexandrov-open. Inversely, a topology is Alexan-drov if any intersection of open sets is open.

Algebraic poset A poset is algebraic if it has a base of compact elements.

Antichain An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elementsx and y such that x y. In other words, the order relation of an antichain is just the identity relation.

Approximates relation See way-below relation.

relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for all elements x, y in X.

antitone function f between posets P and Q is a function for which, for all elements x, y of P, x y (in P) impliesf(y) f(x) (inQ). Another name for this property is order-reversing. In analysis, in the presence of total orders,such functions are often called monotonically decreasing, but this is not a very convenient description whendealing with non-total orders. The dual notion is called monotone or order-preserving.

asymmetric A relation R on a set X is asymmetric, if x R y implies not y R x, for all elements x, y in X.

atom In a poset P with least element 0, an element that is minimal among all elements that are unequal to 0.

atomic poset P with least element 0 is one in which, for every non-zero element x of P, there is an atom a of P witha x.

32

10.2. B 33

10.2 BBase See continuous poset.

Boolean algebra a distributive lattice with least element 0 and greatest element 1, in which every element x has acomplement x, such that x x = 0 and x x = 1.

bounded poset One that has a least element 0 and a greatest element 1.

bounded complete Of poset: if every of its subsets with some upper bound also has a least such upper bound. Thedual notion is not common.

10.3 CChain A chain is a totally ordered set or a totally ordered subset of a poset. See also total order.

Closure operator A closure operator on the poset P is a function C : P P that is monotone, idempotent, andsatises C(x) x for all x in P.

Compact An element x of a poset is compact if it is way below itself, i.e. x

34 CHAPTER 10. APPENDIX:GLOSSARY OF ORDER THEORY

10.4 Ddcpo See directed complete partial order.

dense order poset P is one in which, for all elements x and y in P with x < y, there is an element z in P, such that x< z < y. A subset Q of P is dense in P if for any elements x < y in P, there is an element z in Q such that x < z< y.

Directed A non-empty subset X of a poset P is called directed, if, for all elements x and y of X, there is an elementz of X such that x z and y z. The dual notion is called ltered.

Directed complete partial order A poset D is said to be a directed complete poset, or dcpo, if every directedsubset of D has a supremum.

Distributive A lattice L is called distributive if, for all x, y, and z in L, we nd that x (y z) = (x y) (x z).This condition is known to be equivalent to its order dual. A meet-semilattice is distributive if for all elementsa, b and x, a b x implies the existence of elements a' a and b' b such that a' b' = x. See alsocompletely distributive.

Domain Domain is a general term for objects like those that are studied in domain theory. If used, it requires furtherdenition.

Down-set See lower set.

Dual For a poset (P, ), the dual order (P, ) is dened by setting x y if and only if y x. The dual order of P issometimes denoted by Pop, and is also called opposite or converse order. Any order theoretic notion induces adual notion, dened by applying the original statement to the order dual of a given set. This exchanges and, meets and joins, zero and unit.

10.5 FFilter A subset X of a poset P is called a lter if it is a ltered upper set. The dual notion is called ideal.

Filtered A non-empty subset X of a poset P is called ltered, if, for all elements x and y of X, there is an element zof X such that z x and z y. The dual notion is called directed.

Finite element See compact.

Frame A frame F is a complete lattice, in which, for every x in F and every subset Y of F, the innite distributivelaw x W Y = W {x y | y in Y} holds. Frames are also known as locales and as complete Heyting algebras.

10.6 GGalois connection Given two posets P and Q, a pair of monotone functions F:P Q and G:Q P is called a

Galois connection, if F(x) y is equivalent to x G(y), for all x in P and y in Q. F is called the lower adjointof G and G is called the upper adjoint of F.

Greatest element For a subset X of a poset P, an element a of X is called the greatest element of X, if x a forevery element x in X. The dual notion is called least element.

10.7 HHeyting algebra A Heyting algebra H is a bounded lattice in which the function fa: H H, given by fa(x) = a

x is the lower adjoint of a Galois connection, for every element a of H. The upper adjoint of fa is then denotedby ga, with ga(x) = a x. Every Boolean algebra is a Heyting algebra.

10.8. I 35

10.8 Iideal A subset X of a poset P that is a directed lower set. The dual notion is called lter.

incidence algebra Of a poset: the associative algebra of all scalar-valued functions on intervals, with addition andscalar multiplication dened pointwise, and multiplication dened as a certain convolution; see incidence al-gebra for the details.

inmum For a poset P and a subset X of P, the greatest element in the set of lower bounds of X (if it exists, whichit may not) is called the inmum, meet, or greatest lower bound of X. It is denoted by inf X or V X. Theinmum of two elements may be written as inf{x,y} or x y. If the set X is nite, one speaks of a niteinmum. The dual notion is called supremum.

interval For two elements a, b of a partially ordered set P, the interval [a,b] is the subset {x in P | a x b} of P.If a b does not hold the interval will be empty.

irreexive A relation R on a set X is irreexive, if there is no element x in X such that x R x.

10.9 JJoin See supremum.

10.10 LLattice A lattice is a poset in which all non-empty nite joins (suprema) and meets (inma) exist.

Least element For a subset X of a poset P, an element a of X is called the least element of X, if a x for everyelement x in X. The dual notion is called greatest element.

Linear See total order.

Locale A locale is a complete Heyting algebra. Locales are also called frames and appear in Stone duality andpointless topology.

Locally nite poset A partially ordered set P is locally nite if every interval [a, b] = {x in P | a x b} is a niteset.

Lower bound A lower bound of a subset X of a poset P is an element b of P, such that b x, for all x in X. Thedual notion is called upper bound.

Lower set A subset X of a poset P is called a lower set if, for all elements x in X and p in P, p x implies that p iscontained in X. The dual notion is called upper set.

10.11 MMaximal element A maximal element of a subset X of a poset P is an element m of X, such that m x implies m

= x, for all x in X. The dual notion is called minimal element.

Meet See inmum.

Minimal element A minimal element of a subset X of a poset P is an element m of X, such that x m implies m= x, for all x in X. The dual notion is called maximal element.

36 CHAPTER 10. APPENDIX:GLOSSARY OF ORDER THEORY

Monotone A function f between posets P and Q is monotone if, for all elements x, y of P, x y (in P) implies f(x) f(y) (in Q). Another name for this property is order-preserving. In analysis, in the presence of total orders,such functions are often called monotonically increasing, but this is not a very convenient description whendealing with non-total orders. The dual notion is called antitone or order reversing.

10.12 OOrder-embedding A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x y

(in P) is equivalent to f(x) f(y) (in Q).

Order isomorphism A mapping f: P Q between two posets P and Q is called an order isomorphism, if it isbijective and both f and f1 aremonotone. Equivalently, an order isomorphism is a surjective order embedding.

Order-preserving See monotone.

Order-reversing See antitone.

10.13 Ppartial order A binary relation that is reexive, antisymmetric, and transitive. Since both notions depend on each

other, the term is also used to refer to a