Equivalence classFrom Wikipedia, the free encyclopedia

Contents

1 Equivalence class 11.1 Notation and formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Equivalence relation 62.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Well-denedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.10 Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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2.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.11 Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.12 Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Homogeneous space 153.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Homogeneous spaces as coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Prehomogeneous vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Homogeneous spaces in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Partition of a set 194.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Renement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Quotient category 275.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Quotient ring 296.1 Formal quotient ring construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2.1 Alternative complex planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2.2 Quaternions and alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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6.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.6 Further references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Quotient space (linear algebra) 337.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 Quotient of a Banach space by a subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4.2 Generalization to locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Quotient space (topology) 368.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9 Semigroup 409.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.2 Examples of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.3.1 Identity and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3.2 Subsemigroups and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3.3 Homomorphisms and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.4 Structure of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.5 Special classes of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.6 Structure theorem for commutative semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.7 Group of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.8 Semigroup methods in partial dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 449.9 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.10 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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9.13 Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10 Set (mathematics) 4810.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.15Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 60

10.15.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.15.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.15.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 1

Equivalence class

This article is about equivalency in mathematics. For equivalency in music, see equivalence class (music).In mathematics, when a set has an equivalence relation dened on its elements, there is a natural grouping of

Congruence is an example of an equivalence relation. The two triangles on the left are congruent, while the third and fourth trianglesare not congruent to any other triangle. Thus, the rst two triangles are in the same equivalence class, while the third and fourthtriangles are in their own equivalence class.

elements that are related to one another, forming what are called equivalence classes. Notationally, given a set Xand an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X whichare equivalent to a. It follows from the denition of the equivalence relations that the equivalence classes form apartition of X. The set of equivalence classes is sometimes called the quotient set or the quotient space of X by ~and is denoted by X / ~.When X has some structure, and the equivalence relation is dened with some connection to this structure, thequotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spacesin topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

1.1 Notation and formal denitionAn equivalence relation is a binary relation ~ satisfying three properties:[1]

For every element a in X, a ~ a (reexivity),

For every two elements a and b in X, if a ~ b, then b ~ a (symmetry)

For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

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2 CHAPTER 1. EQUIVALENCE CLASS

The equivalence class of an element a is denoted [a] and is dened as the set

[a] = fx 2 X j a xgof elements that are related to a by ~. An alternative notation [a]R can be used to denote the equivalence class of theelement a, specically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.The set of all equivalence classes in Xwith respect to an equivalence relationR is denoted asX/R and called XmoduloR (or the quotient set of X by R).[2] The surjective map x 7! [x] from X onto X/R, which maps each element to itsequivalence class, is called the canonical surjection or the canonical projection map.When an element is chosen (often implicitly) in each equivalence class, this denes an injective map called a section. Ifthis section is denoted by s, one has [s(c)] = c for every equivalence class c. The element s(c) is called a representativeof c. Any element of a class may be chosen as a representative of the class, by choosing the section appropriately.Sometimes, there is a section that is more natural than the other ones. In this case, the representatives are calledcanonical representatives. For example, in modular arithmetic, consider the equivalence relation on the integersdened by a ~ b if a b is a multiple of a given integer n, called the modulus. Each class contains a unique non-negative integer smaller than n, and these integers are the canonical representatives. The class and its representativeare more or less identied, as is witnessed by the fact that the notation a mod n may denote either the class or itscanonical representative (which is the remainder of the division of a by n).

1.2 Examples If X is the set of all cars, and ~ is the equivalence relation has the same color as. then one particular equivalenceclass consists of all green cars. X/~ could be naturally identied with the set of all car colors (cardinality ofX/~ would be the number of all car colors).

Let X be the set of all rectangles in a plane, and ~ the equivalence relation has the same area as. For eachpositive real number A there will be an equivalence class of all the rectangles that have area A.[3]

Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their dierence x yis an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all evennumbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent thesame element of Z/~.[4]

Let X be the set of ordered pairs of integers (a,b) with b not zero, and dene an equivalence relation ~ on Xaccording to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can beidentied with the rational number a/b, and this equivalence relation and its equivalence classes can be used togive a formal denition of the set of rational numbers.[5] The same construction can be generalized to the eldof fractions of any integral domain.

If X consists of all the lines in, say the Euclidean plane, and L ~M means that L andM are parallel lines, thenthe set of lines that are parallel to each other form an equivalence class as long as a line is considered parallelto itself. In this situation, each equivalence class determines a point at innity.

1.3 PropertiesEvery element x of X is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are eitherequal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongsto one and only one equivalence class.[6] Conversely every partition of X comes from an equivalence relation in thisway, according to which x ~ y if and only if x and y belong to the same set of the partition.[7]

It follows from the properties of an equivalence relation that

x ~ y if and only if [x] = [y].

In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statementsare equivalent:

1.4. GRAPHICAL REPRESENTATION 3

x y [x] = [y] [x] \ [y] 6= ;:

1.4 Graphical representationAny binary relation can be represented by a directed graph and symmetric ones, such as equivalence relations, byundirected graphs. If ~ is an equivalence relation on a set X, let the vertices of the graph be the elements of X andjoin vertices s and t if and only if s ~ t. The equivalence classes are represented in this graph by the maximal cliquesforming the connected components of the graph.[8]

1.5 InvariantsIf ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be an invariant of ~, or well-dened under the relation ~.A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2,then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of nite groups. Some authors use compatible with ~" or just respects ~" instead of invariantunder ~".Any function f : X Y itself denes an equivalence relation on X according to which x1 ~ x2 if and only if f(x1)= f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is theinverse image of f(x). This equivalence relation is known as the kernel of f.More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalentvalues (under an equivalence relation ~Y on Y). Such a function is known as a morphism from ~X to ~Y .

1.6 Quotient space in topologyIn topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relationon a topological space using the original spaces topology to create the topology on the set of equivalence classes.In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebraon the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vectorspace formed by taking a quotient group where the quotient homomorphism is a linear map. By extension, in abstractalgebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotientalgebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a groupaction.The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when theorbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroupon the group by left translations, or respectively the left cosets as orbits under right translation.A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the sensesof topology, abstract algebra, and group actions simultaneously.Although the term can be used for any equivalence relations set of equivalence classes, possibly with further structure,the intent of using the term is generally to compare that type of equivalence relation on a setX either to an equivalencerelation that induces some structure on the set of equivalence classes from a structure of the same kind on X, or to theorbits of a group action. Both the sense of a structure preserved by an equivalence relation and the study of invariantsunder group actions lead to the denition of invariants of equivalence relations given above.

1.7 See also Equivalence partitioning, a method for devising test sets in software testing based on dividing the possible

4 CHAPTER 1. EQUIVALENCE CLASS

program inputs into equivalence classes according to the behavior of the program on those inputs Homogeneous space, the quotient space of Lie groups. Transversal (combinatorics)

1.8 Notes[1] Devlin 2004, p. 122

[2] Wolf 1998, p. 178

[3] Avelsgaard 1989, p. 127

[4] Devlin 2004, p. 123

[5] Maddox 2002, pp. 7778

[6] Maddox 2002, p.74, Thm. 2.5.15

[7] Avelsgaard 1989, p.132, Thm. 3.16

[8] Devlin 2004, p. 123

1.9 References Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8 Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman& Hall/ CRC Press, ISBN 978-1-58488-449-1

Maddox, Randall B. (2002), Mathematical Thinking and Writing, Harcourt/ Academic Press, ISBN 0-12-464976-9

Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematicians Toolbox, Freeman, ISBN 978-0-7167-3050-7

1.10 Further readingThis material is basic and can be found in any text dealing with the fundamentals of proof technique, such as any ofthe following:

Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th Ed.), Thomson (Brooks/Cole) Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4

O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall Lay (2001), Analysis with an introduction to proof, Prentice Hall Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall

1.10. FURTHER READING 5

Cupillari, The Nuts and Bolts of Proofs, Wadsworth Bond, Introduction to Abstract Mathematics, Brooks/Cole Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall Ash, A Primer of Abstract Mathematics, MAA

Chapter 2

Equivalence relation

This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are

members of the same cell within a set that has been partitioned into cells such that every element of the set is amember of one and only one cell of the partition. The intersection of any two dierent cells is empty; the union ofall the cells equals the original set. These cells are formally called equivalence classes.

2.1 NotationAlthough various notations are used throughout the literature to denote that two elements a and b of a set are equivalentwith respect to an equivalence relation R, the most common are "a ~ b" and "a b", which are used when R is theobvious relation being referenced, and variations of "a ~R b", "a R b", or "aRb" otherwise.

2.2 DenitionA given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reexive, symmetric andtransitive. Equivalently, for all a, b and c in X:

a ~ a. (Reexivity)

if a ~ b then b ~ a. (Symmetry)

if a ~ b and b ~ c then a ~ c. (Transitivity)

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is dened as[a] = fb 2 X j a bg .

2.3 Examples

2.3.1 Simple example

Let the set fa; b; cg have the equivalence relation f(a; a); (b; b); (c; c); (b; c); (c; b)g . The following sets are equivalenceclasses of this relation:[a] = fag; [b] = [c] = fb; cg .The set of all equivalence classes for this relation is ffag; fb; cgg .

6

2.4. CONNECTIONS TO OTHER RELATIONS 7

2.3.2 Equivalence relations

The following are all equivalence relations:

Has the same birthday as on the set of all people. Is similar to on the set of all triangles. Is congruent to on the set of all triangles. Is congruent to, modulo n" on the integers. Has the same image under a function" on the elements of the domain of the function. Has the same absolute value on the set of real numbers Has the same cosine on the set of all angles.

2.3.3 Relations that are not equivalences The relation "" between real numbers is reexive and transitive, but not symmetric. For example, 7 5 doesnot imply that 5 7. It is, however, a partial order.

The relation has a common factor greater than 1 with between natural numbers greater than 1, is reexiveand symmetric, but not transitive. (Example: The natural numbers 2 and 6 have a common factor greater than1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).

The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, butnot reexive. (If X is also empty then R is reexive.)

The relation is approximately equal to between real numbers, even if more precisely dened, is not an equiv-alence relation, because although reexive and symmetric, it is not transitive, since multiple small changes canaccumulate to become a big change. However, if the approximation is dened asymptotically, for example bysaying that two functions f and g are approximately equal near some point if the limit of f g is 0 at that point,then this denes an equivalence relation.

2.4 Connections to other relations A partial order is a relation that is reexive, antisymmetric, and transitive. Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set thatis reexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted forone another, a facility that is not available for equivalence related variables. The equivalence classes of anequivalence relation can substitute for one another, but not individuals within a class.

A strict partial order is irreexive, transitive, and asymmetric. A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reexive if and onlyif for all a X, there exists a b X such that a ~ b.

A reexive and symmetric relation is a dependency relation, if nite, and a tolerance relation if innite. A preorder is reexive and transitive. A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraicstructure, and which respects the additional structure. In general, congruence relations play the role of kernelsof homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many importantcases congruence relations have an alternative representation as substructures of the structure on which theyare dened. E.g. the congruence relations on groups correspond to the normal subgroups.

8 CHAPTER 2. EQUIVALENCE RELATION

2.5 Well-denedness under an equivalence relationIf ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be well-dened or a class invariant under the relation ~.A frequent particular case occurs when f is a function from X to another set Y ; if x1 ~ x2 implies f(x1) = f(x2) thenf is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of nite groups. The latter case with the function f can be expressed by a commutative triangle. Seealso invariant. Some authors use compatible with ~" or just respects ~" instead of invariant under ~".More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values(under an equivalence relation ~B). Such a function is known as a morphism from ~A to ~B.

2.6 Equivalence class, quotient set, partitionLet a; b 2 X . Some denitions:

2.6.1 Equivalence classMain article: Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalenceclass of X by ~. Let [a] := fx 2 X j a xg denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

2.6.2 Quotient setMain article: Quotient set

The set of all possible equivalence classes of X by ~, denoted X/ := f[x] j x 2 Xg , is the quotient set of X by~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient spacefor the details.

2.6.3 ProjectionMain article: Projection (relational algebra)

The projection of ~ is the function : X ! X/ dened by (x) = [x] which maps elements of X into theirrespective equivalence classes by ~.

Theorem on projections:[1] Let the function f: X B be such that a ~ b f(a) = f(b). Then there is aunique function g : X/~ B, such that f = g. If f is a surjection and a ~ b f(a) = f(b), then g is abijection.

2.6.4 Equivalence kernelThe equivalence kernel of a function f is the equivalence relation ~ dened by x y () f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

2.6.5 PartitionMain article: Partition of a set

2.7. FUNDAMENTAL THEOREM OF EQUIVALENCE RELATIONS 9

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single elementof P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their unionis X.

Counting possible partitions

Let X be a nite set with n elements. Since every equivalence relation over X corresponds to a partition of X, andvice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which isthe nth Bell number Bn:

Bn =1

e

1Xk=0

kn

k!;

where the above is one of the ways to write the nth Bell number.

2.7 Fundamental theorem of equivalence relationsA key result links equivalence relations and partitions:[2][3][4]

An equivalence relation ~ on a set X partitions X.

Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongsto a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the setof all possible equivalence relations on X and the set of all partitions of X.

2.8 Comparing equivalence relationsIf ~ and are two equivalence relations on the same set S, and a~b implies ab for all a,b S, then is said to be acoarser relation than ~, and ~ is a ner relation than . Equivalently,

~ is ner than if every equivalence class of ~ is a subset of an equivalence class of , and thus every equivalenceclass of is a union of equivalence classes of ~.

~ is ner than if the partition created by ~ is a renement of the partition created by .

The equality equivalence relation is the nest equivalence relation on any set, while the trivial relation that makes allpairs of elements related is the coarsest.The relation "~ is ner than " on the collection of all equivalence relations on a xed set is itself a partial orderrelation.

2.9 Generating equivalence relations Given any set X, there is an equivalence relation over the set [XX] of all possible functions XX. Two suchfunctions are deemed equivalent when their respective sets of xpoints have the same cardinality, correspondingto cycles of length one in a permutation. Functions equivalent in this manner form an equivalence class on[XX], and these equivalence classes partition [XX].

10 CHAPTER 2. EQUIVALENCE RELATION

An equivalence relation ~ on X is the equivalence kernel of its surjective projection : X X/~.[5] Conversely,any surjection between sets determines a partition on its domain, the set of preimages of singletons in thecodomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are threeequivalent ways of specifying the same thing.

The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X X)is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch that a = x1, b = xn, and (xi,xi )R or (xi,xi)R, i = 1, ..., n1.

Note that the equivalence relation generated in this manner can be trivial. For instance, the equivalencerelation ~ generated by:

Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y; Any subset of the identity relation on X has equivalence classes that are the singletons of X.

Equivalence relations can construct new spaces by gluing things together. Let X be the unit Cartesian square[0,1] [0,1], and let ~ be the equivalence relation on X dened by a, b [0,1] ((a, 0) ~ (a, 1) (0, b) ~ (1, b)).Then the quotient space X/~ can be naturally identied (homeomorphism) with a torus: take a square piece ofpaper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder soas to glue together its two open ends, resulting in a torus.

2.10 Algebraic structureMuch of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures themathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics asorder relations, the algebraic structure of equivalences is not as well known as that of orders. The former structuredraws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

2.10.1 Group theoryJust as order relations are grounded in ordered sets, sets closed under pairwise supremum and inmum, equivalencerelations are grounded in partitioned sets, which are sets closed under bijections and preserve partition structure.Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hencepermutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathe-matical structure of equivalence relations.Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denotethe set of bijective functions over A that preserve the partition structure of A: x A g G (g(x) [x]). Then thefollowing three connected theorems hold:[6]

~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations,mentionedabove);

Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the parti-tion;

Given a transformation groupG overA, there exists an equivalence relation ~ over A, whose equivalence classesare the orbits of G.[7][8]

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are theequivalence classes of A under ~.This transformation group characterisation of equivalence relations diers fundamentally from the way lattices char-acterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe

2.11. EQUIVALENCE RELATIONS AND MATHEMATICAL LOGIC 11

A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof bijections, A A.Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a~ b (ab1 H). The equivalence classes of ~also called the orbits of the action of H on Gare the right cosetsof H in G. Interchanging a and b yields the left cosets.Proof.[9] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that x A g G ([g(x)] = [x]), because G satises the following fourconditions:

G is closed under composition. The composition of any two elements of G exists, because the domain andcodomain of any element of G is A. Moreover, the composition of bijections is bijective;[10]

Existence of identity function. The identity function, I(x)=x, is an obvious element of G; Existence of inverse function. Every bijective function g has an inverse g1, such that gg1 = I; Composition associates. f(gh) = (fg)h. This holds for all functions over all domains.[11]

Let f and g be any two elements of G. By virtue of the denition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that[g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function compositionpreserves the partitioning of A. Related thinking can be found in Rosen (2008: chpt. 10).

2.10.2 Categories and groupoidsLet G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing thisequivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there existsa unique morphism from x to y if and only if x~y.The advantages of regarding an equivalence relation as a special case of a groupoid include:

Whereas the notion of free equivalence relation does not exist, that of a free groupoid on a directed graphdoes. Thus it is meaningful to speak of a presentation of an equivalence relation, i.e., a presentation of thecorresponding groupoid;

Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notionof groupoid, a point of view that suggests a number of analogies;

In many contexts quotienting, and hence the appropriate equivalence relations often called congruences, areimportant. This leads to the notion of an internal groupoid in a category.[12]

2.10.3 LatticesThe possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX by convention. The canonical map ker: X^X Con X, relates the monoid X^X of all functions on X and Con X.ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: XX toits kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

2.11 Equivalence relations and mathematical logicEquivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation withexactly two innite equivalence classes is an easy example of a theory which is -categorical, but not categorical forany larger cardinal number.An implication of model theory is that the properties dening a relation can be proved independent of each other(and hence necessary parts of the denition) if and only if, for each property, examples can be found of relationsnot satisfying the given property while satisfying all the other properties. Hence the three dening properties ofequivalence relations can be proved mutually independent by the following three examples:

12 CHAPTER 2. EQUIVALENCE RELATION

Reexive and transitive: The relation on N. Or any preorder; Symmetric and transitive: The relation R on N, dened as aRb ab 0. Or any partial equivalence relation; Reexive and symmetric: The relation R on Z, dened as aRb "a b is divisible by at least one of 2 or 3.Or any dependency relation.

Properties denable in rst-order logic that an equivalence relation may or may not possess include:

The number of equivalence classes is nite or innite; The number of equivalence classes equals the (nite) natural number n; All equivalence classes have innite cardinality; The number of elements in each equivalence class is the natural number n.

2.12 Euclidean relationsEuclid's The Elements includes the following Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing equal by are in relationwith). By relation is meant a binary relation, in which aRb is generally distinct from bRa. An Euclidean relationthus comes in two forms:

(aRc bRc) aRb (Left-Euclidean relation)(cRa cRb) aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem If a relation is (left or right) Euclidean and reexive, it is also symmetric and transitive.

Proof for a left-Euclidean relation

(aRc bRc) aRb [a/c] = (aRa bRa) aRb [reexive; erase T] = bRa aRb. Hence R is symmetric.

(aRc bRc) aRb [symmetry] = (aRc cRb) aRb. Hence R is transitive.

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclideanand reexive. The Elements mentions neither symmetry nor reexivity, and Euclid probably would have deemed thereexivity of equality too obvious to warrant explicit mention.

2.13 See also Partition of a set Equivalence class Up to Conjugacy class Topological conjugacy

2.14. NOTES 13

2.14 Notes[1] Garrett Birkho and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.

[2] Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.

[3] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.

[4] Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 2932, Marcel Dekker

[5] Garrett Birkho and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.

[6] Rosen (2008), pp. 243-45. Less clear is 10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.

[7] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.

[8] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.

[9] Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.

[10] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 22, Th. 6.

[11] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 24, Th. 7.

[12] Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN 0-521-80309-8

2.15 References Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8. Castellani, E., 2003, Symmetry and equivalence in Brading, Katherine, and E. Castellani, eds., Symmetriesin Physics: Philosophical Reections. Cambridge Univ. Press: 422-433.

Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusseshow equivalence relations arise in lattice theory.

Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint. John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31. Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag.Mostly chpts. 9,10.

Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axiomsdening equivalence, pp 4850, John Wiley & Sons.

2.16 External links Hazewinkel, Michiel, ed. (2001), Equivalence relation, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009 Equivalence relation at PlanetMath Binary matrices representing equivalence relations at OEIS.

14 CHAPTER 2. EQUIVALENCE RELATION

Logical matrices of the 52 equivalence relations on a 5-element set (Colored elds, including those in light gray, stand for ones; whiteelds for zeros.)

Chapter 3

Homogeneous space

A torus. The standard torus is homogeneous under its dieomorphism and homeomorphism groups, and the at torus is homogeneousunder its dieomorphism, homeomorphism, and isometry groups.

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneousspace for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements ofG are called the symmetries of X. A special case of this is when the group G in question is the automorphism groupof the space X here automorphism group can mean isometry group, dieomorphism group, or homeomorphismgroup. In this case X is homogeneous if intuitively X looks locally the same at each point, either in the sense ofisometry (rigid geometry), dieomorphism (dierential geometry), or homeomorphism (topology). Some authorsinsist that the action of G be faithful (non-identity elements act non-trivially), although the present article does not.Thus there is a group action of G on X which can be thought of as preserving some geometric structure on X, andmaking X into a single G-orbit.

3.1 Formal denitionLet X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X.[1]Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category,

15

16 CHAPTER 3. HOMOGENEOUS SPACE

then the elements of G are assumed to act as automorphisms in the same category. Thus the maps on X eected byG are structure preserving. A homogeneous space is a G-space on which G acts transitively.Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism:

: G! AutC(X)

into the group of automorphisms of the object X in the category C. The pair (X, ) denes a homogeneous spaceprovided (G) is a transitive group of symmetries of the underlying set of X.

3.1.1 ExamplesFor example, if X is a topological space, then group elements are assumed to act as homeomorphisms on X. Thestructure of a G-space is a group homomorphism : G Homeo(X) into the homeomorphism group of X.Similarly, if X is a dierentiable manifold, then the group elements are dieomorphisms. The structure of a G-spaceis a group homomorphism : G Dieo(X) into the dieomorphism group of X.Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the exampleslisted below.Concrete examples include:

Isometry groups

Positive curvature:

1. Sphere (orthogonal group): Sn1 = O(n)/O(n 1)2. Oriented sphere (special orthogonal group): Sn1 = SO(n)/SO(n 1)3. Projective space (projective orthogonal group): Pn1 = PO(n)/PO(n 1)

Flat (zero curvature):

1. Euclidean space (Euclidean group, point stabilizer is orthogonal group): An E(n)/O(n)

Negative curvature:

1. Hyperbolic space (orthochronous Lorentz group, point stabilizer orthogonal group, corresponding to hyperboloidmodel): Hn O+(1, n)/O(n)

2. Oriented hyperbolic space: SO+(1, n)/SO(n)3. Anti-de Sitter space: AdS = O(2, n)/O(1, n)

Others

Ane space (for ane group, point stabilizer general linear group): An = A(n, K)/GL(n, k). Grassmannian: Gr(r; n) = O(n)/(O(r) O(n r))

3.2 GeometryFrom the point of view of the Erlangen program, onemay understand that all points are the same, in the geometry ofX. This was true of essentially all geometries proposed before Riemannian geometry, in the middle of the nineteenthcentury.

3.3. HOMOGENEOUS SPACES AS COSET SPACES 17

Thus, for example, Euclidean space, ane space and projective space are all in natural ways homogeneous spacesfor their respective symmetry groups. The same is true of the models found of non-Euclidean geometry of constantcurvature, such as hyperbolic space.A further classical example is the space of lines in projective space of three dimensions (equivalently, the spaceof two-dimensional subspaces of a four-dimensional vector space). It is simple linear algebra to show that GL4 actstransitively on those. We can parameterize them by line co-ordinates: these are the 22 minors of the 42 matrix withcolumns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometryof Julius Plcker.

3.3 Homogeneous spaces as coset spacesIn general, if X is a homogeneous space, and Ho is the stabilizer of some marked point o in X (a choice of origin),the points of X correspond to the left cosets G/Ho, and the marked point o corresponds to the coset of the identity.Conversely, given a coset space G/H, it is a homogeneous space for G with a distinguished point, namely the coset ofthe identity. Thus a homogeneous space can be thought of as a coset space without a choice of origin.In general, a dierent choice of origin o will lead to a quotient of G by a dierent subgroup Ho which is related toHo by an inner automorphism of G. Specically,

Ho0 = gHog1 (1)

where g is any element of G for which go = o. Note that the inner automorphism (1) does not depend on which suchg is selected; it depends only on g modulo Ho.If the action of G on X is continuous, then H is a closed subgroup of G. In particular, if G is a Lie group, then H isa Lie subgroup by Cartans theorem. Hence G/H is a smooth manifold and so X carries a unique smooth structurecompatible with the group action.If H is the identity subgroup {e}, then X is a principal homogeneous space.One can go further to double coset spaces, notably CliordKlein forms \G/H, where is a discrete subgroup (ofG) acting properly discontinuously.

3.4 ExampleFor example in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional generallinear group, GL(4), dened by conditions on the matrix entries

h13 = h14 = h23 = h24 = 0,

by looking for the stabilizer of the subspace spanned by the rst two standard basis vectors. That shows that X hasdimension 4.Since the homogeneous coordinates given by theminors are 6 in number, this means that the latter are not independentof each other. In fact a single quadratic relation holds between the six minors, as was known to nineteenth-centurygeometers.This example was the rst known example of a Grassmannian, other than a projective space. There are many furtherhomogeneous spaces of the classical linear groups in common use in mathematics.

3.5 Prehomogeneous vector spacesThe idea of a prehomogeneous vector space was introduced by Mikio Sato.It is a nite-dimensional vector space V with a group action of an algebraic group G, such that there is an orbit of Gthat is open for the Zariski topology (and so, dense). An example is GL(1) acting on a one-dimensional space.

18 CHAPTER 3. HOMOGENEOUS SPACE

The denition is more restrictive than it initially appears: such spaces have remarkable properties, and there is aclassication of irreducible prehomogeneous vector spaces, up to a transformation known as castling.

3.6 Homogeneous spaces in physicsCosmology using the general theory of relativity makes use of the Bianchi classication system. Homogeneous spacesin relativity represent the space part of background metrics for some cosmological models; for example, the threecases of the FriedmannLematreRobertsonWalker metric may be represented by subsets of the Bianchi I (at),V (open), VII (at or open) and IX (closed) types, while the Mixmaster universe represents an anisotropic exampleof a Bianchi IX cosmology.[2]

A homogeneous space of N dimensions admits a set of 12N(N + 1) Killing vectors.[3] For three dimensions, thisgives a total of six linearly independent Killing vector elds; homogeneous 3-spaces have the property that one mayuse linear combinations of these to nd three everywhere non-vanishing Killing vector elds (a)i ,

(a)[i;k] = C

abc

(b)i

(c)k

where the object Cabc , the structure constants, form a constant order-three tensor antisymmetric in its lower twoindices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the covariant dierentialoperator). In the case of a at isotropic universe, one possibility is Cabc = 0 (type I), but in the case of a closedFLRW universe, Cabc = "abc where "abc is the Levi-Civita symbol.

3.7 See also Erlangen program Klein geometry Heap (mathematics) Homogeneous variety

3.8 References[1] We assume that the action is on the left. The distinction is only important in the description of X as a coset space.

[2] Lev Landau and Evgeny Lifshitz (1980), Course of Theoretical Physics vol. 2: The Classical Theory of Fields, Butterworth-Heinemann, ISBN 978-0-7506-2768-9

[3] Steven Weinberg (1972), Gravitation and Cosmology, John Wiley and Sons

Chapter 4

Partition of a set

For the partition calculus of sets, see innitary combinatorics.In mathematics, a partition of a set is a grouping of the sets elements into non-empty subsets, in such a way that

A set of stamps partitioned into bundles: No stamp is in two bundles, and no bundle is empty.

every element is included in one and only one of the subsets.

19

20 CHAPTER 4. PARTITION OF A SET

4.1 DenitionA partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of thesesubsets[1] (i.e., X is a disjoint union of the subsets).Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:[2]

1. P does not contain the empty set.2. The union of the sets in P is equal to X. (The sets in P are said to cover X.)3. The intersection of any two distinct sets in P is empty. (We say the elements of P are pairwise disjoint.)

In mathematical notation, these conditions can be represented as

1. ; /2 P2. SA2P A = X3. if A;B 2 P and A 6= B then A \B = ; ,

where ; is the empty set.The sets in P are called the blocks, parts or cells of the partition.[3]

The rank of P is |X| |P|, if X is nite.

4.2 Examples Every singleton set {x} has exactly one partition, namely { {x} }. For any nonempty set X, P = {X} is a partition of X, called the trivial partition. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U,namely, {A, UA}.

The set { 1, 2, 3 } has these ve partitions: { {1}, {2}, {3} }, sometimes written 1|2|3. { {1, 2}, {3} }, or 12|3. { {1, 3}, {2} }, or 13|2. { {1}, {2, 3} }, or 1|23. { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).

The following are not partitions of { 1, 2, 3 }: { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set. { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block. { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partitionof {1, 2}.

4.3 Partitions and equivalence relationsFor any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from anypartition P of X, we can dene an equivalence relation on X by setting x ~ y precisely when x and y are in the samepart in P. Thus the notions of equivalence relation and partition are essentially equivalent.[4]

The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly oneelement from each part of the partition. This implies that given an equivalence relation on a set one can select acanonical representative element from every equivalence class.

4.4. REFINEMENT OF PARTITIONS 21

4.4 Renement of partitionsA partition of a set X is a renement of a partition of Xand we say that is ner than and that is coarserthan if every element of is a subset of some element of . Informally, this means that is a further fragmentationof . In that case, it is written that .This ner-than relation on the set of partitions of X is a partial order (so the notation "" is appropriate). Each setof elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specically (forpartitions of a nite set) it is a geometric lattice.[5] The partition lattice of a 4-element set has 15 elements and isdepicted in the Hasse diagram on the left.Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a nite set cor-responds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions withn 2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of acomplete graph. The matroid closure of a set of atomic partitions is the nest common coarsening of them all; ingraph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of thesubgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the graphic matroidof the complete graph.Another example illustrates the rening of partitions from the perspective of equivalence relations. If D is the set ofcards in a standard 52-card deck, the same-color-as relation on D which can be denoted ~C has two equivalenceclasses: the sets {red cards} and {black cards}. The 2-part partition corresponding to ~C has a renement that yieldsthe same-suit-as relation ~S, which has the four equivalence classes {spades}, {diamonds}, {hearts}, and {clubs}.

4.5 Noncrossing partitionsA partition of the set N = {1, 2, ..., n} with corresponding equivalence relation ~ is noncrossing provided that forany two 'cells C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all theelements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cellcalled C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a nite sethas recently taken on importance because of its role in free probability theory. These form a subset of the lattice ofall partitions, but not a sublattice, since the join operations of the two lattices do not agree.

4.6 Counting partitionsThe total number of partitions of an n-element set is the Bell number Bn. The rst several Bell numbers are B0 =1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, and B6 = 203 (sequence A000110 in OEIS). Bell numbers satisfy therecursion Bn+1 =

Pnk=0

nk

Bk

and have the exponential generating function

1Xn=0

Bnn!

zn = eez1:

The Bell numbers may also be computed using the Bell triangle in which the rst value in each row is copied fromthe end of the previous row, and subsequent values are computed by adding the two numbers to the left and above leftof each position. The Bell numbers are repeated along both sides of this triangle. The numbers within the trianglecount partitions in which a given element is the largest singleton.The number of partitions of an n-element set into exactly k nonempty parts is the Stirling number of the second kindS(n, k).The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by

Cn =1

n+ 1

2n

n

:

22 CHAPTER 4. PARTITION OF A SET

4.7 See also Exact cover Cluster analysis Weak ordering (ordered set partition) Equivalence relation Partial equivalence relation Partition renement List of partition topics Lamination (topology)

Rhyme schemes by set partition

4.8 Notes[1] Naive Set Theory (1960). Halmos, Paul R. Springer. p. 28. ISBN 9780387900926.

[2] Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littleeld. p. 187. ISBN 9780912675732.

[3] Brualdi, pp. 4445

[4] Schechter, p. 54

[5] Birkho, Garrett (1995), Lattice Theory, Colloquium Publications 25 (3rd ed.), American Mathematical Society, p. 95,ISBN 9780821810255.

4.9 References Brualdi, Richard A. (2004). Introductory Combinatorics (4th edition ed.). Pearson Prentice Hall. ISBN 0-13-100119-1.

Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.

4.9. REFERENCES 23

The 52 partitions of a set with 5 elements

24 CHAPTER 4. PARTITION OF A SET

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

The traditional Japanese symbols for the chapters of the Tale of Genji are based on the 52 ways of partitioning ve elements.

4.9. REFERENCES 25

Partitions of a 4-set ordered by renement

26 CHAPTER 4. PARTITION OF A SET

Construction of the Bell triangle

Chapter 5

Quotient category

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Thenotion is similar to that of a quotient group or quotient space, but in the categorical setting.

5.1 DenitionLet C be a category. A congruence relation R on C is given by: for each pair of objects X, Y in C, an equivalencerelation RX,Y on Hom(X,Y), such that the equivalence relations respect composition of morphisms. That is, if

f1; f2 : X ! Y

are related in Hom(X, Y) and

g1; g2 : Y ! Z

are related in Hom(Y, Z) then g1f1, g1f2, g2f1 and g2f2 are related in Hom(X, Z).Given a congruence relation R on C we can dene the quotient category C/R as the category whose objects are thoseof C and whose morphisms are equivalence classes of morphisms in C. That is,

HomC/R(X;Y ) = HomC(X;Y )/RX;Y :

Composition of morphisms in C/R is well-dened since R is a congruence relation.There is also a notion of taking the quotient of an Abelian category A by a Serre subcategory B. This is done asfollows. The objects of A/B are the objects of A. Given two objects X and Y of A, we dene the set of morphismsfrom X to Y in A/B to be lim!HomA(X

0; Y /Y 0) where the limit is over subobjects X 0 X and Y 0 Y such thatX/X 0; Y 0 2 B . Then A/B is an Abelian category, and there is a canonical functor Q : A ! A/B . This Abelianquotient satises the universal property that if C is any other Abelian category, and F : A ! C is an exact functorsuch that F(b) is a zero object of C for each b 2 B , then there is a unique exact functor F : A/B ! C such thatF = F Q . (See [Gabriel].)

5.2 PropertiesThere is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functoris bijective on objects and surjective on Hom-sets (i.e. it is a full functor).

27

28 CHAPTER 5. QUOTIENT CATEGORY

5.3 Examples Monoids and group may be regarded as categories with one object. In this case the quotient category coincideswith the notion of a quotient monoid or a quotient group.

The homotopy category of topological spaces hTop is a quotient category of Top, the category of topologicalspaces. The equivalence classes of morphisms are homotopy classes of continuous maps.

5.4 See also Subobject

5.5 References Gabriel, Pierre, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448. Mac Lane, Saunders (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathe-matics 5). Springer-Verlag.

Chapter 6

Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, dierence ring[1] or residueclass ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linearalgebra.[2][3] One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I,whose elements are the cosets of I in R subject to special + and operations.Quotient rings are distinct from the so-called 'quotient eld', or eld of fractions, of an integral domain as well asfrom the more general 'rings of quotients obtained by localization.

6.1 Formal quotient ring constructionGiven a ring R and a two-sided ideal I in R, we may dene an equivalence relation ~ on R as follows:

a ~ b if and only if a b is in I.

Using the ideal properties, it is not dicult to check that ~ is a congruence relation. In case a ~ b, we say that a andb are congruent modulo I. The equivalence class of the element a in R is given by

[a] = a + I := { a + r : r in I }.

This equivalence class is also sometimes written as a mod I and called the residue class of a modulo I".The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of Rmodulo I, if one denes

(a + I) + (b + I) = (a + b) + I; (a + I)(b + I) = (a b) + I.

(Here one has to check that these denitions are well-dened. Compare coset and quotient group.) The zero-elementof R/I is (0 + I) = I, and the multiplicative identity is (1 + I).The map p from R to R/I dened by p(a) = a + I is a surjective ring homomorphism, sometimes called the naturalquotient map or the canonical homomorphism.

6.2 Examples The quotient R/{0} is naturally isomorphic to R, and R/R is the zero ring {0}, since, by our denition, for anyr in R , we have that [r]=r +{0}:={r+b : b in {0}} (where {0} is the zero ring), which is isomorphic to R itself. This ts with the general rule of thumb that the larger the ideal I, the smaller the quotient ring R/I. If I is aproper ideal of R, i.e., I R, then R/I is not the zero ring.

29

30 CHAPTER 6. QUOTIENT RING

Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z. Then the quotient ring Z/2Zhas only two elements, zero for the even numbers and one for the odd numbers; applying the denition again,[z]=z+2Z:={z+2z: 2z in {2Z}}, where {2Z} is the ideal of even numbers. It is naturally isomorphic to thenite eld with two elements, F2. Intuitively: if you think of all the even numbers as 0, then every integer iseither 0 (if it is even) or 1 (if it is odd and therefore diers from an even number by 1). Modular arithmetic isessentially arithmetic in the quotient ring Z/nZ (which has n elements).

Now consider the ring R[X] of polynomials in the variable X with real coecients, and the ideal I = (X2 + 1)consisting of all multiples of the polynomial X2 + 1. The quotient ring R[X]/(X2 + 1) is naturally isomorphicto the eld of complex numbers C, with the class [X] playing the role of the imaginary unit i. The reason: weforced X2 + 1 = 0, i.e. X2 = 1, which is the dening property of i.

Generalizing the previous example, quotient rings are often used to construct eld extensions. Suppose K issome eld and f is an irreducible polynomial in K[X]. Then L = K[X]/(f) is a eld whose minimal polynomialover K is f, which contains K as well as an element x = X + (f).

One important instance of the previous example is the construction of the nite elds. Consider for instancethe eld F3 = Z/3Z with three elements. The polynomial f(X) = X2 + 1 is irreducible over F3 (since it hasno root), and we can construct the quotient ring F3[X]/(f). This is a eld with 32=9 elements, denoted by F9.The other nite elds can be constructed in a similar fashion.

The coordinate rings of algebraic varieties are important examples of quotient rings in algebraic geometry. Asa simple case, consider the real variety V = {(x,y) | x2 = y3 } as a subset of the real plane R2. The ring ofreal-valued polynomial functions dened on V can be identied with the quotient ring R[X,Y]/(X2 Y3), andthis is the coordinate ring of V. The variety V is now investigated by studying its coordinate ring.

SupposeM is a C-manifold, and p is a point ofM. Consider the ring R = C(M) of all C-functions denedonM and let I be the ideal in R consisting of those functions f which are identically zero in some neighborhoodU of p (where U may depend on f). Then the quotient ring R/I is the ring of germs of C-functions on M atp.

Consider the ring F of nite elements of a hyperreal eld *R. It consists of all hyperreal numbers diering froma standard real by an innitesimal amount, or equivalently: of all hyperreal numbers x for which a standardinteger n with n < x < n exists. The set I of all innitesimal numbers in *R, together with 0, is an ideal in F,and the quotient ring F/I is isomorphic to the real numbers R. The isomorphism is induced by associating toevery element x of F the standard part of x, i.e. the unique real number that diers from x by an innitesimal.In fact, one obtains the same result, namely R, if one starts with the ring F of nite hyperrationals (i.e. ratioof a pair of hyperintegers), see construction of the real numbers.

6.2.1 Alternative complex planesThe quotients R[X]/(X), R[X]/(X + 1), and R[X]/(X 1) are all isomorphic to R and gain little interest at rst.But note that R[X]/(X2) is called the dual number plane in geometric algebra. It consists only of linear binomialsas remainders after reducing an element of R[X] by X2. This alternative complex plane arises as a subalgebrawhenever the algebra contains a real line and a nilpotent.Furthermore, the ring quotient R[X]/(X2 1) does split into R[X]/(X + 1) and R[X]/(X 1), so this ring is oftenviewed as the direct sumRR. Nevertheless, an alternative complex number z = x + y j is suggested by j as a root ofX2 1, compared to i as root of X2 + 1 = 0. This plane of split-complex numbers normalizes the direct sum RRby providing a basis {1, j } for 2-space where the identity of the algebra is at unit distance from the zero. With thisbasis a unit hyperbola may be compared to the unit circle of the ordinary complex plane.

6.2.2 Quaternions and alternativesSuppose X and Y are two, non-commuting, indeterminates and form the free algebra RhX;Y i: Then Hamiltonsquaternions of 1843 can be cast as

6.3. PROPERTIES 31

RhX;Y i/(X2 + 1; Y 2 + 1; XY + Y X):

If Y2 1 is substituted for Y2 + 1, then one obtains the ring of split-quaternions. Substituting minus for plus in boththe quadratic binomials also results in split-quaternions. The anti-commutative property YX = XY implies that XYhas for its square

(XY)(XY) = X(YX)Y = X(XY)Y = XXYY = 1.

The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminatesRX,Y,Z and constructing appropriate ideals.

6.3 PropertiesClearly, if R is a commutative ring, then so is R/I; the converse however is not true in general.The natural quotient map p has I as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, wecan state that two-sided ideals are precisely the kernels of ring homomorphisms.The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows:the ring homomorphisms dened on R/I are essentially the same as the ring homomorphisms dened on R that vanish(i.e. are zero) on I. More precisely: given a two-sided ideal I in R and a ring homomorphism f : R S whose kernelcontains I, then there exists precisely one ring homomorphism g : R/I S with gp = f (where p is the natural quotientmap). The map g here is given by the well-dened rule g([a]) = f(a) for all a in R. Indeed, this universal propertycan be used to dene quotient rings and their natural quotient maps.As a consequence of the above, one obtains the fundamental statement: every ring homomorphism f : R S inducesa ring isomorphism between the quotient ring R/ker(f) and the image im(f). (See also: fundamental theorem onhomomorphisms.)The ideals of R and R/I are closely related: the natural quotient map provides a bijection between the two-sided idealsof R that contain I and the two-sided ideals of R/I (the same is true for left and for right ideals). This relationshipbetween two-sided ideal extends to a relationship between the corresponding quotient rings: ifM is a two-sided idealin R that contains I, and we write M/I for the corresponding ideal in R/I (i.e. M/I = p(M)), the quotient rings R/Mand (R/I)/(M/I) are naturally isomorphic via the (well-dened!) mapping a + M (a+I) + M/I.In commutative algebra and algebraic geometry, the following statement is often used: If R {0} is a commutativering and I is a maximal ideal, then the quotient ring R/I is a eld; if I is only a prime ideal, then R/I is only an integraldomain. A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I.The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwisecoprime ideals I1,...,Ik, then the quotient ring R/I is isomorphic to the product of the quotient rings R/Ip, p=1,...,k.

6.4 See also Residue eld

Goldies theorem

6.5 Notes[1] Jacobson, Nathan (1984). Structure of Rings (revised ed.). American Mathematical Soc. ISBN 0-821-87470-5.

[2] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.

[3] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

32 CHAPTER 6. QUOTIENT RING

6.6 Further references F. Kasch (1978) Moduln und Ringe, translated by DAR Wallace (1982) Modules and Rings, Academic Press,page 33.

Neal H. McCoy (1948) Rings and Ideals, 13 Residue class rings, page 61, Carus Mathematical Monographs#8, Mathematical Association of America.

Joseph Rotman (1998). Galois Theory (2nd edition). Springer. pp. 213. ISBN 0-387-98541-7. B.L. van der Waerden (1970) Algebra, translated by Fred Blum and John R Schulenberger, Frederick UngarPublishing, New York. See Chapter 3.5, Ideals. Residue Class Rings, pages 47 to 51.

6.7 External links Hazewinkel, Michiel, ed. (2001), Quotient ring, Encyclopedia ofMathematics, Springer, ISBN978-1-55608-010-4

Ideals and factor rings from John Beachys Abstract Algebra Online Quotient ring at PlanetMath.org.

Chapter 7

Quotient space (linear algebra)

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by collapsing N tozero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).

7.1 DenitionFormally, the construction is as follows (Halmos 1974, 21-22). Let V be a vector space over a eld K, and let Nbe a subspace of V. We dene an equivalence relation ~ on V by stating that x ~ y if x y N. That is, x is relatedto y if one can be obtained from the other by adding an element of N. From this denition, one can deduce that anyelement of N is related to the zero vector; more precisely all the vectors in N get mapped into the equivalence classof the zero vector.The equivalence class of x is often denoted

[x] = x + N

since it is given by

[x] = {x + n : n N}.

The quotient space V/N is then dened as V/~, the set of all equivalence classes over V by ~. Scalar multiplicationand addition are dened on the equivalence classes by

[x] = [x] for all K, and [x] + [y] = [x+y].

It is not hard to check that these operations are well-dened (i.e. do not depend on the choice of representative).These operations turn the quotient space V/N into a vector space over K with N being the zero class, [0].The mapping that associates to v V the equivalence class [v] is known as the quotient map.

7.2 ExamplesLet X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Ycan be identied with the space of all lines in X which are parallel to Y. That is to say that, the elements of the setX/Y are lines in X parallel to Y. This gives one way in which to visualize quotient spaces geometrically.Another example is the quotient of Rn by the subspace spanned by the rst m standard basis vectors. The space Rnconsists of all n-tuples of real numbers (x1,,xn). The subspace, identied with Rm, consists of all n-tuples such thatthe last n-m entries are zero: (x1,,xm,0,0,,0). Two vectors of Rn are in the same congruence class modulo the

33

34 CHAPTER 7. QUOTIENT SPACE (LINEAR ALGEBRA)

subspace if and only if they are identical in the last nm coordinates. The quotient space Rn/ Rm is isomorphic toRnm in an obvious manner.More generally, if V is an (internal) direct sum of subspaces U andW,

V = U W

then the quotient space V/U is naturally isomorphic toW (Halmos 1974, Theorem 22.1).An important example of a functional quotient space is a Lp space.

7.3 PropertiesThere is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. Thekernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exactsequence

0 ! U ! V ! V /U ! 0:

If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may beconstructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, thedimension of V is the sum of the dimensions ofU and V/U. If V is nite-dimensional, it follows that the codimensionof U in V is the dierence between the dimensions of V and U (Halmos 1974, Theorem 22.2):

codim(U) = dim(V /U) = dim(V ) dim(U):

Let T : V W be a linear operator. The kernel of T, denoted ker(T), is the set of all x V such that Tx = 0. Thekernel is a subspace of V. The rst isomorphism theorem of linear algebra says that the quotient space V/ker(T)is isomorphic to the image of V in W. An immediate corollary, for nite-dimensional spaces, is the rank-nullitytheorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of theimage (the rank of T).The cokernel of a linear operator T : V W is dened to be the quotient spaceW/im(T).

7.4 Quotient of a Banach space by a subspaceIf X is a Banach space andM is a closed subspace of X, then the quotient X/M is again a Banach space. The quotientspace is already endowed with a vector space structure by the construction of the previous section. We dene a normon X/M by

k[x]kX/M = infm2M

kxmkX :

The quotient space X/M is complete with respect to the norm, so it is a Banach space.

7.4.1 Examples

Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm.Denote the subspace of all functions f C[0,1] with f(0) = 0 byM. Then the equivalence class of some function g isdetermined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R.If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.

7.5. SEE ALSO 35

7.4.2 Generalization to locally convex spacesThe quotient of a locally convex space by a closed subspace is again locally convex (Dieudonn 1970, 12.14.8).Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {p | A} where A is an index set. Let M be a closed subspace, and dene seminorms q by on X/M

q([x]) = infx2[x]

p(x):

Then X/M is a locally convex space, and the topology on it is the quotient topology.If, furthermore, X is metrizable, then so is X/M. If X is a Frchet space, then so is X/M (Dieudonn 1970, 12.11.3).

7.5 See also quotient set quotient group quotient module quotient space (topology)

7.6 References Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 978-0-387-90093-3. Dieudonn, Jean (1970), Treatise on analysis, Volume II, Academic Press.

Chapter 8

Quotient space (topology)

For quotient spaces in linear algebra, see quotient space (linear algebra).In topology and related areas of mathematics, a quotient space (also called an identication space) is, intuitively

Illustration of quotient space, S2, obtained by gluing the boundary (in blue) of the disk D2 together to a single point.

speaking, the result of identifying or gluing together certain points of a given topological space. The points to beidentied are specied by an equivalence relation. This is commonly done in order to construct new spaces fromgiven ones. The quotient topology consists of all sets with an open preimage under the canonical projection mapthat maps each element to its equivalence class.

36

8.1. DEFINITION 37

8.1 DenitionLet (X, X) be a topological space, and let ~ be an equivalence relation on X. The quotient space, Y = X / ~ is denedto be the set of equivalence classes of elements of X:

Y = f[x] : x 2 Xg = ffv 2 X : v xg : x 2 Xg;equipped with the topology where the open sets are dened to be those sets of equivalence classes whose unions areopen sets in X:

Y =

8

38 CHAPTER 8. QUOTIENT SPACE (TOPOLOGY)

8.4 Properties

Quotient maps q : X Y are characterized among surjective maps by the following property: if Z is any topologicalspace and f : Y Z is any function, then f is continuous if and only if f q is continuous.

X

q

Y Z

f q

fCharacteristic property of the quotient topology

The quotient space X/~ together with the quotient map q : X X/~ is characterized by the following universalproperty: if g : X Z is a continuous map such that a ~ b implies g(a) = g(b) for all a and b in X, then there exists aunique continuous map f : X/~ Z such that g = f q. We say that g descends to the quotient.The continuous maps dened on X/~ are therefore precisely those maps which arise from continuous maps denedon X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). Thiscriterion is constantly used when studying quotient spaces.Given a continuous surjection q : X Y it is useful to have criteria by which one can determine if q is a quotientmap. Two sucient criteria are that q be open or closed. Note that these conditions are only sucient, not necessary.It is easy to construct examples of quotient maps that are neither open nor closed.

8.5. COMPATIBILITY WITH OTHER TOPOLOGICAL NOTIONS 39

8.5 Compatibility with other topological notions Separation

In general, quotient spaces are ill-behaved with respect to separation axioms. The separation propertiesof X need not be inherited by X/~, and X/~ may have separation properties not shared by X.

X/~ is a T1 space if and only if every equivalence class of ~ is closed in X. If the quotient map is open, then X/~ is a Hausdor space if and only if ~ is a closed subset of the productspace XX.

Connectedness If a space is connected or path connected, then so are all its quotient spaces. A quotient space of a simply connected or contractible space need not share those properties.

Compactness If a space is compact, then so are all its quotient spaces. A quotient space of a locally compact space need not be locally compact.

Dimension The topological dimension of a quotient space can be more (as well as less) than the dimension of theoriginal space; space-lling curves provide such examples.

8.6 See also

8.6.1 Topology Topological space Subspace (topology) Product space Disjoint union (topology) Final topology Mapping cone

8.6.2 Algebra Quotient group Quotient space (linear algebra) Quotient category Mapping cone (homological algebra)

8.7 References Willard, Stephen (1970). General Topology. Reading, MA: Addison-Wesley. ISBN 0-486-43479-6. Quotient space at PlanetMath.org.

Chapter 9

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.The binary operation of a semigroup is most often denoted multiplicatively: xy, or simply xy, denotes the result ofapplying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (xy)z = x(yz)for all x, y and z in the semigroup.The name semigroup originates in the fact that a semigroup generalizes a group by preserving only associativityand closure under the binary operation from the axioms dening a group.[note 1] From the opposite point of view (ofadding rather than removing axioms), a semigroup is an associative magma. As in the case of groups or magmas,the semigroup operation need not be commutative, so xy is not necessarily equal to yx; a typical example of asso-ciative but non-commutative operation is matrix multiplication. If the semigroup operation is commutative, then thesemigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called anabelian semigroup.Amonoid is an algebraic structure intermediate between groups and semigroups, and is a semigroup having an identityelement, thus obeying all but one of the axioms of a group; existence of inverses is not required of a monoid. A naturalexample is strings with concatenation as the binary operation, and the empty string as the identity element. Restrictingto non-empty strings gives an example of a semigroup that is not a monoid. Positive inte