Equivalence relation
All definitions tacitly require the homogeneous relationA term's definition may require additional properties that are not listed in this table.is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive.The following sets are equivalence classes of this relation:is said to be well-defined or a class invariant under the relationThis occurs, e.g. in the character theory of finite groups.Some key definitions and terminology follow: A subsetThe equivalence kernel of an injection is the identity relation.Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number Bn: A key result links equivalence relations and partitions:[5][6][7] In both cases, the cells of the partition of X are the equivalence classes of X by ~.Since each element of X belongs to 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 ~." on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice.[8] Much of mathematics is grounded in the study of equivalences, and order relations.Lattice theory captures the mathematical structure of order relations.Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders.The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under bijections that preserve partition structure.Hence permutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathematical structure of equivalence relations.Let G denote the set of bijective functions over A that preserve the partition structure of A, meaning that for allThen the following three connected theorems hold:[10] In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are the equivalence classes of A under ~.This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations.The arguments of the lattice theory operations meet and join are elements of some universe A.Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a set of bijections, A → A.The equivalence classes of ~—also called the orbits of the action of H on G—are the right cosets of H in G. Interchanging a and b yields the left cosets.Related thinking can be found in Rosen (2008: chpt.Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.The advantages of regarding an equivalence relation as a special case of a groupoid include: The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X 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.Equivalence relations are a ready source of examples or counterexamples.For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples: Properties definable in first-order logic that an equivalence relation may or may not possess include:
Doctrine of equivalentsEquivalenceTransitivebinary relationsSymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveAsymmetricPreorder (Quasiorder)Partial orderTotal preorderTotal orderPrewellorderingWell-quasi-orderingWell-orderingLatticeJoin-semilatticeMeet-semilatticeStrict partial orderStrict total orderhomogeneous relationlogical matricesmathematicsbinary relationequipollencepartitionequivalence classesif and only ifreflexivitysymmetrytransitivitysetoidrelational algebracomposite relationfunctional compositionidentity functionsimilartrianglescongruentmodulointegersfunctiondomaincommon factornatural numbersempty relationvacuouslyEqualityalgebraic expressionssubstitutedpartial equivalence relationternary equivalence relationdependency relationtolerance relationpreordercongruence relationalgebraic structurekernelsnormal subgroupsapartness relationconstructive mathematicslaw of excluded middleEuclideanwell-definedinvariantEquivalence classQuotient settopological spaceQuotient spaceProjection (relational algebra)projectionssurjectionbijectioninjectionidentity relationPartition of a setpairwise disjointBell numberDobinski's formulageometric latticefixpointscardinalitypermutationsurjectivepreimagessingletonscodomainsubsetnatural numberCartesian squarehomeomorphismorder relationsLattice theorygroup theorycategoriesgroupoidsordered setssupremuminfimumpartitioned setsbijectionspermutationspermutation groupstransformation groupsuniversetransformation grouplatticescompositioninversesubgroupactioncosetsgroupoidfree groupoiddirected graphgroup actionscongruencescategoryset inclusioncomplete latticemonoidfunctions