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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.
For example, "being isomorphic" is an equivalence relation on groups, and the equivalence classes, called isomorphism classes, are not sets. The set of all equivalence classes in X {\displaystyle X} with respect to an equivalence relation R {\displaystyle R} is denoted as X / R , {\displaystyle X/R,} and is called X {\displaystyle X} modulo R ...
A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective. For example, the green relation in the diagram is a surjection, but the red, blue and black ones are not. A bijection [d] A function that is injective and surjective.
The theory of an equivalence relation with exactly 2 infinite equivalence classes is an easy example of a theory which is ω-categorical but not categorical for any larger cardinal. The equivalence relation ~ should not be confused with the identity symbol '=': if x=y then x~y, but the converse is not necessarily true. Theories of equivalence ...
A generalization of the previous example is the following: Suppose a topological group acts continuously on a space . One can form an equivalence relation on by saying points are equivalent if and only if they lie in the same orbit.
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics.
In mathematics, given a category C, a quotient of an object X by an equivalence relation: is a coequalizer for the pair of maps , =,, where R is an object in C and "f is an equivalence relation" means that, for any object T in C, the image (which is a set) of : = (,) () is an equivalence relation; that is, a reflexive, symmetric and transitive relation.
An equivalent formulation in this context is the following: [4] A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A. The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel.