a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle S} into equivalence classes. These...

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments...

In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X {\displaystyle f:R\to X\times X} is a coequalizer...

coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction...

left quotient is m ∖ S = { u ∈ M | m u ∈ S } . {\displaystyle m\setminus S=\{u\in M\;\vert \;mu\in S\}.} The syntactic quotient induces an equivalence relation...

mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of...

mathematics, a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are...

congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. The definition...

partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that...

science, a quotient type is a data type which respects a user-defined equality relation. A quotient type defines an equivalence relation ≡ {\displaystyle...

In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X × X (in the product topology)...

satisfy the equivalence relation because their difference vectors belong to Y. This gives a way to visualize quotient spaces geometrically. (By re-parameterising...

A over a ring R, and a submodule B of A, the quotient space A/B is defined by the equivalence relation a ∼ b {\displaystyle a\sim b} if and only if b...

"quotient" to describe structures built by breaking larger structures into pieces. Given a set with an equivalence relation defined on it, a "quotient...

theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q...

define algebraic spaces: they can be defined as either quotients of schemes by étale equivalence relations, or as sheaves on a big étale site that are...

is short for "modulo". Formally this is a special case of quotient by an equivalence relation, where g ∼ h {\displaystyle g\sim h} iff g = h n {\displaystyle...

congruence relation R on C we can define the quotient category C/R as the category whose objects are those of C and whose morphisms are equivalence classes...

an equivalence relation Quotient group Quotient ring Quotient module Quotient space (linear algebra) Quotient space (topology), by an equivalence relation...

Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. MR1304906 ISBN 3-540-56963-4 Quotient by an equivalence relation Quotient stack v t e...

indistinguishability is an equivalence relation on any topological space X, we can form the quotient space KX = X/≡. The space KX is called the Kolmogorov quotient or T0...

points is an equivalence relation. No matter what topological space X might be to begin with, the quotient space under this equivalence relation is always...

codomain A {\displaystyle A} , we define an equivalence relation by u ≡ v {\displaystyle u\equiv v} if there exists an isomorphism ϕ : S → T {\displaystyle...

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that...

characterize the equivalence relation. In this case, the quotient by the equivalence relation is denoted by X / K {\displaystyle X/K} (usually read as " X {\displaystyle...

subset, function, and quotient by an equivalence relation. When studying groups, one uses instead subgroups, homomorphisms, and quotient groups. These are...

complete quotient in the regular continued fraction expansions of x and y be denoted by ζ k and ψ k, respectively, Then x ~ y (under the equivalence defined...

has edge set E and vertex set V and R is the equivalence relation induced by the partition, then the quotient graph has vertex set V/R and edge set {([u]R...

F=\{(i,x,y):i\in I,x\in U_{i},y\in F\}} and then forms the quotient by the equivalence relation ( j , x , y ) ∼ ( i , x , t i j ( x ) ⋅ y ) ∀ x ∈ U i ∩ U...

order), an equivalence relation, or a relation with any other special properties, if and only if R is. For example, R is an ordering ≤ and S an ordering...