Difference between revisions of "Quotient set"
(New page: A '''quotient set''' is a set derived from another by an equivalence relation. Let <math>S</math> be a set, and let <math>\mathcal{R}</math> be an equivalence relation. The set o...) |
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Let <math>S</math> be a set, and let <math>\mathcal{R}</math> be an equivalence relation. The set of [[equivalence class]]es of <math>S</math> with respect to <math>\mathcal{R}</math> is called the ''quotient of <math>S</math> by <math>\mathcal{R}</math>'', and is denoted <math>S/\mathcal{R}</math>. | Let <math>S</math> be a set, and let <math>\mathcal{R}</math> be an equivalence relation. The set of [[equivalence class]]es of <math>S</math> with respect to <math>\mathcal{R}</math> is called the ''quotient of <math>S</math> by <math>\mathcal{R}</math>'', and is denoted <math>S/\mathcal{R}</math>. | ||
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| + | A [[subset]] <math>A</math> of <math>S</math> is said to be ''saturated'' with respect to <math>\mathcal{R}</math> if for all <math>x,y \in S</math>, <math>x\in A</math> and <math>\mathcal{R}(x,y)</math> imply <math>y\in A</math>. Equivalently, <math>A</math> is saturated if it is the union of a family of equivalence classes with respect to <math>\mathcal{R}</math>. The ''saturation of <math>A</math> with respect to <math>\mathcal{R}</math>'' is the least saturated subset <math>A'</math> of <math>S</math> that contains <math>A</math>. | ||
== Compatible relations; derived relations; quotient structure == | == Compatible relations; derived relations; quotient structure == | ||
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Let <math>P(x)</math> be a relation. The relation <math>P'(y)</math> on the elements of <math>S/\mathcal{R}</math>, defined as | Let <math>P(x)</math> be a relation. The relation <math>P'(y)</math> on the elements of <math>S/\mathcal{R}</math>, defined as | ||
| − | <cmath> \ | + | <cmath> \exists x \in y, P(x) </cmath> |
is called the relation ''derived from <math>P</math> by passing to the quotient.'' | is called the relation ''derived from <math>P</math> by passing to the quotient.'' | ||
Latest revision as of 08:42, 7 June 2016
A quotient set is a set derived from another by an equivalence relation.
Let 
be a set, and let 
be an equivalence relation. The set of equivalence classes of with respect to is called the quotient of by , and is denoted 
.
A subset 
of is said to be saturated with respect to if for all 
, 
and 
imply 
. Equivalently, is saturated if it is the union of a family of equivalence classes with respect to . The saturation of with respect to is the least saturated subset 
of that contains .
Compatible relations; derived relations; quotient structure
Let 
be a relation, and let be an equivalence relation. If and together imply 
, then 
is said to be compatible with .
Let be a relation. The relation 
on the elements of , defined as
![\[\exists x \in y, P(x)\]](http://latex.artofproblemsolving.com/a/4/c/a4cac3697895d47459807b3b691836e4e2b6b129.png)
is called the relation derived from by passing to the quotient.
Let be a structure, , an equivalence relation. If the equivalence classes form a structure of the same species as under relations derived from passing to quotients, is said to be compatible with the structure on , and this structure on the equivalence classes of is called the quotient structure, or the derived structure, of .
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