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 09:42, 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
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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