Difference between revisions of "Quotient set"
(added note on saturation) |
(→Compatible relations; derived relations; quotient structure) |
||
Line 10: | Line 10: | ||
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 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 .
This article is a stub. Help us out by expanding it.