# Quotient set

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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