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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