Quotient set

A quotient set is a set derived from another by an equivalence relation.

Let $S$ be a set, and let $\mathcal{R}$ be an equivalence relation. The set of equivalence classes of $S$ with respect to $\mathcal{R}$ is called the quotient of $S$ by $\mathcal{R}$ , and is denoted $S/\mathcal{R}$ .

Compatible relations; derived relations; quotient structure

Let $P(x)$ be a relation, and let $\mathcal{R}$ be an equivalence relation. If $\mathcal{R}(x,y)$ and $P(x)$ together imply $P(y)$ , then $P$ is said to be compatible with $\mathcal{R}$ .

Let $P(x)$ be a relation. The relation $P'(y)$ on the elements of $S/\mathcal{R}$ , defined as

\[\exist x\in y, P(x)\] (Error compiling LaTeX. Unknown error_msg)

is called the relation derived from $P$ by passing to the quotient.

Let $S$ be a structure, $\mathcal{R}$ , an equivalence relation. If the equivalence classes form a structure of the same species as $S$ under relations derived from passing to quotients, $\mathcal{R}$ is said to be compatible with the structure on $S$ , and this structure on the equivalence classes of $S$ is called the quotient structure, or the derived structure, of $S/\mathcal{R}$ .

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Quotient set

Compatible relations; derived relations; quotient structure