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 $S/\mathcal{R}$.

A subset $A$ of $S$ is said to be saturated with respect to $\mathcal{R}$ if for all $x,y \in S$, $x\in A$ and $\mathcal{R}(x,y)$ imply $y\in A$. Equivalently, $A$ is saturated if it is the union of a family of equivalence classes with respect to $\mathcal{R}$. The saturation of $A$ with respect to $\mathcal{R}$ is the least saturated subset $A'$ of $S$ that contains $A$.

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 \[\exists x \in y, P(x)\] 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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