# Definition:Closure (Abstract Algebra)/Algebraic Structure

(Redirected from Definition:Closed Operation)

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Then $S$ has the property of closure under $\circ$ if and only if:

$\forall \tuple {x, y} \in S \times S: x \circ y \in S$

$S$ is said to be closed under $\circ$, or just that $\struct {S, \circ}$ is closed.

## Also known as

Some authors use stable under $\circ$ for closed under $\circ$.

It is sometimes more convenient to express this property the other way about, as:

$\circ$ is closed in (or on) $S$.

## Examples

### Numbers of form $2^m 3^n$ under Multiplication

Let $S$ be the set defined as:

$S := \set {2^m 3^n: m, n \in \Z}$

Then the algebraic structure $\struct {S, \times}$ is closed.

## Also see

• Results about algebraic closure can be found here.

## Internationalization

Closure is translated:

 In German: Abgeschlossenheit (literally: seclusion)