Definition:Closure (Abstract Algebra)/Algebraic Structure

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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)


Sources

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