Definition:Closure (Abstract Algebra)/Algebraic Structure

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure.

Then $S$ has the property of closure under $\circ$ iff:


 * $\forall \left({x, y}\right) \in S \times S: x \circ y \in S$

$S$ is said to be closed under $\circ$, or just that $\left({S, \circ}\right)$ 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 $S$.

Also see

 * Definition:Magma: an algebraic structure which has the property of closure as defined here.


 * Definition:Submagma: how the concept of closure is applied to a subset of an algebraic structure.