Definition:Closure (Abstract Algebra)

Definition

Algebraic Structures

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.

Scalar Product

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

Let $T \subseteq S$ such that $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$.

Then $T$ is closed for scalar product.

If $T$ is also closed for operations on $S$, then it is called a closed subset of $S$.