# Definition:Closure (Abstract Algebra)

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## Definition

### Algebraic Structures

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**.

### Scalar Product

Let $\struct {S, \circ}_R$ be an $R$-algebraic structure over a ring $R$.

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 $S$.