Definition:Closure (Abstract Algebra)

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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 $S$.