Definition:Closure (Abstract Algebra)
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Definition
Algebraic Structure
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 $S$.