Definition:Closure (Abstract Algebra)

From ProofWiki
Jump to navigation Jump to search

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.

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 $S$.