Definition:Closure (Abstract Algebra)/Algebraic Structure

Definition

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.

Also known as

Some authors use stable under $\circ$ for closed under $\circ$.

It is sometimes more convenient to express this property the other way about, as $\circ$ is closed in $S$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.1: \ S 1$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 8$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.5$: Theorem $7$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 27$