Definition:Closure (Abstract Algebra)/Algebraic Structure
Jump to navigation
Jump to search
Definition
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.
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 (or on) $S$.
Examples
Numbers of form $2^m 3^n$ under Multiplication
Let $S$ be the set defined as:
- $S := \set {2^m 3^n: m, n \in \Z}$
Then the algebraic structure $\struct {S, \times}$ is closed.
Also see
- Definition:Magma: an algebraic structure which has the property of closure as defined here.
- Definition:Submagma: how the concept of closure is applied to a subset of an algebraic structure.
- Results about algebraic closure can be found here.
Internationalization
Closure is translated:
In German: | Abgeschlossenheit | (literally: seclusion) |
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.1$. Subsets closed to an operation: $S 1$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.5$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Operations
- 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $1$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 27$. Binary operations
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $1$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): closed: 1. (of a set under an operation)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): closed (under an operation)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): operation