Definition:Ordered Set
Definition
An ordered set is a relational structure $\struct {S, \preceq}$ such that the relation $\preceq$ is an ordering.
Such a structure may be:
depending on whether the ordering $\preceq$ is:
Ordered Class
The concept carries naturally over into class theory:
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation.
Let $A$ be a subclass of the field of $\RR$.
Let the restriction of $\RR$ to $A$ be an ordering on $A$.
Then $A$ is described as being ordered under $\RR$.
Also known as
Some sources call this an ordered structure, but this often has a more specialized meaning.
Some call this a poset, a partially ordered set or a partly ordered set, but we tend to avoid these on $\mathsf{Pr} \infty \mathsf{fWiki}$.
The term order structure is also sometimes encountered.
Some sources refer to $\struct {S, \preceq}$ as a partial order, calling $\preceq$ a partial order relation.
Also defined as
Some sources reserve the term ordered set for what is known on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a totally ordered set.
Also see
- Results about ordered sets can be found here.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Orderings
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.) ... (next): $\S \text I.1$
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.1$: Partially ordered sets
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 3$: Equivalence relations and quotient sets: Binary relations
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 1982: Peter T. Johnstone: Stone Spaces ... (previous) ... (next): Chapter $\text I$: Preliminaries, Definition $1.2$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ordered set
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.3$