Definition:Poset
Contents |
Definition
A poset (convenient abbreviation for partially ordered set) is a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is a partial ordering.
The poset $\left({S, \preceq}\right)$ is said to be partially ordered by $\preceq$.
In general, a poset can also be a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is an ordering which may or may not be partial.
Alternative Name
Some sources call this an ordered set, and prefer not to use the term partial.
However, according to ProofWiki house style, the advantage of being able to specify a difference between the various types of ordering outweighs any possible perceived inaccuracy in terminology.
Thus, on ProofWiki, poset and partially ordered set are the names to be used.
Also see
References
- ↑ T.S. Blyth: Set Theory and Abstract Algebra (1975):
"What we have defined as an order on a set many authors call a partial order and what we have called an ordered set is often called a partially ordered set or poset. As there is nothing which can even remotely be called 'partial' in the definition of an order relation, we shall not use this terminology."
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
- A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis (1968): $\S 3.1$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$
- Stanford handout on Lattice theory
Rationalise the links to this resource so as to conform it to house style. Ultimately we would like to eliminate links to papers like these and replace them with citations to documents which are attributed to authors and if possible official publications. Course handouts and lecture notes are often ephemeral, ad hoc and (in many cases, in my experience) exhibit the idiosyncrasies of particular lecturers which have been known to cause considerable confusion.
You can help Proof Wiki by completing it.
To discuss it in more detail, feel free to use the talk page.
When this has been done, {{WIP}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
