Types and Realizations of Posets |

**In General** > s.a. Hasse Diagram.

* __Well founded__: A poset *P*
is well founded if no infinite decreasing sequences occur in *P*.

* __Well partially ordered__: A well
founded poset containing no infinite antichains.

* __Locally finite__: A poset such that
every interval in it is finite.

* __Prime poset__: One such that all
its autonomous subsets are trivial.

@ __General references__: Bosi et al Ord(01) [interval orders, numerical representation];
Hubička & Nešetřil EJC(05) [universal partial order].

@ __Types__: Ng Ord(05) [linear discrepancy *n*−2];
Malicki & Rutkowski Ord(05) [well partially ordered];
Pouzet & Zaguia Ord(09),
Boudabbous et al Ord(10) [prime];
Dzhafarov Ord(11) [infinite saturated orders];
Bonanzinga & Matveev Ord(11) [thin posets];
Pouzet et al EJC(14)
[posets embeddable in a product of finitely many scattered chains];
> s.a. Relations [*k*-homogeneous].

**Sphere Order** > s.a. 2D geometries [Lorentzian].

* __Idea__: A poset realization
defined by sphere containment in Euclidean space.

* __Result__: Every circle order
has a normal representation, ∩_{i}
int(*P*_{i}) ≠ 0.

* __Result__: For all *n*
> 2, ∃ *P* with dim(*P*) = *n* which is not a
circle order; __Example__: [2] × [3] × \(\mathbb N\) is not,
but perhaps all 3D finite posets are; Some infinite posets are not sphere orders in any dimension!
[@ Fon-Der-Flaass Ord(93)].

@ __References__: Fishburn Ord(88);
Scheinerman & Wierman Ord(88);
Hurlbert Ord(88);
Sidney et al Ord(88);
Brightwell & Winkler Ord(89);
Lin Ord(91),
Scheinerman Ord(92) [circle orders];
Meyer Ord(93);
Scheinerman & Tanenbaum Ord(97);
Fishburn & Trotter Ord(98) [survey];
Vatandoost & Bahrampour JMP(11) [and spacetime].

**Other Realizations**

* __Path order__: A representation
by finite oriented paths ordered by the existence of homomorphism between them;
Any countable (finite or infinite) partially ordered set can be realized this way.

* __Function order__: A proper set of
functions *f*_{i}: [0,1] →
\(\mathbb R\) (two functions intersect, and cross, only a finite no of times,
but not at 0 or 1), with *p*_{i}
< *p*_{j} iff
*f*_{i}(*x*)
< *f*_{j}(*x*) for all
*x* ∈ [0,1]; Every poset can be represented as a function order.

@ __References__:
Sydney, Sydney & Urrutia Ord(88) [function orders];
Hubička & Nešetřil Ord(04),
Ord(05) [path orders];
Laison Ord(04)
[(*n*, *i*, *f*)-tube orders];
Berman & Blok Ord(06) [as algebras].

**Special Posets**
> s.a. set theory [directed set]; lattices.

* __Discrete poset__: One in which no two elements are comparable.

* __Binomial poset__: It is
locally finite; All maximal chains between two points have the same length;
Any two *n*-intervals contain the same number of maximal chains.

* __Linear order__:
*L*_{n} = The totally
ordered *n*-element poset; They are all 1-dimensional.

* __Standard n-dimensional
poset__:

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**Random Posets** > s.a. Relation.

* __(i) Uniform random orders__:
As *n* → ∞, most *n*-element orders have 3
levels; This is used for poset enumeration; 2004, We do not have a good way
to generate those random individual posets; 2015, Algorithm for generating uniform random orders.

* __(ii) Random k-dimensional orders__:
[Winkler Ord(85)]

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**Examples and Applications**
> s.a. Clustering; connection;
Filters; logic; topological
spaces; Well Ordering.

* __On the set of partitions of an
integer m__: A partition

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send feedback and suggestions to bombelli at olemiss.edu – modified 12 jun 2020