The
Set of Posets |

**In General** > s.a. posets and types
of posets.

* __As a poset__: It is partially ordered by containment, with each *P*_{n} at
one level in the minimal ranking of *P*.

* __As a commutative semi-ring__: With Cartesian product and cardinal
sum as operations.

* __As a metric space__: > see types of distances.

* __Topology__: *P*_{n} has
a topology from a distance, and *P* can
be given one as a disjoint union, or from a partial order.

@ __References__: Arhangel'skii & Buzyakova T&A(09)
[linear orders, topology
of
pointwise
convergence].

**Operations on a Poset** > s.a. Wikipedia page on the way-below
relation [domain theory].

* __Subposet__: Any subset, with the induced order.

* __Covering poset__: The
set *C*(*P*) of all covering pairs
in *P*, with (*a*,* b*) < (*a'*, *b'*)
iff (*a*,* b*) = (*a'*,* b'*) or *b* < *a'* [@
Behrendt DM(88)].

* __Duality__: The dual *P**
of a poset *P* has the same underlying
set, but the relations are reversed, in the sense that *x* < *y* in
*P** iff *y *< *x* in *P*; Duality is the only
non-identity automorphism of the ordered set of isomorphism types of finite
posets, and of the lattice of universal classes of posets.

* __Extension__: Given an *n*-element
poset (*P*, <_{P}),
an extension of it is an order <_{L} on *P*,
such that *x* <_{P}* y* implies *x* <_{L} *y*;
A linear extension is one in which *P* is totally ordered.

* __Set of antichains__:
Various possible orders can be defined; Used in quantum gravity.

@ __Duality__: Banaschewski & Bruns Ord(88);
Navarro Ord(90);
Jezek & McKenzie Ord(10).

@ __Extensions__: Edelman et al Ord(89),
Brightwell & Winkler Ord(91)
[number of linear extensions];
Canfield & Williamson Ord(95)
[loop-free algorithm for linear extensions]; Corrêa & Szwarcfiter DM(05)
[set of extensions].

@ __Transitive closure__: Ma & Spinrad Ord(91).

@ __Completion__: Banaschewski ZMLGM(56); in
Bombelli & Meyer PLA(89);
Nation & Pogel Ord(97);
> s.a. limits.

@ __Exponentiation__: & Birkhoff; McKenzie Ord(99),
Ord(00) [*A*^{P} = *B*^{P} implies *A* = *B*].

@ __Other operations__: McKenzie Ord(03) [decompositions, + history]; Pach et al JCTA(13) [new operation, rotation of a finite poset].

**Binary Operations on Posets** > s.a. Star Product.

* __Cardinal sum__: *P*_{1} + *P*_{2} is the disjoint union of the two posets.

* __Cartesian product__: *P*_{1} × *P*_{2} is
the set of ordered pairs, ordered by (*x*_{1}, *x*_{2}) < (*y*_{1},
*y*_{2}) iff *x*_{1} < *y*_{1}
and *x*_{2} < *y*_{2}.

* __Intersection__: Given two different orders <_{1} and <_{2}
defined on the same underlying (labelled) set *P*, their intersection is defined by *a* <_{1
and 2} *b* iff *a* <_{1}
*b* and *a* <_{2} *b*;
It is used to represent an order of dimension *k* as intersection of *k* linear orders.

* __Ordinal sum__: *P*_{1} ⊕ *P*_{2} is "*P*_{1} sitting
on top of *P*_{2}",
or *P*_{1} ∪ *P*_{2}
with all elements of *P*_{1} preceding
all those of *P*_{2}.

**Operations on Families of Posets** > see Inductive
Limit.

**Generalizations** > s.a. Preorder; Quasiorder;
set theory [directed set].

* __Semiorder__: A set with
a transitive, reflexive but not necessarily antisymmetric relation; Basically a poset except for the fact that it may have closed loops; > s.a. Wikipedia page.

* __n____-Poset__: Any of several concepts that generalize posets in higher category theory; *n*-posets are the same as (*n*−1,*n*)-categories; For example, a 0-poset is a truth value, and a 1-poset or (0,1)-category is simply a poset; > s.a. nLab page.

@ __General references__: Brightwell Ord(89)
[semiorders, linear extensions]; Fishburn & Woodall Ord(99)
[cycle orders]; Voutsadakis Ord(07)
[*n*-ordered sets, completion]; Besnard JGP(09) ["non-commutative" ordered
spaces]; Mayburov IJTP(10) [fuzzy ordered sets]; Balof et al Ord(13) [representation polyhedron of a semiorder].

@ __Generalized ordered spaces__:
Bennett et al Ord(01) [cleavability], T&A(05) [separability and monotone
Lindelöf property].

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send feedback and suggestions to bombelli at olemiss.edu – modified 25
oct 2015