The Set of Posets| The Set of Posets |
In General
> s.a. posets and types of posets.
* As a poset: It is partially
ordered by containment, with each Pn
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:
Pn 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)
[AP = BP
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:
P1 + P2
is the disjoint union of the two posets.
* Cartesian product:
P1 × P2
is the set of ordered pairs, ordered by (x1,
x2) < (y1,
y2) iff x1
< y1 and x2
< y2.
* 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: P1
⊕ P2 is "P1
sitting on top of P2", or
P1 ∪ P2
with all elements of P1 preceding all those of
P2.
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.
* Quantum poset: A hereditarily
atomic von Neumann algebra equipped with a quantum partial order in Weaver's sense.
@ 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];
Kornell et al a2101 [quantum posets].
@ Generalized ordered spaces: Bennett et al Ord(01) [cleavability],
T&A(05)
[separability and monotone Lindelöf property].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 28 jan 2021