|The Set of Posets|
In General > s.a. posets and types
* 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.
@ 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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