Linear extension: Difference between revisions

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In [[order theory]], a branch of [[mathematics]], a '''linear extension''' of a [[partial order]] is a [[total order]] (or linear order) that is compatible with the partial order. As a classic example, the [[lexicographic order]] of totally ordered sets is a linear extension of their [[product order]].

== Definitions ==

In [[functional analysis]] and [[linear algebra]], a '''linear extension''' of a [[Function (mathematics)|function]] <math>f</math> refers to an [[Extension of a function|extension]] of <math>f</math> to some larger [[vector space]] that is also a [[linear map]].

==Definitions==

Given any partial orders <math>\,\leq\,</math> and <math>\,\leq^*\,</math> on a set <math>X,</math> <math>\,\leq^*\,</math> is a linear extension of <math>\,\leq\,</math> exactly when (1) <math>\,\leq^*\,</math> is a [[total order]] and (2) for every <math>x, y \in X,</math> if <math>x \leq y,</math> then <math>x \leq^* y.</math> It is that second property that leads mathematicians to describe <math>\,\leq^*\,</math> as '''extending''' <math>\,\leq.</math>

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Alternatively, a linear extension may be viewed as an [[order-preserving]] [[bijection]] from a partially ordered set <math>P</math> to a [[Total order#Chains|chain]] <math>C</math> on the same ground set.

== Order-extension principle ==

{{further|Szpilrajn extension theorem}}

The statement that every partial order can be extended to a total order is known as the '''order-extension principle'''. A proof using the [[axiom of choice]] was first published by [[Edward Marczewski]] in 1930. Marczewski writes that the theorem had previously been proven by [[Stefan Banach]], [[Kazimierz Kuratowski]], and [[Alfred Tarski]], again using the axiom of choice, but that the proofs had not been published.<ref>{{citation

| last = Marczewski | first = Edward | authorlink = Edward Marczewski

| journal = Fundamenta Mathematicae

| language = French

| pages = 386–389

| title = Sur l'extension de l'ordre partiel

| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm16125.pdf

| volume = 16

| year = 1930| doi = 10.4064/fm-16-1-386-389 | doi-access = free
}}.</ref>

In modern [[axiomatic set theory]] the order-extension principle is itself taken as an axiom, of comparable ontological status to the axiom of choice. The order-extension principle is implied by the [[Boolean prime ideal theorem]] or the equivalent [[compactness theorem]],<ref>{{citation

| last = Jech | first = Thomas | author-link = Thomas Jech

| isbn = 978-0-486-46624-8

| orig-year=originally published in 1973

| publisher = [[Dover Publications]]

| title = The Axiom of Choice

| year = 2008}}.</ref> but the reverse implication doesn't hold.<ref>{{citation

| last1 = Felgner | first1 = U.

| last2 = Truss | first2 = J. K.

| date = March 1999

| doi = 10.2307/2586759

| issue = 1

| journal = The Journal of Symbolic Logic

| pages = 199–215

| title = The Independence of the Prime Ideal Theorem from the Order-Extension Principle

| volume = 64

| jstor = 2586759| citeseerx = 10.1.1.54.8336
}}.</ref>

Applying the order-extension principle to a partial order in which every two elements are incomparable shows that (under this principle) every set can be linearly ordered. This assertion that every set can be linearly ordered is known as the '''ordering principle''', OP, and is a weakening of the [[well-ordering theorem]]. However, there are [[Model theory|models of set theory]] in which the ordering principle holds while the order-extension principle does not.<ref>{{citation

| last = Mathias | first = A. R. D.

| contribution = The order extension principle

| editor1-last = Scott | editor1-first = Dana S.

| editor2-last = Jech | editor2-first = Thomas J.

| pages = 179–183

| publisher = American Mathematical Society

| series = Proceedings of Symposia in Pure Mathematics

| title = Axiomatic Set Theory (University of California, Los Angeles, Calif., July 10 – August 5, 1967)

| volume = 13

| year = 1971}}.</ref>

== Related results ==

The order extension principle is [[Constructive proof|constructively provable]] for {{em|finite}} sets using [[topological sorting]] algorithms, where the partial order is represented by a [[directed acyclic graph]] with the set's elements as its [[Vertex (graph theory)|vertices]]. Several algorithms can find an extension in [[linear time]].<ref>{{citation

| last1 = Cormen | first1 = Thomas H. | author1-link = Thomas H. Cormen

| last2 = Leiserson | first2 = Charles E. | author2-link = Charles E. Leiserson

| last3 = Rivest | first3 = Ronald L. | author3-link = Ronald L. Rivest

| last4 = Stein | first4 = Clifford | author4-link = Clifford Stein

| contribution = Section 22.4: Topological sort

| edition = 2nd

| isbn = 978-0-262-03293-3

| pages = 549–552

| publisher = MIT Press

| title = Introduction to Algorithms

| year = 2001| title-link = Introduction to Algorithms }}.</ref> Despite the ease of finding a single linear extension, the problem of counting all linear extensions of a finite partial order is [[Sharp-P-complete|#P-complete]]; however, it may be estimated by a [[fully polynomial-time randomized approximation scheme]].<ref>{{citation

| last1 = Brightwell | first1 = Graham R.

| last2 = Winkler | first2 = Peter | author2-link = Peter Winkler

| doi = 10.1007/BF00383444

| issue = 3

| journal = [[Order (journal)|Order]]

| pages = 225–242

| title = Counting linear extensions

| volume = 8

| year = 1991| s2cid = 119697949
}}</ref><ref>

{{citation

| last1 = Bubley | first1 = Russ

| last2 = Dyer | first2 = Martin | author2-link = Martin Dyer

| doi = 10.1016/S0012-365X(98)00333-1

| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]

| pages =81–88

| title = Faster random generation of linear extensions

| volume = 201

| issue = 1–3

| year = 1999| s2cid = 2942330

| doi-access = free

|doi-access = free }}.</ref> Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of linear extensions are [[semiorder]]s.<ref>{{citation

| last1 = Fishburn | first1 = Peter C. | author1-link = Peter C. Fishburn

| last2 = Trotter | first2 = W. T.

| doi = 10.1016/0012-365X(92)90036-F

|mr = 1171114

|issue mr = 11171114

| issue = 1

| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]

| pages = 25–40

| title = Linear extensions of semiorders: a maximization problem

|volume = 103

| volume = 103
| year = 1992| doi-access = free
}}.</ref>

The [[order dimension]] of a partial order is the minimum cardinality of a set of linear extensions whose intersection is the given partial order; equivalently, it is the minimum number of linear extensions needed to ensure that each [[Critical pair (order theory)|critical pair]] of the partial order is reversed in at least one of the extensions.

[[Antimatroid]]s may be viewed as generalizing partial orders; in this view, the structures corresponding to the linear extensions of a partial order are the basic words of the antimatroid.<ref>{{citation

| last1 = Björner

| first1 = Anders

| last2 = Ziegler

| first2 = Günter M.

| author2-link = Günter M. Ziegler

| contribution = Introduction to Greedoids

| editor-last = White

| editor-first = Neil

| isbn = 978-0-521-38165-9

| pages = [https://archive.org/details/matroidapplicati0000unse/page/284 284–357]

| publisher = Cambridge University Press

| series = Encyclopedia of Mathematics and its Applications

| title = Matroid Applications

| volume = 40

| year = 1992

| url = https://archive.org/details/matroidapplicati0000unse/page/284
}}. See especially item (1) on {{nowrap|p. 294.}}</ref>

This area also includes one of order theory's most famous open problems, the [[1/3–2/3 conjecture]], which states that in any finite partially ordered set <math>P</math> that is not [[Total order|totally ordered]] there exists a pair <math>(x, y)</math> of elements of <math>P</math> for which the linear extensions of <math>P</math> in which <math>x < y</math> number between 1/3 and 2/3 of the total number of linear extensions of <math>P.</math><ref>{{citation|author=Kislitsyn, S. S.|year=1968|title=Finite partially ordered sets and their associated sets of permutations|journal=Matematicheskie Zametki|volume=4|pages=511–518}}.</ref><ref>{{citation

| last = Brightwell | first = Graham R.

| doi = 10.1016/S0012-365X(98)00311-2

| issue = 1–3

| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]

| pages = 25–52

| title = Balanced pairs in partial orders

| volume = 201

| year = 1999| doi-access = free

|year = 1999|doi-access = free }}.</ref> An equivalent way of stating the conjecture is that, if one chooses a linear extension of <math>P</math> uniformly at random, there is a pair <math>(x, y)</math> which has probability between 1/3 and 2/3 of being ordered as <math>x < y.</math> However, for certain infinite partially ordered sets, with a canonical probability defined on its linear extensions as a limit of the probabilities for finite partial orders that cover the infinite partial order, the 1/3–2/3 conjecture does not hold.<ref>{{citation

|last1 = Brightwell|first1 = G. R.

|last2 last1 = FelsnerBrightwell |first2 first1 = SG. R.

|last3 last2 = TrotterFelsner |first3 =first2 W.= TS.

| last3 = Trotter | first3 = W. T.

| doi = 10.1007/BF01110378

|mr = 1368815

|issue mr = 41368815

| issue = 4
| journal = [[Order (journal)|Order]]

| pages = 327–349

| title = Balancing pairs and the cross product conjecture

| volume = 12

| year = 1995| citeseerx = 10.1.1.38.7841

| s2cid = 14793475
}}.</ref>

== Algebraic combinatorics ==

Counting the number of linear extensions of a finite poset is a common problem in [[algebraic combinatorics]]. This number is given by the leading coefficient of the [[order polynomial]] multiplied by <math>|P|!.</math>

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[[Young tableau]] can be considered as linear extensions of a finite [[Ideal (order theory)|order-ideal]] in the infinite poset <math>\N \times \N,</math> and they are counted by the [[hook length formula]].

== References ==

{{reflist}}