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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]].
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.
==
{{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
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
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
==
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
{{citation
| 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.
|mr = 1171114
▲ | doi = 10.1016/0012-365X(92)90036-F
▲ | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
▲ | pages = 25–40
▲ | title = Linear extensions of semiorders: a maximization problem
▲ | volume = 103
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
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
|last1 = Brightwell|first1 = G. R.
▲ }}.</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
|mr = 1368815
▲ | doi = 10.1007/BF01110378
▲ | 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
==
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]].
==
{{reflist}}
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