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| [[Image:Order dimension.svg|thumb|240px|A partial order of dimension 4 (shown as a [[Hasse diagram]]) and four total orderings that form a realizer for this partial order.]]
| | I am Oscar and I completely dig that name. Body developing is what my family members and I enjoy. Hiring is her day job now and she will not change it anytime quickly. California is exactly where her home is but she needs to move simply because of her family.<br><br>Here is my webpage - [http://www.alemcheap.fi/people/anhipkiss http://www.alemcheap.fi/people/anhipkiss] |
| In [[mathematics]], the '''dimension''' of a [[partially ordered set]] (poset) is the smallest number of [[total order]]s the [[Intersection (set theory)|intersection]] of which gives rise to the partial order.
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| This concept is also sometimes called the '''order dimension''' or the '''Dushnik–Miller dimension''' of the partial order.
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| {{harvtxt|Dushnik|Miller|1941}} first studied order dimension; for a more detailed treatment of this subject than provided here, see {{harvtxt|Trotter|1992}}.
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| ==Formal definition==
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| The dimension of a poset ''P'' is the least integer ''t'' for which there exists a family
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| :<math>\mathcal R=(<_1,\dots,<_t)</math>
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| of [[linear extension]]s of ''P'' so that, for every ''x'' and ''y'' in ''P'', ''x'' precedes ''y'' in ''P'' if and only if it precedes ''y'' in each of the linear extensions. That is,
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| :<math>P=\bigcap\mathcal R=\bigcap_{i=1}^t <_i.</math>
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| ==Realizers==
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| A family <math>\mathcal R=(<_1,\dots,<_t)</math> of linear orders on ''X'' is called a '''realizer''' of a poset ''P'' = (''X'', <<sub>''P''</sub>) if
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| :<math><_P = \bigcap\mathcal R</math>,
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| which is to say that for any ''x'' and ''y'' in ''X'',
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| ''x'' <<sub>''P''</sub> ''y'' precisely when ''x'' <<sub>1</sub> ''y'', ''x'' <<sub>2</sub> ''y'', ..., and ''x'' <<sub>''t''</sub> ''y''.
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| Thus, an equivalent definition of the dimension of a poset ''P'' is "the least [[cardinality]] of a realizer of ''P''."
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| It can be shown that any nonempty family ''R'' of linear extensions is a realizer of a finite partially ordered set ''P'' if and only if, for every [[Critical pair (order theory)|critical pair]] (''x'',''y'') of ''P'', ''y'' <<sub>''i''</sub> ''x'' for some order
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| <<sub>''i''</sub> in ''R''.
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| == Example ==
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| Let ''n'' be a positive integer, and let ''P'' be the partial order on the elements ''a<sub>i</sub>'' and ''b<sub>i</sub>'' (for 1 ≤ ''i'' ≤ ''n'') in which ''a<sub>i</sub>'' ≤ ''b<sub>j</sub>'' whenever ''i'' ≠ ''j'', but no other pairs are comparable. In particular, ''a<sub>i</sub>'' and ''b<sub>i</sub>'' are incomparable in ''P''; ''P'' can be viewed as an oriented form of a [[crown graph]]. The illustration shows an ordering of this type for ''n'' = 4.
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| Then, for each ''i'', any realizer must contain a linear order that begins with all the ''a<sub>j</sub>'' except ''a<sub>i</sub>'' (in some order), then includes ''b<sub>i</sub>'', then ''a<sub>i</sub>'', and ends with all the remaining ''b<sub>j</sub>''. This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have ''a<sub>i</sub>'' preceding ''b<sub>i</sub>'', which would contradict the incomparability of ''a<sub>i</sub>'' and ''b<sub>i</sub>'' in ''P''. And conversely, any family of linear orders that includes one order of this type for each ''i'' has ''P'' as its intersection. Thus, ''P'' has dimension exactly ''n''. In fact, ''P'' is known as the ''standard example'' of a poset of dimension ''n'', and is usually denoted by ''S<sub>n</sub>''.
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| ==Order dimension two==
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| The partial orders with order dimension two may be characterized as the partial orders whose [[comparability graph]] is the [[complement (graph theory)|complement]] of the comparability graph of a different partial order {{harv|Baker|Fishburn|Roberts|1971}}. That is, ''P'' is a partial order with order dimension two if and only if there exists a partial order ''Q'' on the same set of elements, such that every pair ''x'', ''y'' of distinct elements is comparable in exactly one of these two partial orders. If ''P'' is realized by two linear extensions, then partial order ''Q'' complementary to ''P'' may be realized by reversing one of the two linear extensions. Therefore, the comparability graphs of the partial orders of dimension two are exactly the [[permutation graph]]s, graphs that are both themselves comparability graphs and complementary to comparability graphs.
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| The partial orders of order dimension two include the [[series-parallel partial order]]s {{harv|Valdes|Tarjan|Lawler|1982}}. They are exactly the partial orders whose [[Hasse diagram]]s have [[dominance drawing]]s, which can be obtained by using the positions in the two permutations of a realizer as Cartesian coordinates.
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| ==Computational complexity==
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| It is possible to determine in [[polynomial time]] whether a given finite partially ordered set has order dimension at most two, for instance, by testing whether the comparability graph of the partial order is a permutation graph. However, for any ''k'' ≥ 3, it is [[NP-complete]] to test whether the order dimension is at most ''k'' {{harv|Yannakakis|1982}}.
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| ==Incidence posets of graphs==
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| The incidence poset of any [[undirected graph]] ''G'' has the vertices and edges of ''G'' as its elements; in this poset, ''x'' ≤ ''y'' if either ''x'' = ''y'' or ''x'' is a vertex, ''y'' is an edge, and ''x'' is an endpoint of ''y''. Certain kinds of graphs may be characterized by the order dimensions of their incidence posets: a graph is a [[path graph]] if and only if the order dimension of its incidence poset is at most two, and according to [[Schnyder's theorem]] it is a [[planar graph]] if and only if the order dimension of its incidence poset is at most three {{harv|Schnyder|1989}}.
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| == k-dimension and 2-dimension ==
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| A generalization of dimension is the notion of ''k''-dimension (written <math>\textrm{dim}_k</math>) which is the minimal number of [[chain (order theory)|chains]] of length at most ''k'' in whose product the partial order can be embedded. In particular, the 2-dimension of an order can be seen as the size of the smallest set such that the order embeds in the [[containment order]] on this set.
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| == See also ==
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| * [[Interval dimension]]
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| == References ==
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| *{{citation
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| | last1 = Baker | first1 = K. A.
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| | last2 = Fishburn | first2 = P. | author2-link = Peter C. Fishburn
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| | last3 = Roberts | first3 = F. S. | author3-link = Fred S. Roberts
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| | doi = 10.1002/net.3230020103
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| | issue = 1
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| | journal = Networks
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| | pages = 11–28
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| | title = Partial orders of dimension 2
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| | volume = 2
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| | year = 1971}}.
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| *{{citation
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| | last1 = Dushnik | first1 = Ben
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| | last2 = Miller | first2 = E. W.
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| | doi = 10.2307/2371374
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| | issue = 3
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| | journal = [[American Journal of Mathematics]]
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| | pages = 600–610
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| | title = Partially ordered sets
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| | volume = 63
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| | year = 1941
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| | jstor = 2371374}}.
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| *{{citation
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| | last = Schnyder | first = W.
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| | doi = 10.1007/BF00353652
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| | issue = 4
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| | journal = [[Order (journal)|Order]]
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| | pages = 323–343
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| | title = Planar graphs and poset dimension
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| | volume = 5
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| | year = 1989}}.
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| *{{citation
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| | last = Trotter | first = William T.
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| | isbn = 978-0-8018-4425-6
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| | publisher = The Johns Hopkins University Press
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| | series = Johns Hopkins Series in the Mathematical Sciences
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| | title = Combinatorics and partially ordered sets: Dimension theory
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| | year = 1992}}.
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| *{{citation
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| | last1 = Valdes | first1 = Jacobo
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| | last2 = Tarjan | first2 = Robert E. | author2-link = Robert Tarjan
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| | last3 = Lawler | first3 = Eugene L. | author3-link = Eugene Lawler
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| | doi = 10.1137/0211023
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| | issue = 2
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| | journal = [[SIAM Journal on Computing]]
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| | pages = 298–313
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| | title = The recognition of series parallel digraphs
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| | volume = 11
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| | year = 1982}}.
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| *{{citation
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| | doi = 10.1137/0603036
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| | last = Yannakakis | first = Mihalis | author-link = Mihalis Yannakakis
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| | issue = 3
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| | journal = SIAM Journal on Algebraic and Discrete Methods
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| | pages = 351–358
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| | title = The complexity of the partial order dimension problem
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| | volume = 3
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| | year = 1982}}.
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| [[Category:Order theory]]
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| [[Category:Dimension theory]]
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I am Oscar and I completely dig that name. Body developing is what my family members and I enjoy. Hiring is her day job now and she will not change it anytime quickly. California is exactly where her home is but she needs to move simply because of her family.
Here is my webpage - http://www.alemcheap.fi/people/anhipkiss