Dimension of Partially Ordered Sets

Wednesday, November 12, 2014 - 3:30pm - 3:55pm
Keller 3-180
Attila Por (Western Kentucky University)
Let $P$ be a poset. A set of linear extensions $LL = { L_1, dots , L_d }$ forms a realizer if $P=L_1 cap dots cap L_d$.
The dimension (Dushnik-Miller dimension) of the poset $P$ is the minimum cardinality of a realizer.
The standard example $S_n$ is the poset on all the $ and $(n-1)$ element subsets of an $n$ element set with respect to the $subset$ relation.
The dimension of a poset on n$ points can be at most $n$ and if it is equal to $n$ than it must be the standard example $S_n$. \

We investigate the connection between the dimension of the poset $P$ and the size of the largest standard example it contains as a sub-poset.
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