Dimension of a partially ordered set
dimension of a poset
Let be a partially ordered set. A linear extension
of
is a total ordering
(see also Totally ordered set) on
such that
is order preserving, i.e. such that
in
implies
in
. The existence of linear extensions was proved by E. Szpilrain, [a4], in 1931.
The dimension of a poset is the least
for which there exists a family of linear extensions
such that
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A chain in is a partially ordered subset of
(a sub-poset) that is totally ordered. An anti-chain in
is a set of elements
such that no
,
,
, are comparable. The height of
is the maximal length of a chain. The width of
is the maximal size of an anti-chain. The Dilworth decomposition theorem says that if
, then
decomposes as the disjoint sum of
chains (cf. Disjoint sum of partially ordered sets; cf. also (the editorial comments to) Partially ordered set). Using this, T. Hiraguchi [a2] proved
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The concept was introduced by B. Dushnik and E.W. Miler [a1].
References
[a1] | B. Dushnik, E.W. Miller, "Partially ordered sets" Amer. J. Math. , 63 (1941) pp. 600–610 |
[a2] | T. Hiraguchi, "On the dimension of partially ordered sets" Sci. Rep. Kanazawa Univ. , 1 (1951) pp. 77–94 |
[a3] | D. Kelley, W.T. Trotter, "Dimension theory for ordered sets" I. Rival (ed.) , Ordered Sets , Reidel (1987) pp. 171–212 |
[a4] | E. [E. Szpilrain] Szpilrajn, "Sur l'extension de l'ordre partiel" Fund. Math. , 16 (1930) pp. 386–389 |
[a5] | W.T. Trotter, "Partially ordered sets" R.L. Graham (ed.) M. Grötschel (ed.) L. Lovász (ed.) , Handbook of Combinatorics , I , North-Holland (1995) pp. 433–480 |
Dimension of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dimension_of_a_partially_ordered_set&oldid=37445