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- W2074390982 abstract "The dimension of a partially ordered set ( X , P ) is the smallest positive integer t for which there exists a function ƒ which assigns to each x ∈ X a sequence {ƒ(x)(i): 1⩽i⩽t} of real numbers so that xles ; y in P if and only if ƒ(x)(i)⩽ƒ(y)(i) for each i =1,2,…, t . The interval dimension of ( X , P ) is the smallest integer t for which there exists a function F which assigns to each x ∈ X a sequence { F ( x )( i ): 1⩽ i ⩽ t } of closed intervals of the real line R so that x < y in P if and only if a < b in R for every a ∈ F ( x )( i ), b ∈ F ( y )( i ), and i = 1,2…, t . For t ⩾2, a partially ordered set (poset) is said to be t -irreducible (resp. t -interval irreducible) if it has dimension t (resp. interval dimension t ), and every proper subposet has dimension (resp. interval dimension) less than t . The only 2-irreducible poset is a two element anti-chain, and the only 2-interval irreducible poset is the free sum of two chains each having two points. In sharp contrast, the collection R of all 3-irreducible posets consists of 9 infinite families and 18 odd examples, and the collection R I of all 3-interval irreducible posets is sufficiently complex to have avoided complete determination as of this date. Trotter and Moore determined R from Gallai's forbidden subgraph characterization of comparability graphs. David Kelly independently determined R by a lattice theoretic argument combined with the characterization of planar lattices Kelly and Ivan Rival had previously obtained. In this paper, we introduce a new operation called a stack which we will apply to posets of height one. In some ways the stack operation is an inverse of the split operation on posets previously defined by Kimble. These operations behave predictably with respect to dimension and interval dimension. In particular, the stack of a poset of height one plays a role in interval dimension theory which is analogous to the role played by the completion by cuts in dimension theory. As a consequence, we can exploit the similarities to Kelly's approach to the determination of R to produce a relatively compact argument to determine the collection R ( I , 1) of all 3-interval irreducible posets of height one. This characterization problem has immediate combinatorial connections with a wide range of well-known forbidden subgraph problems including interval graphs, rectangle graphs, and circular arc graphs." @default.
- W2074390982 created "2016-06-24" @default.
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- W2074390982 date "1981-01-01" @default.
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- W2074390982 title "Stacks and splits of partially ordered sets" @default.
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- W2074390982 doi "https://doi.org/10.1016/0012-365x(81)90211-9" @default.
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