Inequalities in Dimension Theory for Posets
نویسنده
چکیده
The dimension of a poset (X, P), denoted dim (A", P), is the minimum number of linear extensions of P whose intersection is P. It follows from Dilworth's decomposition theorem that dim (X, P)& width (X, P). Hiraguchi showed that dim(X, P)s \X\/Z In this paper, A denotes an antichain of (A", P) and E the set of maximal elements. We then prove that dim {X, P) s |X A\; dim(X, P) < 1 + width (X E); and dim (A", P) s 1+2 width (A"— A). We also construct examples to show that these inequalities are sharp.
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