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On the dimension of amalgamated ordered sets. (English) Zbl 0792.06002

If \(A\) is an ordered set then \(\dim A\) denotes its dimension. In the paper the connections between the dimensions of ordered sets \(A\) and \(B\) and the dimension of their amalgam \(A\vee B\) are investigated. In general, it is not true that \(\dim A\vee B= \dim A+ \dim B\). The author studies situations when the above equality is valid and gives some of its specifications. For instance (Theorem 3), if \(A\cap B\) is a chain then \(\dim A\vee B\leq \max\{\dim A, \dim B\}+ 2\), and (Theorem 8) if a distributive lattice \(D\) is the amalgam of its sublattices \(A\) and \(B\) such that \(A\cap B\) is also a sublattice of \(D\) then \(\dim D\leq \max\{\dim A, \dim B\}+ 1\).

MSC:

06A06 Partial orders, general
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