Elsevier

Discrete Mathematics

Volume 195, Issues 1–3, 28 January 1999, Pages 111-126
Discrete Mathematics

Split semiorders

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Abstract

A poset P = (X, ≺) is a split semiorder if there are maps a, f : XR with a(x) ⩽ f(x) ⩽ a(x) + 1 for every x ϵ X such that xy if and only if f(x) < a(y) and a(x) + 1 < f(y). A split interval order is defined similarly with a(x) + 1 replaced by b(x), a(x) ⩽ f(x) ⩽ b(x), such that xy if and only if f(x) < a(y) and b(x) < f(y). We investigate these generalizations of semiorders and interval orders through aspects of their numerical representations, three notions of poset dimensionality, minimal forbidden posets, and inclusion relationships to other classes of posets, including several types of tolerance orders.

Keywords

Split semiorder
Interval order
Tolerance order
Poset dimension

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