Retracts of Posets: The Chain_gap Property and the Selection Property are Independant

2006/12/15 by Dwight Duffus, Duffus, Dwight, Claude Laflamme +3
Mathematics · #06A #06B #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:06A #msc:06B

paper · pdf · doi:10.48550/arxiv.math/0612458

10 pages

arxiv created 2006/12/15 · arxiv updated 2009/12/01

Abstract

Posets which are retract of products of chains are characterized by means of two properties: the chain-gap property and the selection property (Rival and Wille, 1981 \cite R-W). Examples of posets with the selection property and not the chain-gap property are easy to find. To date, the Boolean lattice \Power (ω₁)/Fin was the sole example of lattice without the selection property \cite R-W. We prove that it does not have the chain-gap property. We provide an example of a lattice which has the chain-gap property but not the selection property. This answer questions raised in \cite R-W.

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