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

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

2006-12-15

arxiv.org/abs/math/0612458v1

Mathematics

Combinatorics

10 pages

Scientific paper

Posets which are retract of products of chains are characterized by means of two properties: \emph{the chain-gap property} and \emph{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 $\Power (\omega_1)/Fin$ 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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