%0 Journal Article
%T Retracts of Posets: The Chain_gap Property and the Selection Property are Independant
%A Dwight Duffus
%A Claude Laflamme
%A Maurice Pouzet
%J Mathematics
%D 2006
%I arXiv
%X 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}.
%U http://arxiv.org/abs/math/0612458v1