Retracts of posets: the chain-gap property and the selection property are independent
Résumé
Posets which are retracts of products of chains are characterized by means of two properties: \emph{the chain-gap property} and \emph{ the selection property} (Rival and Wille \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 $\Power (\omega_1)/Fin$ has been the sole example of a lattice without the selection property \cite {R-W}. We prove that it also fails to have the chain-gap property. In addition, we provide an example of a lattice which has the chain-gap property but not the selection property. This answers questions raised in \cite {R-W}