Resumen de Retracts of posets: the chain-gap property and the selection property are independent

Dwight Duffus, Claude Laflamme, Maurice Pouzet

• Posets which are retracts of products of chains are characterized by means of two properties: the chain-gap property and the selection property (Rival and Wille [9]). Examples of posets with the selection property and not the chain-gap property are easy to find. To date, the Boolean lattice $${\mathcal{P}}(?_{1})$$/ Fin has been the sole example of a lattice without the selection property [9]. 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 [9].