Retracts of Posets: The Chaingap Property and the Selection Property are Independant
Dwight Duffus, Claude Laflamme, Maurice Pouzet
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 R-W). Examples of posets with the selection property and not the chain-gap property are easy to find. To date, the Boolean lattice (ω1)/Fin was the sole example of lattice without the selection property 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 R-W.
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