Abstract.
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 \({\mathcal{P}}(ω_{1})\)/ 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].
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Dedicated to George Grätzer and E. Tamás Schmidt on their 70th birthdays
Received December 17, 2006; accepted in final form July 8, 2008.
The second author was supported by NSERC of Canada Grant # 690404. The research of the third author was done within the Intas programme on Universal Algebra and Lattice Theory, and completed while the author visited the Mathematics and Statistics Department of the University of Calgary in July 2006; the support provided is gratefully acknowledged.
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Duffus, D., Laflamme, C. & Pouzet, M. Retracts of posets: the chain-gap property and the selection property are independent. Algebra univers. 59, 243–255 (2008). https://doi.org/10.1007/s00012-008-2125-6
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DOI: https://doi.org/10.1007/s00012-008-2125-6