I'm looking for an explicit example of a uniquely complemented lattice that is non-modular, since neither of the two non-modular lattices described here at wikipedia have this property.
Thanks.
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asked Dec 16, 2013 at 2:22
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$\begingroup$ An atomic uniquely complemented lattice is boolean, so your example would have to be non-atomic. There are exercises in Gratzer's Lattice Theory book in Chapter 3 (latest edition) about this. I don't have it in front of me so I can't give you the exact page. Sorry. $\endgroup$– William DeMeoDec 17, 2013 at 0:57
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We know that uniquely complemented lattices that are non-modular exist. This follows from the celebrated result by Dilworth (pdf). As far as I know, we do not know how to exhibit a concrete example, all the known constructions use some sort of complicated colimit. See the Graetzers paper in Notices (pdf) for a general overview of results concerning unique complementation and much more.
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$\begingroup$ Thanks, I'm reading Gratzer's article right now. $\endgroup$ Dec 17, 2013 at 15:49