8

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.

goblin GONE
  • 67,744
  • 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. – William DeMeo Dec 17 '13 at 00:57

1 Answers1

9

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.

Gejza Jenča
  • 1,563