Unique Complemented lattice which is not distributive lattice!

Question

I know that : "if a distributive lattice is also complemented lattice then lattice should only have unique complement" but i am not able to find a case where A lattice is uniquely complemented but not distributive lattice. I am also aware about the Dilworth's theorem that "Every lattice is a sub-lattice of some uniquely complemented lattice", but even so, if we take any non distributive lattice (say L) which has non unique complement and according to Dilworth's theorem, L is part of a bigger uniquely complemented lattice(say L'), but still we cannot say anything about the lattice L' being non-distributive lattice, it may nor may not be distributive lattice, then how are we sure that lattice L' is uniquely complemented but not distributive lattice.?, i am not able to find an example to prove this! I hope i was clear with my question!

Answer 1

There are (at least two) related questions on this site.

The first one asks for a weaker property: is there a non-distributive lattice in which every element has at most one complement?

The second one, proposed by @goblinGONE asks for an apparently stronger property: is there a uniquely complemented lattice that is non-modular? The positive answer given by Gejza Jenča, that I invite you to upvote, gives a key reference, namely Grätzer's survey paper in Notices (pdf). In this paper, Theorem 1 states that a uniquely complemented modular lattice is distributive. It follows from this theorem that your question is actually a duplicate of @goblinGONE's question. Grätzer gives several solutions to your question in Section 2, but, as he doesn't know any simple example, he concludes with the following remarks:

All known examples of nondistributive uniquely complemented lattices are freely generated, one way or another. Is there a construction of a nondistributive uniquely complemented lattice that is different?

In the same vein, is there a “natural” example of a nondistributive uniquely complemented lattice from geometry, topology, or whatever else?