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!
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If $L'$ were distributive then $L$ would be distributive too since it's a sublattice. – Eric Wofsey Mar 08 '22 at 05:32
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@Eric Wofsey, sorry, edited my question, I meant to say that how are we sure that L' is non distributive lattice given that its sub-lattice(L) is non distributive but non uniquely complemented lattice! – yash Mar 08 '22 at 05:45
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It is immediate from the definitions that if $L'$ were distributive then $L$ would be distributive too, given that $L$ is a sublattice of $L'$. – Eric Wofsey Mar 08 '22 at 05:49
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It is given to us that L is non-distributive and is non uniquely complemented lattice, and because L is sub-lattice of L', i want to ask, how can we conclude that L' is also non-distributive lattice or distributive lattice!? – yash Mar 08 '22 at 05:54
1 Answers
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?
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I think it is remarkable the paragraph in that Grätzer's paper before your citation: "It is interesting how little we know about a subject on which we have published so many papers. Ask any question and probably we do not know the answer. Let me mention a (very) few of my favorite ones." It's a testimony to the difficulty of the task... – amrsa Mar 09 '22 at 20:27