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I came across following fact while reading online:

If a given Hasse diagram contains following structures, then it wont be the distributive lattice
enter image description here

I am unsure if the above fact is correct and if it is, then whether are these exhaustive substructures required to be contained in given Hasse diagram in order for a lattice not a distributive lattice. In other words, if the fact is true, then can there be a non distributive lattice with Hasse diagram not containing any of these structures?

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From Wikipedia's article on distributive lattices: "A lattice is distributive if and only if none of its sublattices is isomorphic to $M_3$ or $N_5$."

Yes, it is correct. It is a complete characterization of distributive lattices.
The proof is not difficult, but it is rather elaborated.

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  • $\begingroup$ ohh the diagram was a copy paste from the wikipedia itself (which I too copied from my source 😅)!!! $\endgroup$ – anir Feb 6 '18 at 20:27
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    $\begingroup$ Ok, so maybe you overlooked the content of the page? Anyway, if you're interest, and in case you don't have access to much bibliography, a reference book on Universal Algebra which has a good introduction to Lattice Theory is A Course in Universal Algebra, where you can find the proof of this characterization (Theorems I.3.5 and I.3.6). Good luck! :) $\endgroup$ – amrsa Feb 6 '18 at 20:37
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    $\begingroup$ JB Nation's "Notes on Lattice Theory" is another excellent (and freely available) resource. $\endgroup$ – William DeMeo Feb 7 '18 at 5:35

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