This comes from a dynamical system I'm studying. The space is [0,1], and the elements in what I was pretty sure was a distributive lattice look like $\{0\},\{1\},\{\frac {1} {2k+1}\}$ and any closed interval between two such points, as well as arbitrary unions of such elements. So intervals starting at 0 or ending at 1 or starting/ending at 1 over an odd integer. Thus, we have both singletons and the connection between singletons as intervals as lattice elements, along with the empty set as the bottom element.
Join is union, meet is intersection. This seemed to set up a lattice just fine, a subset of the powerset algebra. But I was having trouble with the Priestley duology distinguishing $[\frac {1} {5},\frac 1 7]$ from $\{\frac 1 5,\frac 1 7\}$.
That's when I realized that this apparently fails the $M_3,N_5$ theorem, as the sublattice $[0,1], [\frac {1} {5},\frac 1 7],\{\frac 1 5,\frac 1 7\},\{\frac 1 {11}\},\emptyset $ looks to be a copy of $N_5$. So, this would imply I don't have a distributive lattice.
Is that the case? And if so, how did I start with a subset of the powerset on $[0,1]$, closed under unions and intersections, with meet and join just being set intersection and union, and end up with a nondistributive lattice?
Best Answer
Those five elements that make a lattice isomorphic to $N_5$ are not a sub-lattice of the original lattice.
For that to happen, both meets and joins in the subset would have to agree with those in the original lattice.
As you pointed out, in your lattice joins are given by union, so for example
$$\{1/11\} \vee \{1/5,1/7\} = \{1/5,1/7,1/11\} \neq [0,1].$$
So the join of those two elements doesn't even belong to the set whence the set is not a sub-lattice (the same thing that happens in every algebra; for example, groups).
In general, every lattice in which the operations are given by intersection and union is distributive, since these operations distribute over each other on sets.