I would like to create a lattice of intervals of integers, but I don't know how to 'draw'(Hasse diagram) it.
An interval looks like: [1,4] or ]-Inf,3] for example.
I'm having difficulties deciding what should be bottom and top, and what should be in between.
Some join/meet rules that should hold:
join
[0,10]⊔[15,20]=[0,20]
[0,10]⊔[5,7]=[0,10]
meet
]−∞, 10] ⊔ [5, 15] = ]−∞, 15]
[0,10]⊓[15,20]= Bottom
[0,10]⊓[5,7]=[5,7]
]−∞,10]⊓[5,15]=[5,10]
any pointers would be greatly appreciated.
The part of the lattice consisting of finite sets has a Hasse diagram as follows:
where the bottom node is the empty interval and is covered by all singleton intervals. Every other interval has that $[a,b]$ is covered by $[a-1,b]$ and $[a,b+1]$, creating the visible grid that we see. Notice that through each element, we can draw two lines following the grid - that is, through $[a,b]$, we can draw a line consisting of $\{\ldots,[a-2,b],[a-1,b],[a,b],[a+1,b],[a+2,b],\ldots,[b,b],\emptyset\}$ and a similar one where we modify the second endpoint. If we take two elements, and draw these lines, then the meet and the join lie at the two intersections of these lines (and it should be obvious which is which).
For the infinite elements, we end up with two infinite "chains" (i.e. posets isomorphic to $\mathbb Z$) which we have to imagine as lying diagonally above, in either direction, the poset. That is, we consider that if we start at a point $[a,b]$ and travel $[a-1,b],[a-2,b],\ldots$ we eventually get up to this diagonal limit to $(-\infty,b]$, placing us between $(-\infty,b-1]$ and $(-\infty,b+1]$ - these sort of acts like "points at infinity" on the grid. Moreover, these two chains have their meet $\mathbb Z$, thrust way in the middle of nowhere, above everything else - and it covers no elements, but acts as the result of traveling upwards forever through the grid.