Then, parameterizing the choice situation and / or introducing strategic dependecies between actors leads to analyzing sub/supermodular correspondences on these sets. In fact, the most general version would make use of quasi-supermodularity. I think Milgrom&Roberts 94 show that this is the most general way, one can think of coherent (rational) decision theory.
Preference relations of rational agents are, anyway. When it comes to us mere mortals, well, see https://en.wikipedia.org/wiki/Intransitivity#Occurrences_in_....
If it were so, then human behavior would be impossible to analyze. Instead, if we focus on causes and consequences of non transitive preferences - like in behavioral economics - we may regain the ability to do analyzes.
Without lattices, we would never have had the ambiguous phrase "computer scientists commonly choose models which have bottoms, but prefer them topless" occurring in a serious textbook.
Edit: agreed, intellectual descent is only partially ordered, but the interval between aristotle's example syllogism and lattices contains a chain. Guess I should've said "also comparable (by."
The semantics of determinate CCP programs (and by extension, LVars programs) can be described really beautifully in terms of closure operators on the underlying domain. I'm not sure this made it into the initial published thesis, but this is an accessible paper on the topic: http://www.lix.polytechnique.fr/comete/stages/references/ccp...
This book is quite famous in the field and all about lattices as you can infer from its cover: https://dl.acm.org/doi/book/10.5555/1965094
It's full of pseudocode and algorithms.
A monotonic operation is one that’s either non-decreasing or non-increasing. That is, it never reverses direction.
So in the context of conflict-free replicated data types, you can think of something like an append-only database as a basic example. Over time, the database can only grow or stay the same. It can never shrink. This allows you to split the database into pieces and distribute them. You always know they can be merged back together because they’re append-only. This makes it a join semilattice with the union of corresponding tables being the least upper bound operation.
For example, CvRDTs (a class of conflict-free replicated data types), are made up of the semilattices over a monotonic operation. If you can prove a few basic properties about your data + the merge operation over it, you can construct a CRDT. It won't necessarily be efficient, but it's a good starting point for reasoning about the problem.
People spend a lot of time trying to sell abstract algebra using monads, but in my opinion it's a lot easier to grasp the applications of lattices.