That's the first time I've seen order theory used to describe hierarchical structures like a city. Continuing the analogy, what would the join of a nonempty finite subset of parts of a city be? Say the set was an apartment block, road and grocery store, would the join of all these concepts be the "local neighborhood" that makes up part of a town?
Open to counter-arguments though!
My feeling is that it's not really fruitful to describe "real life" phenomena in terms of abstract algebraic mathematical structures. It's more of a gut feeling than anything else, but I think there are two points that one can make here:
* Generally, it's better to first observe a phenomenon, understand it, and try to find a good vocabulary for it, mathematical or otherwise, rather than take an out-of-the-box fancy sounding math concept and try to make it fit your phenomenon.
* It seems as soon as you try to describe a phenomenon with an algebraic object, you'll soon get stuck trying to make sense of your functions/axioms: what's the meet? what's the join, is there a bottom element, etc. If you rather work with a more “geometric” notion (graphs, metric spaces,…) you don't have as many things to match, and you can focus on studying your objects.
To be clear, I don't have any strong point here, it's mostly gut feeling/rumblings.
I disagree, however I don't think you're enterily wrong. We just need a physical theory to bridge the abstract mathematics with the real world. as a starting point it occurs to me to wonder what does throwing a ball have to do with F = ma
also, the whole point of math is to do everything with such a level of abstraction that you can apply the abstractions to many things.