Is lattice theory the most underrated topic in math?
>>7666487
Yes. So much so that no one wanted to contribute to this thread
>>7666487
Yes.
And universal algebras come in second.
What are the advantages?
Nope. Type theory is.
>>7666595
They're everywhere. The set of substructures of an algebraic structure (subgroups of a group or ideals of rings) are lattices.
The family of open sets of a topological space is a special kind of lattice called Heyting algebras. And if the space is compact, Hausdorff and totally disconnected (or equivalently, profinite) it's actually a boolean algebra.
Riesz spaces are vector spaces with an order which induces a lattice structure compatible with the vector space structure. Topological Riesz spaces are important in measure theory.
There's also combinatoric aspects of it that I am less familiar with, but you can look it in Combinatorics: The Rota Way.
>>7666601
Type theory is getting a lot of attention right now, it's not underrated.
What's a good book on lattice theory?
Concept lattices are pretty dank.
>>7666595
Really useful in the study of abstract interpretation of programming languages too. If for instance you study the possible values of a variable at a given line of your code, a statement can be "the variable at this point can be greater than 5, or 0, or smaller than -10". If you only have a finite possible number of values to choose from for your variables (e.g., 8 bit signed integers), then the set of all possible "knoweldges" of a variable's state is basically the power set of the variable's range, and forms a lattice just as in OP's picture. Furthermore, you can define "abstraction" of that variable's range by simplifying it (for instance into {"positive","negative","null"}), and use the properties of Galois connections to transport your potentially complex study of the precise knowledge of the possible states of the variable into a simpler study of the knowledge of the possible states of the variable's sign. Galois connections work on CPOs, and lattices one of the nicest forms of CPOs, so it applies rather well.
convince me that lattice theory and category theory aren't just terminology math.
that they aren't just disciplines where you call call things something else.
for instance what problems do we want to be able to solve that we came up with lattice theory or category theory in order to solve?
what powerful statements can we now make thanks to lattice theory and type theory that could not be made before.
what are some open problems in lattice theory or type theory that it would be very useful to if we were to finally solve?
for example interuniversal teichmuller theory I don't understand but obviously will be able ot make big statements in number theory and be able to solve some big problems that could not be solved before (when its independently understood and verified).
but what part of lattice theory or category theory is necessary rather than gratuitous, meta, terminology maths?
>>7668371
>not doing math just for the beauty of it
>>7668371
>lattice theory
It can be used to simulate relations between complex numbers in a 4 dimensional grid.
>>7666487
Recently got the book Introduction to Lattices and Order, self-studying now. It's pretty chill.
>>7667399
All of that is just category theory tbqh.
>>7668371
All of math is morphic :^)
Can someone explain what this lattice theory is?
Well I guess, at least I never paid much attention to them.
I have always thought of a lattice as the kind of structure that has so little properties that you cannot say anything interesting about it (kind of like a magma). I only now realize that it is actually widely studied (in algebra, logic, topology, CS, etc)
>>7668371
the thing is, calling something by another name is often super helpful and makes your work go from nearly impossible to trivial at times.
you can come up with a bajillion duality theorems for lattices independently as particular algebraic statements about particular lattices, but taking them as a special case of points of topoi or isbell duality is just more straightforwards, and you can nearly state them all in the same breath.
many monads allow you to nearly automatically create function spaces where you have collections which themselves are indexed by varying sets. Barr points out how mind-bogglingly hard it is to verify algebraic properties of such collections with sets and elements.
you could do a lot of sheaf theory just with etale spaces, but the functor definition is gonna help you move things around a lot faster (like the pushforward). many relationships that are obscure without the new terminology pop. for example, the relation between finding subtopoi and subspaces would be totally unclear without pointless topology and functors.
also, lattice theory in general is boss. almost all of my research has benefitted from it.
>>7668371
This is exactly why I rejected group theory. In fact, this is why I rejected the natural numbers and what not, after I found out they're just silly names people assign to some specific sets. I only deal with sets now.
>>7669761
you're braindead if you think of groups as "just" sets
>>7669766
I'm stoned and I still understand sarcasm better than you
>>7669777
i'm also stoned
what's you're point?
>>7669782
That you're very bad at understanding sarcasm, since I understand it better than you whilst impaired.
>>7668560
i mean, you can say that for just about anything in order theory because basic category theory is p much just the categorification of orders. not really a fair point.
>>7668371
i'm only gonna give an example or two for each
>for instance what problems do we want to be able to solve that we came up with lattice theory or category theory in order to solve?
finding topological invariants is nearly impossible without the language of functors and natural transformations
>what powerful statements can we now make thanks to lattice theory and type theory that could not be made before.
I would look up the "niceness" properties of locales vs topological spaces. there are a lot of theorems about topology that only need the open sets, and it turns out that the localic constructions preserve a lot more good properties.
>what are some open problems in lattice theory or type theory that it would be very useful to if we were to finally solve?
https://en.wikipedia.org/wiki/Finite_lattice_representation_problem
but what part of lattice theory or category theory is necessary rather than gratuitous, meta, terminology maths?
>>7669816
sorry, post fucked up:
>but what part of lattice theory or category theory is necessary rather than gratuitous, meta, terminology maths?
it is rare to find categorical facts that cannot be turned back into set theory facts, and in fact when they can't it is due to dumb problems like size. (such as here https://amathew.wordpress.com/2012/01/26/homotopy-is-not-concrete/)
but that's not really the right question to be asking. just because set theory came first, that is no reason to give it priority until something "even more subsuming" comes along (which, incidentally, the theory of topoi does pretty much subsume it, and even meta-facts about it; see maclane and moerdijk for example for a condensed version of cohen's proof totally inside of the category of grothendieck topoi), in order to consider it more than "mere terminology". set theory itself was set up to help better state functions and logico-model facts, but it turns out that category theory just does so even better for the majority of algebraic (and increasingly, spatial) problems.
>>7668368
i half understand what you;re saying. any practical uses?
>>7668371
Try to figure out what products of schemes or cokernels of sheaf morphisms should look like without having a general categorical framework. Representable functors play a huge role in fiber products of schemes. Basic category theory is not that far fetched when you start working with objects which are more than just sets with some additional structure.
