I learned some universal algebra stuff
Jan. 12th, 2023 02:00 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySome things I learned at a talk recently by Bernardo Rossi! A lot of this is basic universal algebra concepts that I wasn't aware of because I don't do universal algebra. :P (I mean, I knew some basic stuff, like congruence relations, but...)
1. Two useful bits of terminology: A function made from an algebra's operations is a "term function". If you also allow arbitrary constants, then they call it a "polynomial" (quite a generalization, yes).
2. Universal algebraists often think of an algebra in terms of its whole set (or rather, clone) of term functions, rather than the specific defined operations in the signature; this allows switching between different signatures for the same algebra.
3. One particularly nice (as universal algebra goes, anyway! Or as nice as universal algebra goes while still being called that) class of algebras is Mal'cev algebras (not to be confused with those Mal'cev algebras or those Mal'cev algebras), which are algebras such that there exists a term function q such that, for all x and y, q(x,y,y)=x=q(y,y,x). So e.g. this includes groups and rings (and Moufang loops and alternative algebras!).
4. There's something called the "term condition commutator", an operation on congruence relations. The definition of this appears to be complicated, and a lot of this material looks like it's in actual books which are of course more effort than just searching the web. Anyway the term condition commutator takes in several congruence relations and returns a new one which is contained in their intersection. I say "several" because apparently the multiary one does not in all contexts agree with the iterated binary one? Huh. Also apparently there are other notions of commutator too that satisfy similar properties.
Anyway it's called the commutator because, for groups, if you put in two normal subgroups, it gives you their commutator. But for rings, if you put in two ideals, it gives you their product! Weird.
5. Anyway the actual theorem that Rossi was there to talk about was, he and his coauthors were looking at the question of, over what algebras is the union of two algebraic sets algebraic? (An algebraic set being as usual the set of solutions to a set of polynomial equations.) We're all familiar with how this is true over integral domains because there you can express the union with a product. But what about in general -- not just for rings, for any Mal'cev algebra?
Well, the theorem they proved is that the union of two algebraic sets over a given Mal'cev algebra is always again algebraic iff, for any two nonzero congruence relations on that algebra, their term condition commutator is also nonzero! (Where the "zero" congruence relation means equality.) Huh. Pretty neat!
...I'm less writing this down for the theorem and more writing this down because, hey, universal algebra concepts I wasn't aware of!
-Harry
1. Two useful bits of terminology: A function made from an algebra's operations is a "term function". If you also allow arbitrary constants, then they call it a "polynomial" (quite a generalization, yes).
2. Universal algebraists often think of an algebra in terms of its whole set (or rather, clone) of term functions, rather than the specific defined operations in the signature; this allows switching between different signatures for the same algebra.
3. One particularly nice (as universal algebra goes, anyway! Or as nice as universal algebra goes while still being called that) class of algebras is Mal'cev algebras (not to be confused with those Mal'cev algebras or those Mal'cev algebras), which are algebras such that there exists a term function q such that, for all x and y, q(x,y,y)=x=q(y,y,x). So e.g. this includes groups and rings (and Moufang loops and alternative algebras!).
4. There's something called the "term condition commutator", an operation on congruence relations. The definition of this appears to be complicated, and a lot of this material looks like it's in actual books which are of course more effort than just searching the web. Anyway the term condition commutator takes in several congruence relations and returns a new one which is contained in their intersection. I say "several" because apparently the multiary one does not in all contexts agree with the iterated binary one? Huh. Also apparently there are other notions of commutator too that satisfy similar properties.
Anyway it's called the commutator because, for groups, if you put in two normal subgroups, it gives you their commutator. But for rings, if you put in two ideals, it gives you their product! Weird.
5. Anyway the actual theorem that Rossi was there to talk about was, he and his coauthors were looking at the question of, over what algebras is the union of two algebraic sets algebraic? (An algebraic set being as usual the set of solutions to a set of polynomial equations.) We're all familiar with how this is true over integral domains because there you can express the union with a product. But what about in general -- not just for rings, for any Mal'cev algebra?
Well, the theorem they proved is that the union of two algebraic sets over a given Mal'cev algebra is always again algebraic iff, for any two nonzero congruence relations on that algebra, their term condition commutator is also nonzero! (Where the "zero" congruence relation means equality.) Huh. Pretty neat!
...I'm less writing this down for the theorem and more writing this down because, hey, universal algebra concepts I wasn't aware of!
-Harry
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Date: 2023-01-16 02:28 am (UTC)no subject
Date: 2023-01-16 03:20 am (UTC)