I didn't realize that in categorical algebra one would be considered a «loser» for using the word variety to refer to a category which is merely equivalent to a full subcategory of Alg T specified by equations rather than the classical case of a variety in the sense of universal algebra. To be honest, I've never seen any textbook discuss a «loser version» of a definition before.
#UniversalAlgebra
A fundamental result in universal algebra is the Subdirect Representation Theorem, which tells us how to decompose an algebra \(A\) into its "basic parts". Formally, we say that \(A\) is a subdirect product of \(A_1\), \(A_2\), ..., \(A_n\) when \(A\) is a subalgebra of the product
\[
A_1\times A_2\times\cdots\times A_n
\]
and for each index \(1\le i\le n\) we have for the projection \(\pi_i\) that \(\pi_i(A)=A_i\). In other words, a subdirect product "uses each component completely", but may be smaller than the full product.
A trivial circumstance is that \(\pi_i:A\to A_i\) is an isomorphism for some \(i\). The remaining components would then be superfluous. If an algebra \(A\) has the property than any way of representing it as a subdirect product is trivial in this sense, we say that \(A\) is "subdirectly irreducible".
Subdirectly irreducible algebras generalize simple algebras. Subdirectly irreducible groups include all simple groups, as well as the cyclic \(p\)-groups \(\mathbb{Z}_{p^n}\) and the Prüfer groups \(\mathbb{Z}_{p^\infty}\).
In the case of lattices, there is no known classification of the finite subdirectly irreducible (or simple) lattices. This page (https://math.chapman.edu/~jipsen/posets/si_lattices92.html) by Peter Jipsen has diagrams showing the 92 different nontrivial subdirectly irreducible lattices of order at most 8. See any patterns?
We know that every finite subdirectly irreducible lattice can be extended to a simple lattice by adding at most two new elements (Lemma 2.3 from Grätzer's "The Congruences of a Finite Lattice", https://arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.
#UniversalAlgebra #combinatorics #logic #math #algebra #AbstractAlgebra
Just found an English translation of Emmy Noether's 1921 "Idealtheorie in Ringbereichen" ("Ideal Theory in Rings"): https://arxiv.org/abs/1401.2577
(while editing the wikipedia page on subdirect products - my first wiki edit to add an Emmy Noether reference! Turns out there's a direct lineage from Noether to Birkhoff's introduction of subdirect products in universal algebra. Just one more way in which she really revolutionized algebra.)
Microsoft's Outlook on mobile won't allow me to attach this 618kb jpg to an email, so I'm posting it on Mastodon in order to have a copy of it on my laptop. This is related to the thing about Tarski's High School Algebra Problem I posted a while ago. The long identity is called the Wilkie Identity.
#Microsoft #Outlook #math #algebra #AbstractAlgebra #UniversalAlgebra #logic
I've found a citation of my own work on Wikipedia for the first time!
https://en.wikipedia.org/wiki/Commutative_magma
Naturally, I read this page before I wrote my rock-paper-scissors paper. It's neat to see that my own work is now the citation for something that was unsourced "original research" on Wikipedia.
#math #research #Wikipedia #algebra #games #RockPaperScissors #AbstractAlgebra #UniversalAlgebra #combinatorics #GameTheory
But! TIL there's a categorical definition that supposedly agrees w/ "surjection" on any variety of algebras:
h is "categorically surjective" (a term I just made up) if for any factorization h=fg with f monic, f must be an iso.
(h/t Knoebel's book https://doi.org/10.1007/978-0-8176-4642-4)
Are there categorical definitions that agree w/ injective (resp. surjective) on all concrete categories?
The notion of epimorphism can be quite different from surjection, e.g. in Rings.
Though I recently learned epimorphisms can be characterized in terms of Isbell's zig-zags: https://en.wikipedia.org/wiki/Isbell%27s_zigzag_theorem.
Whereas monic seems to capture the notion of "injective" quite well in a categorical def. And indeed the two agree on any variety of algebras in the sense of universal algebra.
My fourteenth Math Research Livestream is now available on YouTube:
https://www.youtube.com/watch?v=pVoFfZAyXzk
I talked about some topics related to my recent preprint (https://arxiv.org/abs/2409.12923) about topological lattices.
I decided to skip streaming today because I wanted to talk about polyhedral products, but I haven't found the old calculation that I wanted to talk about yet. Shocking I couldn't find something I did like six years ago in the ten minutes before I would start streaming. I'll look for it now, so hopefully I'll be ready next week.
#math #topology #algebra #AbstractAlgebra #UniversalAlgebra #combinatorics #LatticeTheory
Apparently I missed that Zhuk posted a *simplified* proof of the CSP Dichotomy Conjecture back in January: https://arxiv.org/abs/2404.01080
I'd really love to understand all of this!
#ComputationalComplexity #complexity #math #UniversalAlgebra
I'll be streaming again in 20 minutes at twitch.tv/charlotteaten. I'll be talking about my recent preprint (https://arxiv.org/abs/2409.12923) about topological lattices!
#math #topology #algebra #AbstractAlgebra #UniversalAlgebra #combinatorics #LatticeTheory
I posted a new paper on the arXiv!
https://arxiv.org/abs/2409.12923
In "Higher-dimensional book-spaces" I show that for each \(n\) there exists an \(n\)-dimensional compact simplicial complex which is a topological modular lattice but cannot be endowed with the structure of topological distributive lattice. This extends a result of Walter Taylor, who did the \(2\)-dimensional case.
I think this kind of result is interesting because we can see that whether spaces continuously model certain equations is a true topological invariant. All of the spaces that I discuss here are contractible, but only some can have a distributive lattice structure.
A similar phenomenon happens with H-spaces. The \(7\)-sphere is an H-space, and it is even a topological Moufang loop, but it cannot be made into a topological group, even though our homotopical tools tell us that it "looks like a topological group".
This is (a cleaned up version of) something I did during my second year of graduate school. It only took me about six years to post it.
#math #topology #algebra #AbstractAlgebra #UniversalAlgebra #combinatorics #LatticeTheory
My thirteenth Math Research Livestream is now available on YouTube:
In this one, I mention that 13 is a lucky number in math, and then keep talking about topological lattices as a continuation of my stream from the previous week.
I'm taking this week off from streaming, but I expect to be back next week at the same time!
#math #livestream #Twitch #topology #research #UniversalAlgebra #AbstractAlgebra #algebra
This is a friendly reminder that
((1+𝑥)ʸ+(1+𝑥+𝑥²)ʸ)ˣ⋅((1+𝑥³)ˣ+(1+𝑥²+𝑥⁴)ˣ)ʸ=((1+𝑥)ˣ+(1+𝑥+𝑥²)ˣ)ʸ⋅((1+𝑥³)ʸ+(1+𝑥²+𝑥⁴)ʸ)ˣ for all natural numbers \(x\) and \(y\), but this formula is impossible to obtain by using only those arithmetic laws taught in high school. Credit for this goes to Alex Wilkie, who found this in the 1980s.
I'll be streaming again in 15 minutes at https://www.twitch.tv/charlotteaten. I'm gonna keep talking about topological lattices, since I've realized some new things since last week.
#math #livestream #Twitch #topology #research #UniversalAlgebra #AbstractAlgebra #algebra #LatticeTheory #CategoryTheory
My twelfth Math Research Livestream is now available on YouTube:
This time, I talked about this paper (https://arxiv.org/abs/1602.00034) by George Bergman. I have something related which I've finally decided to post to the arXiv, so hopefully I'll be ready to talk about that preprint next week.
#math #livestream #Twitch #topology #research #UniversalAlgebra #AbstractAlgebra #algebra #LatticeTheory #CategoryTheory
I'll be streaming again in 20 minutes at https://www.twitch.tv/charlotteaten. This week I'm going to switch gears and talk about this paper (https://arxiv.org/abs/1602.00034) of George Bergman. I have something related which I've finally decided to post to the arXiv, so hopefully this will prepare me to talk about that new preprint next week.
#math #livestream #Twitch #topology #research #UniversalAlgebra #AbstractAlgebra #algebra #LatticeTheory #CategoryTheory
The Cayley table below has an infinite amount of structure in the following sense: For any finite list of equations that hold for this operation, there will always be another equation which holds but is not a consequence of the given ones. In other words, the \(3\)-element magma below is not finitely based.
\[
\begin{array}{r|ccc}
& 0 & 1 & 2 \\ \hline
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 \\
2 & 0 & 2 & 2
\end{array}
\]
In 1951, Lyndon showed that every \(2\)-element algebra is finitely based, so three is the smallest order of a non-finitely based algebra. This example was found by Murskiĭ in 1965.
My eleventh Math Research Livestream is now available on YouTube:
This time, I continued my work from the previous week and produced a higher-dimensional version of the formula for the number of Latin squares given in this paper (https://www.sciencedirect.com/science/article/pii/0012365X9290722R). It turned out to be quite similar!
The only real difference in the higher-dimensional case was the need for an analogue of the permanent of a matrix for a rank \(d\) hypermatrix. This can be obtained by summing over all \((d-1)\)-ary quasigroups, which specializes to the usual unary quasigroups (i.e. permutations) in the \(d=2\) case.
#math #livestream #Twitch #YouTube #research #combinatorics #LinearAlgebra #AbstractAlgebra #UniversalAlgebra
Earlier this summer I did this livestream (https://youtu.be/XwdgxMARr9c), in which I ended up finding a lot of examples of simple quasigroups showing up. I took a look at Bruck's 1944 paper on the subject (https://www.ams.org/journals/bull/1944-50-10/S0002-9904-1944-08236-0/S0002-9904-1944-08236-0.pdf), and I saw an unusual pronoun show up: her.
Now there are a few usual suspects for women in early abstract algebra, but not too many. In order of decreasing proximity to quasigroup theory, we have Ruth Moufang (https://en.wikipedia.org/wiki/Ruth_Moufang), Hanna Neumann (https://en.wikipedia.org/wiki/Hanna_Neumann), and Emmy Noether (https://en.wikipedia.org/wiki/Emmy_Noether). The woman in question was new to me: Harriet Griffin (https://en.wikipedia.org/wiki/Harriet_Griffin).
Strangely, Bruck refers to Griffin as "Miss Griffin" rather than "Dr. Griffin", although he references her PhD thesis work. I'm not sure what his intent was in specifying her gender.
In any case, I'm always happy to discover another woman who was an early pioneer in non-associative algebra.
#math #algebra #WomenInSTEM #WomenInAcademia #AbstractAlgebra #UniversalAlgebra
My ninth Math Research Livestream is now available on YouTube:
This time, I intended to work on improving the presentation of the main theorem in my quasigroup manifolds paper (https://arxiv.org/abs/2110.05660), but instead I ended up finding an upper bound for the number of elements in a quasigroup whose serenation has a surface of genus n as a connected component. Many of these were prime for small n (under 1000), but no related sequence showed up on OEIS.
#math #livestream #Twitch #research #AbstractAlgebra #algebra #topology #UniversalAlgebra #CategoryTheory #combinatorics #OEIS
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