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.

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.)

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.

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.

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!

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.

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.

I have a new YouTube channel! You can find the intro video for it at https://youtu.be/3jmO0IixHZw

Make sure you subscribe to this one if you want to see my future Math Research Livestream videos, or any other video I post in the future.

You can find a video explaining the change on my old channel (https://youtu.be/PCBvt7vNB8I). The tl;dr is that my old account was tied to my undergrad email, and I want something which is not managed by another organization.

I'll be leaving all the old videos up so that links don't die, and I will also repost them to my new account over the coming weeks.

My talk from yesterday on a categorical semantics for neural nets for the New York Category Theory Seminar has already been posted on YouTube! You can find it at https://www.youtube.com/watch?v=FKkpVKuspmA and you can see more about this seminar at https://www.sci.brooklyn.cuny.edu/~noson/Seminar/. The preprint I mention is https://arxiv.org/abs/2308.00677 and the talk by Joyal which was mentioned at the end can be found at https://www.youtube.com/watch?v=MxClaWFiGKw.

I have a new #icanhazpdf request!

I have an ongoing project (https://arxiv.org/abs/2110.05660) on the construction of manifolds from quasigroups and I believe that the book "Finite embedding theorems for partial designs and algebras" by Curt Lindner and Trevor Evans (MR0460213) may be helpful to me. I was wondering if anyone knows where I could obtain a copy, as it seems to be out of print.

Ideally, I would like to buy my own physical copy from someone, or even better obtain a pdf of the whole book.

I have tried many of the usual places online, as well as asking the #UniversalAlgebra Google group, to no avail. I still need to try using the access to #Cambridge Core which I now have as a member of the Association for Symbolic #Logic. Ironically, I am in Cambridge right now, but my credentials for this are back in Colorado and I don't remember them. I'm not sure how likely it is that they'll transfer an entire book to me digitally, though.

I made my first post to the #AI section of the arXiv this week! You can find the preprint "Discrete neural nets and polymorphic learning" at https://arxiv.org/abs/2308.00677.

In this paper a learning algorithm based on polymorphisms of finite structures is described. This provides a systematic way to choose activation functions for neural nets whose neurons can only act on a fixed finite set of values. These polymorphisms preserve any specified constraints imposed by the learning task in question.

This paper is the result of a 2021 REU at the University of #Rochester. I am working with a great group of students on a follow-up project right now, so videos of talks and a sequel preprint should be out soon!

After posting an answer on this MathOverflow question https://mathoverflow.net/questions/450930/is-there-an-identity-between-the-associative-identity-and-the-constant-identity , I wonder if it might be a suitable graduate research project to see if current generation #ProofAssistant / #MachineLearning / #AI tools can be used to determine the logical relationship between various universal equational laws that could be satisfied by a single binary operation + on a set (i.e., by a magma). For instance, in the answer to this related question https://mathoverflow.net/questions/450890/is-there-an-identity-between-the-commutative-identity-and-the-constant-identity?noredirect=1&lq=1 it was shown (by a slightly intricate argument) that the law (𝑥+𝑥)+𝑦=𝑦+𝑥 implies the commutative law 𝑥+𝑦=𝑦+𝑥, but not conversely, while I showed that the law 𝑥+(𝑦+𝑧)=(𝑥+𝑦)+𝑤 is strictly intermediate between the triple constant law 𝑥+(𝑦+𝑧)=(𝑤+𝑢)+𝑣 and the associative law 𝑥+(𝑦+𝑧)=(𝑥+𝑦)+𝑧. It seems that this is a restrictive enough fragment of #mathematics (or even of #UniversalAlgebra) that automated tools should function rather well, without being so trivial as to be completely solvable by brute force.

I'm Charlotte Aten, a mathematician who studies combinatorics, universal algebra, and category theory. I'm a postdoc at the University of Denver where I work with @ProfKinyon. I write software as part of my research and have been applying my skills to the real world a bit more lately. This is my first foray into "real" social media in over a decade. I said I'd never do it again, but I can't pass up a decentralized system which is getting some mainstream traction.