Livello Segreto

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Charlotte AtenA 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 (<a href="https://math.chapman.edu/~jipsen/posets/si_lattices92.html" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://math.chapman.edu/~jipsen/posets/si_lattices92.html</a>) 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", <a href="https://arxiv.org/pdf/2104.06539" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://arxiv.org/pdf/2104.06539</a>), so there must be oodles of finite simple lattices out there.<a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/combinatorics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#combinatorics</a> <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AbstractAlgebra</a>

Joshua GrochowJust found an English translation of Emmy Noether's 1921 "Idealtheorie in Ringbereichen" ("Ideal Theory in Rings"): <a href="https://arxiv.org/abs/1401.2577" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://arxiv.org/abs/1401.2577</a>(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.)<a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/AlgebraicGeomtry" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AlgebraicGeomtry</a> <a href="https://mathstodon.xyz/tags/Algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Algebra</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a>

Charlotte AtenI've found a citation of my own work on Wikipedia for the first time!<a href="https://en.wikipedia.org/wiki/Commutative_magma" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://en.wikipedia.org/wiki/Commutative_magma</a>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.<a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/research" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#research</a> <a href="https://mathstodon.xyz/tags/Wikipedia" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Wikipedia</a> <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/games" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#games</a> <a href="https://mathstodon.xyz/tags/RockPaperScissors" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#RockPaperScissors</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AbstractAlgebra</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/combinatorics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#combinatorics</a> <a href="https://mathstodon.xyz/tags/GameTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#GameTheory</a>

Joshua GrochowBut! 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 <a href="https://doi.org/10.1007/978-0-8176-4642-4" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://doi.org/10.1007/978-0-8176-4642-4</a>) Are there categorical definitions that agree w/ injective (resp. surjective) on all concrete categories?<a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CategoryTheory</a> <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a>

Joshua GrochowThe 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: <a href="https://en.wikipedia.org/wiki/Isbell%27s_zigzag_theorem" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://en.wikipedia.org/wiki/Isbell%27s_zigzag_theorem</a>.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.<a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#CategoryTheory</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a>

Joshua GrochowApparently I missed that Zhuk posted a *simplified* proof of the CSP Dichotomy Conjecture back in January: <a href="https://arxiv.org/abs/2404.01080" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://arxiv.org/abs/2404.01080</a>I'd really love to understand all of this!<a href="https://mathstodon.xyz/tags/ComputationalComplexity" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ComputationalComplexity</a> <a href="https://mathstodon.xyz/tags/complexity" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#complexity</a> <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a>

Charlotte AtenThis 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.<a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AbstractAlgebra</a>

Charlotte AtenThe 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.<a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AbstractAlgebra</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#logic</a>

Charlotte AtenEarlier this summer I did this livestream (<a href="https://youtu.be/XwdgxMARr9c" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://youtu.be/XwdgxMARr9c</a>), 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 (<a href="https://www.ams.org/journals/bull/1944-50-10/S0002-9904-1944-08236-0/S0002-9904-1944-08236-0.pdf" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://www.ams.org/journals/bull/1944-50-10/S0002-9904-1944-08236-0/S0002-9904-1944-08236-0.pdf</a>), 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 (<a href="https://en.wikipedia.org/wiki/Ruth_Moufang" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://en.wikipedia.org/wiki/Ruth_Moufang</a>), Hanna Neumann (<a href="https://en.wikipedia.org/wiki/Hanna_Neumann" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://en.wikipedia.org/wiki/Hanna_Neumann</a>), and Emmy Noether (<a href="https://en.wikipedia.org/wiki/Emmy_Noether" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://en.wikipedia.org/wiki/Emmy_Noether</a>). The woman in question was new to me: Harriet Griffin (<a href="https://en.wikipedia.org/wiki/Harriet_Griffin" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://en.wikipedia.org/wiki/Harriet_Griffin</a>).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.<a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/WomenInSTEM" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#WomenInSTEM</a> <a href="https://mathstodon.xyz/tags/WomenInAcademia" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#WomenInAcademia</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AbstractAlgebra</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a>

Terence TaoAfter posting an answer on this MathOverflow question <a href="https://mathoverflow.net/questions/450930/is-there-an-identity-between-the-associative-identity-and-the-constant-identity" rel="nofollow noopener noreferrer" target="_blank">https://mathoverflow.net/questions/450930/is-there-an-identity-between-the-associative-identity-and-the-constant-identity</a> , I wonder if it might be a suitable graduate research project to see if current generation <a href="https://mathstodon.xyz/tags/ProofAssistant" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ProofAssistant</a> / <a href="https://mathstodon.xyz/tags/MachineLearning" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#MachineLearning</a> / <a href="https://mathstodon.xyz/tags/AI" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AI</a> 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 <a href="https://mathoverflow.net/questions/450890/is-there-an-identity-between-the-commutative-identity-and-the-constant-identity?noredirect=1&lq=1" rel="nofollow noopener noreferrer" target="_blank">https://mathoverflow.net/questions/450890/is-there-an-identity-between-the-commutative-identity-and-the-constant-identity?noredirect=1&lq=1</a> 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 <a href="https://mathstodon.xyz/tags/mathematics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#mathematics</a> (or even of <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#UniversalAlgebra</a>) that automated tools should function rather well, without being so trivial as to be completely solvable by brute force.

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