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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" 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" 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" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/combinatorics" class="mention hashtag" rel="nofollow noopener" target="_blank">#combinatorics</a> <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#logic</a> <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#AbstractAlgebra</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" 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" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/research" class="mention hashtag" rel="nofollow noopener" target="_blank">#research</a> <a href="https://mathstodon.xyz/tags/Wikipedia" class="mention hashtag" rel="nofollow noopener" target="_blank">#Wikipedia</a> <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/games" class="mention hashtag" rel="nofollow noopener" target="_blank">#games</a> <a href="https://mathstodon.xyz/tags/RockPaperScissors" class="mention hashtag" rel="nofollow noopener" target="_blank">#RockPaperScissors</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#AbstractAlgebra</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/combinatorics" class="mention hashtag" rel="nofollow noopener" target="_blank">#combinatorics</a> <a href="https://mathstodon.xyz/tags/GameTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#GameTheory</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" target="_blank">#math</a> <a href="https://mathstodon.xyz/tags/algebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#algebra</a> <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#logic</a> <a href="https://mathstodon.xyz/tags/UniversalAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#UniversalAlgebra</a> <a href="https://mathstodon.xyz/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#AbstractAlgebra</a>

Lain(e/o/a) Taffin AltmanSo I’d like to take some time to talk about a <a href="https://social.treehouse.systems/tags/math" class="mention hashtag" rel="nofollow noopener" target="_blank">#math</a> thing I did today—specifically in <a href="https://social.treehouse.systems/tags/abstractalgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#abstractalgebra</a>. I actually formalized it in <a href="https://social.treehouse.systems/tags/Agda" class="mention hashtag" rel="nofollow noopener" target="_blank">#Agda</a>!But before we get to that, I need to take you on a journey, one that will involve building an algebraic structure that looks completely unfamiliar and strange and then discovering that it’s actually a very familiar structure indeed, just looked at from a new angle.

aliivibriofollowing intro up with some hashtagsThings I do actively!Programming: <a href="https://mathstodon.xyz/tags/csharp" class="mention hashtag" rel="nofollow noopener" target="_blank">#csharp</a> <a href="https://mathstodon.xyz/tags/fsharp" class="mention hashtag" rel="nofollow noopener" target="_blank">#fsharp</a> <a href="https://mathstodon.xyz/tags/functionalprogramming" class="mention hashtag" rel="nofollow noopener" target="_blank">#functionalprogramming</a> <a href="https://mathstodon.xyz/tags/idris" class="mention hashtag" rel="nofollow noopener" target="_blank">#idris</a> <a href="https://mathstodon.xyz/tags/clojure" class="mention hashtag" rel="nofollow noopener" target="_blank">#clojure</a> <a href="https://mathstodon.xyz/tags/typescript" class="mention hashtag" rel="nofollow noopener" target="_blank">#typescript</a> Gamedev: <a href="https://mathstodon.xyz/tags/bitsy" class="mention hashtag" rel="nofollow noopener" target="_blank">#bitsy</a> <a href="https://mathstodon.xyz/tags/twine" class="mention hashtag" rel="nofollow noopener" target="_blank">#twine</a> <a href="https://mathstodon.xyz/tags/interactivefiction" class="mention hashtag" rel="nofollow noopener" target="_blank">#interactivefiction</a> <a href="https://mathstodon.xyz/tags/pico8" class="mention hashtag" rel="nofollow noopener" target="_blank">#pico8</a> <a href="https://mathstodon.xyz/tags/roguelike" class="mention hashtag" rel="nofollow noopener" target="_blank">#roguelike</a> <a href="https://mathstodon.xyz/tags/rotjs" class="mention hashtag" rel="nofollow noopener" target="_blank">#rotjs</a> Music: <a href="https://mathstodon.xyz/tags/guitar" class="mention hashtag" rel="nofollow noopener" target="_blank">#guitar</a> <a href="https://mathstodon.xyz/tags/lute" class="mention hashtag" rel="nofollow noopener" target="_blank">#lute</a> <a href="https://mathstodon.xyz/tags/earlymusic" class="mention hashtag" rel="nofollow noopener" target="_blank">#earlymusic</a> <a href="https://mathstodon.xyz/tags/mbira" class="mention hashtag" rel="nofollow noopener" target="_blank">#mbira</a> <a href="https://mathstodon.xyz/tags/livecoding" class="mention hashtag" rel="nofollow noopener" target="_blank">#livecoding</a> <a href="https://mathstodon.xyz/tags/sonicpi" class="mention hashtag" rel="nofollow noopener" target="_blank">#sonicpi</a> <a href="https://mathstodon.xyz/tags/dungeonsynth" class="mention hashtag" rel="nofollow noopener" target="_blank">#dungeonsynth</a>Interests! Math: <a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> <a href="https://mathstodon.xyz/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#logic</a> <a href="https://mathstodon.xyz/tags/tessellations" class="mention hashtag" rel="nofollow noopener" target="_blank">#tessellations</a> <a href="https://mathstodon.xyz/tags/theoreticalcomputerscience" class="mention hashtag" rel="nofollow noopener" target="_blank">#theoreticalcomputerscience</a> <a href="https://mathstodon.xyz/tags/polyhedra" class="mention hashtag" rel="nofollow noopener" target="_blank">#polyhedra</a> <a href="https://mathstodon.xyz/tags/abstractalgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#abstractalgebra</a> Nature <a href="https://mathstodon.xyz/tags/ferns" class="mention hashtag" rel="nofollow noopener" target="_blank">#ferns</a> <a href="https://mathstodon.xyz/tags/fungi" class="mention hashtag" rel="nofollow noopener" target="_blank">#fungi</a> <a href="https://mathstodon.xyz/tags/slimemold" class="mention hashtag" rel="nofollow noopener" target="_blank">#slimemold</a> <a href="https://mathstodon.xyz/tags/lichen" class="mention hashtag" rel="nofollow noopener" target="_blank">#lichen</a> <a href="https://mathstodon.xyz/tags/invertebrates" class="mention hashtag" rel="nofollow noopener" target="_blank">#invertebrates</a> Other: <a href="https://mathstodon.xyz/tags/bicycling" class="mention hashtag" rel="nofollow noopener" target="_blank">#bicycling</a> <a href="https://mathstodon.xyz/tags/swordandsorcery" class="mention hashtag" rel="nofollow noopener" target="_blank">#swordandsorcery</a>

John TerwiskeJust trying some tags to see what pops up.<a href="https://mstdn.social/tags/AbstractAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#AbstractAlgebra</a> <a href="https://mstdn.social/tags/bioinformatics" class="mention hashtag" rel="nofollow noopener" target="_blank">#bioinformatics</a> <a href="https://mstdn.social/tags/numbertheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#numbertheory</a> <a href="https://mstdn.social/tags/EmbarcaderoDelphi" class="mention hashtag" rel="nofollow noopener" target="_blank">#EmbarcaderoDelphi</a> <a href="https://mstdn.social/tags/cplusplus" class="mention hashtag" rel="nofollow noopener" target="_blank">#cplusplus</a>

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