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298 results for “UniversalBlue”
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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.
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Recieved the mail from your blog with more schemas detailed and your text.
Excellent job as usual !!
[You should maybe put the link to it here ..]
#FleetAdmiral DuckDAWorld =/\= #S31 #STO #UniversalFleet #Toholl
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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
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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. -
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.
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My next Math Research Livestream starts in about 10 minutes on Twitch! Check it out at https://www.twitch.tv/charlotteaten. This week I'm going to work on improving the presentation of the main theorem in my quasigroup manifolds paper (https://arxiv.org/abs/2110.05660).
#math #livestream #Twitch #research #AbstractAlgebra #algebra #topology #UniversalAlgebra #CategoryTheory
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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.
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My seventh Math Research Livestream is now available on YouTube:
In this one I decided to pick up with where I ended the previous week, writing code for my discrete neural nets paper (https://arxiv.org/abs/2308.00677). This time I did actually get a decent amount of code written!
#math #livestream #Twitch #algebra #AbstractAlgebra #AI #MachineLearning #NeuralNets #combinatorics #UniversalAlgebra #CategoryTheory
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My next Math Research Livestream starts in about 30 minutes on Twitch! Check it out at https://www.twitch.tv/charlotteaten. I decided to pick up with where I ended last week, writing code for my discrete neural nets paper (https://arxiv.org/abs/2308.00677).
#math #livestream #Twitch #algebra #AbstractAlgebra #AI #MachineLearning #NeuralNets #combinatorics #UniversalAlgebra #CategoryTheory
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My sixth Math Research Livestream is now available on YouTube:
In this one I was coding live for the first time, since I needed to get back to the code for my discrete neural nets paper (https://arxiv.org/abs/2308.00677). I actually spent most of the time refamiliarizing myself with the state of the code and discussing the context of the work, but I did start on the task I had in mind near the end of the stream.
#math #livestream #Twitch #algebra #AbstractAlgebra #AI #MachineLearning #NeuralNets #combinatorics #UniversalAlgebra #CategoryTheory
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My next Math Research Livestream starts in about 30 minutes on Twitch! Check it out at https://www.twitch.tv/charlotteaten. I'm going to be coding live for the first time, since I need to get back to the code for my discrete neural nets paper (https://arxiv.org/abs/2308.00677).
#math #livestream #Twitch #algebra #AbstractAlgebra #AI #MachineLearning #NeuralNets #combinatorics #UniversalAlgebra #CategoryTheory
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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.
#CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #UniversalAlgebra #NeuralNets
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By the way, you can see the poster I presented in 2017 here: https://aten.cool/documents/NCUWM_poster.pdf
This kind of idea appears in my work with Semin Yoo on constructing manifolds from quasigroups, so I'm still up to some of the same things seven years later.
Quasigroup manifolds paper: https://arxiv.org/abs/2110.05660
#algebra #topology #manifolds #UniversalAlgebra #combinatorics
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John Chrysostom, archbishop of Constantinople, on responding to appeals for help. Conversation could in some cases “recover” the person as well as money. He says the biblical commands to give, period, may be God’s design to create opportunities for connection, and learning of people’s misfortunes.
Some today are using the state to prevent these human connections.
How can you talk to someone in misfortune today?
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John Chrysostom, archbishop of Constantinople, on responding to appeals for help. Conversation could in some cases “recover” the person as well as money. He says the biblical commands to give, period, may be God’s design to create opportunities for connection, and learning of people’s misfortunes.
Some today are using the state to prevent these human connections.
How can you talk to someone in misfortune today?
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John Chrysostom, archbishop of Constantinople, on responding to appeals for help. Conversation could in some cases “recover” the person as well as money. He says the biblical commands to give, period, may be God’s design to create opportunities for connection, and learning of people’s misfortunes.
Some today are using the state to prevent these human connections.
How can you talk to someone in misfortune today?
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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
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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
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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
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#ESTRENO: El agente Nolan y su equipo están de regreso en #TheRookie temporada 7 ya disponible con su primer episodio en exclusiva por #UniversalPlus.
Nuevo episodio cada miércoles!#NathanFillion #EricWinter #AlyssaDiaz #MekiaCox #MelissaONeil #RichardJones #ShawnAshmore #ABCTelevisionNetwork
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I'm pleased to announce that I will be doing my second postdoc at CU Boulder! I'll be working with Keith Kearnes, so I'm remaining in the same general area both geographically and mathematically.
You may be wondering what @ProfKinyon and I have been up to during my first postdoc. Rest assured that you will see those results soon™. Seriously though, we should have a preprint posted before I start at Boulder.
#CUBoulder #Boulder #Denver #UniversalAlgebra #combinatorics #logic
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Recieved the mail from your blog with more schemas detailed and your text.
Excellent job as usual !!
[You should maybe put the link to it here ..]
#FleetAdmiral DuckDAWorld =/\= #S31 #STO #UniversalFleet #Toholl
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Gustave Courbet, L'apôtre Jean Journet partant pour la conquête de l'harmonie universalle, 1850 #artsmia #gustavecourbet https://collections.artsmia.org/art/43484/
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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
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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
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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
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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
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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
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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
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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