#universalalgebra — Public Fediverse posts

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.

#algebra #CategoryTheory #math #UniversalAlgebra #logic

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

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

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

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

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.)

#math #AlgebraicGeomtry #Algebra #UniversalAlgebra

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

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

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

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

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

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

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

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

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?

#algebra #CategoryTheory #math #UniversalAlgebra

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.

#algebra #CategoryTheory #UniversalAlgebra #math

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

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:

https://youtu.be/j7mHxTUkRn4

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.

#math #algebra #logic #UniversalAlgebra #AbstractAlgebra

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:

https://youtu.be/W3oXOS4Yx6k

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.

#algebra #AbstractAlgebra #UniversalAlgebra #logic

My eleventh Math Research Livestream is now available on YouTube:

https://youtu.be/P-sQSm1_0Ww

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:

https://youtu.be/XwdgxMARr9c

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

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

My eighth Math Research Livestream is now available on YouTube:

https://youtu.be/aWNn_5ob448

This time, I took another look at my expository article explaining free algebras and presentations by way of division by zero. My goal was to put a new, more mathematically mature spin on my old construction from way back when, and I made some interesting progress.

#math #livestream #Twitch #algebra #UniversalAlgebra #AbstractAlgebra #CategoryTheory

My next Math Research Livestream starts in about 30 minutes on Twitch! Check it out at https://www.twitch.tv/charlotteaten. This week I'm taking another look at my expository article explaining free algebras and presentations by way of division by zero. I'm going to mess around with putting a new, more mathematically mature spin on my old construction from way back when.

#math #livestream #Twitch #algebra #UniversalAlgebra #AbstractAlgebra #CategoryTheory

My seventh Math Research Livestream is now available on YouTube:

https://youtu.be/gKm-5vkb4yc

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

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

My sixth Math Research Livestream is now available on YouTube:

https://youtu.be/SN2zNj-CXdI

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

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

My fourth Math Research Livestream is now available on YouTube:

https://youtu.be/Xwdzzai7NrM

In this one, I started an expository article explaining free algebras and presentations by way of a discussion of division by zero. Throughout the video I find a much cleaner description of a magma I studied as a teenager.

#math #livestream #Twitch #algebra #UniversalAlgebra #AbstractAlgebra #CategoryTheory

My next Math Research Livestream starts in about 15 minutes on Twitch! Check it out at https://www.twitch.tv/charlotteaten. I'll be starting a new expository article explaining free algebras and presentations of algebras by way of a discussion about division by zero. (And yes, I will divide by zero.)

#math #livestream #Twitch #algebra #UniversalAlgebra #AbstractAlgebra #CategoryTheory

I've been getting some awesome feedback from my weekly livestreams and I'm looking forward to trying something a bit different on my next one. I mentioned previously on Mastodon that I wanted to write a paper using division by zero to explain some notions related to free algebras in universal algebra. Since this is a good expository topic for which I cannot be scooped, I can actually show the start-to-finish of writing a paper.

In previous streams, I've gotten a lot of discussion and questions, so I end up writing on the board most of the time. I'm gonna try to show a fair amount of actual writing in this next one. I may complement this with some light music, if I can find a good no-copyright option.

#algebra #UniversalAlgebra #Twitch #livestream #math

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

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

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

I will be giving a talk on a functorial semantics for neural nets on Friday at the #Cambridge computer lab's Logic and Semantics Seminar. If you happen to be in the area at the time, consider checking it out!

This will be my first talk on a synthesis of my paper on discrete neural nets (https://arxiv.org/abs/2308.00677) and the categorified invariant theory appearing in my thesis (https://aten.cool/documents/thesis.pdf).

Abstract and other info: https://www.cst.cam.ac.uk/seminars/list/208831

#CategoryTheory #AppliedCategoryTheory #AI #MachineLearning #ComputerScience #logic #math #combinatorics #UniversalAlgebra

The new version of my paper with Semin Yoo, "Orientable triangulable manifolds are essentially quasigroups" is now available on arXiv! You can find the preprint at https://arxiv.org/abs/2110.05660 and you can find some videos of me talking about it on my YouTube channel (https://www.youtube.com/channel/UCT0qXiThOxzbCO36U-iXNTQ).

In addition to new images which illustrate our constructions we also have filled a gap in the proof of the main theorem. In order to show that all orientable triangulable manifolds could be created from an \(n\)-ary quasigroup by our construction, we needed to make an appropriate \(n\)-quasigroup for each manifold. What we actually did in the original paper was give a presentation of such an algebraic structure, which is not quite enough to prove the desired result. This new version contains an explicit description of such an \(n\)-quasigroup.

You can look forward to hearing more from me on connections between #quasigroups and #topology in the future!

#UniversalAlgebra #combinatorics #AlgebraicTopology

The name for the algebraic structure consisting of a set \(A\) equipped with a binary operation \(f\colon A^2\to A\) which is not assumed to be commutative, associative, etc. has an interesting history. As far as I can tell, this sort of algebra was originally known as a "groupoid", and you can find recent literature using the term in that way. However, Nicolas Bourbaki called a set with a single binary operation a "magma" in his Éléments de mathématique and this name came to be used when there might be confusion with the newer topological use of "groupoid" to mean a category whose morphisms are all invertible. A third contender is "binar". I don't know the history of this one but it seems reasonable to me.

I prefer "magma" myself. I don't know what a "lava" or a "volcano" should be nor have I had the audacity to try to name something like this in the literature. One reason I prefer "magma" is that I can talk about a set \(A\) equipped with a single \(n\)-ary operation \(f\colon A^n\to A\) as an "\(n\)-ary magma" (or just "\(n\)-magma" or even "magma").

#algebra #UniversalAlgebra #topology #bourbaki #terminology