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  1. My eighth Math Research Livestream is now available on YouTube:

    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

  2. My next Math Research Livestream starts in about 30 minutes on Twitch! Check it out at 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

  3. My next Math Research Livestream starts in about 15 minutes on Twitch! Check it out at 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

  4. 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

  5. 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 arxiv.org/abs/2110.05660 and you can find some videos of me talking about it on my YouTube channel (youtube.com/channel/UCT0qXiThO).

    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

  6. Gustave Courbet, L'apôtre Jean Journet partant pour la conquête de l'harmonie universalle, 1850 #artsmia #gustavecourbet collections.artsmia.org/art/43

  7. Today in #Science (@sciencemagazine), Claudia Arevalo, Scott Hensley, & al develop an #mRNA#LipidNanoparticle #vaccine encoding #hemagglutinin antigens from all 20 known #influenza A & B virus subtypes/lineages! #Vaccination protected animals challenged w/ matched & mismatched viral strains!

    In the same issue, Alyson Kelvin & Darryl Falzarano write a wonderful Perspective on this important next step towards a #UniversalFluVaccine!

    Article: bit.ly/Sci_abm0271
    PS: bit.ly/Sci_adf0900

    #IAV #IBV

  8. Today in #Science (@sciencemagazine), Claudia Arevalo, Scott Hensley, & al develop an #mRNA#LipidNanoparticle #vaccine encoding #hemagglutinin antigens from all 20 known #influenza A & B virus subtypes/lineages! #Vaccination protected animals challenged w/ matched & mismatched viral strains!

    In the same issue, Alyson Kelvin & Darryl Falzarano write a wonderful Perspective on this important next step towards a #UniversalFluVaccine!

    Article: bit.ly/Sci_abm0271
    PS: bit.ly/Sci_adf0900

    #IAV #IBV

  9. Today in #Science (@sciencemagazine), Claudia Arevalo, Scott Hensley, & al develop an #mRNA#LipidNanoparticle #vaccine encoding #hemagglutinin antigens from all 20 known #influenza A & B virus subtypes/lineages! #Vaccination protected animals challenged w/ matched & mismatched viral strains!

    In the same issue, Alyson Kelvin & Darryl Falzarano write a wonderful Perspective on this important next step towards a #UniversalFluVaccine!

    Article: bit.ly/Sci_abm0271
    PS: bit.ly/Sci_adf0900

    #IAV #IBV

  10. 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 (math.chapman.edu/~jipsen/poset) 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", arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.

    #UniversalAlgebra #combinatorics #logic #math #algebra #AbstractAlgebra

  11. 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 (math.chapman.edu/~jipsen/poset) 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", arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.

    #UniversalAlgebra #combinatorics #logic #math #algebra #AbstractAlgebra

  12. 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 (math.chapman.edu/~jipsen/poset) 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", arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.

    #UniversalAlgebra #combinatorics #logic #math #algebra #AbstractAlgebra

  13. 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 (math.chapman.edu/~jipsen/poset) 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", arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.

    #UniversalAlgebra #combinatorics #logic #math #algebra #AbstractAlgebra

  14. @RJB_Mallacore

    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

  15. @RJB_Mallacore

    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

  16. @RJB_Mallacore

    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

  17. @RJB_Mallacore

    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

  18. #BreakingNews #ESTRENO

    La saga de acción que empezó con la película Olympus Has Fallen, ahora llega a París en esta nueva serie; Sean Harris interpreta a un villano que busca venganza, y amenaza a la “ciudad de la luz” con un violento ataque terrorista.
    ¿Podrán detenerlo el agente de protección Vicent Taleb, y su equipo? ¡Descúbrelo en #ParisBajoFuego (#ParisHasFallen) ya disponible solo en #UniversalPlus!

  19. #BreakingNews #ESTRENO

    La saga de acción que empezó con la película Olympus Has Fallen, ahora llega a París en esta nueva serie; Sean Harris interpreta a un villano que busca venganza, y amenaza a la “ciudad de la luz” con un violento ataque terrorista.
    ¿Podrán detenerlo el agente de protección Vicent Taleb, y su equipo? ¡Descúbrelo en #ParisBajoFuego (#ParisHasFallen) ya disponible solo en #UniversalPlus!