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.
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.)
This paper is a translation of the paper "Idealtheorie in Ringbereichen", written by Emmy Noether in 1920, from the original German into English. It in particular brings the language used into the modern world so that it is easily understandable by the mathematicians of today. The paper itself deals with ideal theory, and was revolutionary in its field, that is modern algebra. Topics covered include: the representation of an ideal as the least common multiple of irreducible ideals; the representation of an ideal as the least common multiple of maximal primary ideals; the association of prime ideals with primary ideals; the representation of an ideal as the least common multiple of relatively prime irreducible ideals; isolated ideals; the representation of an ideal as the product of coprime irreducible ideals; equivalent concepts regarding modules.
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
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?
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.
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
We develop a new theory of strong subalgebras and linear congruences that are defined globally. Using this theory we provide a new proof of the correctness of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the new theory we prove that composing a weak near-unanimity operation of an odd arity $n$ we can derive an $n$-ary operation that is symmetric on all two-element sets. Thus, CSP over a constraint language $Γ$ on a finite domain is tractable if and only if there exist infinitely many polymorphisms of $Γ$ that are symmetric on all two-element sets.
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
In 2017, Walter Taylor showed that there exist $2$-dimensional simplicial complexes which admit the structure of topological modular lattice but not topological distributive lattice. We give a positive answer to his question as to whether $n$-dimensional simplicial complexes with the same property exist. We do this by giving, for each $n\ge2$, an infinite family of compact simplicial complexes which admit the structure of topological modular lattice but not topological distributive lattice.
Charlotte Aten
Joshua Grochow