Questions tagged [universal-algebra]

The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).

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Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
William DeMeo's user avatar
19 votes
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554 views

What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?

Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
Noah Schweber's user avatar
17 votes
0 answers
434 views

The free complete lattice on three generators, beyond ZF

This was originally asked at MSE; although it is still under bounty it seems unlikely to be answered there. $\mathsf{ZF}$ proves that there is no free complete lattice on three generators since any ...
Noah Schweber's user avatar
17 votes
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2k views

Why did Bourbaki not use universal algebra?

I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra? Well, universal algebra is not much older than category ...
13 votes
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When does HSP reduce to SPH?

This is actually a poorly camouflaged attempt to use the answers to When is the opposite of the category of algebras of a Lawvere theory extensive? (all very interesting) for the purposes of my ...
მამუკა ჯიბლაძე's user avatar
13 votes
0 answers
239 views

Birkhoff's HSP theorem in categories other than $\mathbf{Set}$

Fix a category $C$ with finite products and a set $L$ of function symbols (each equipped with an arity in $\mathbb N$). An $L$-algebra in $C$, $\mathbf A=(A,(f^\mathbf{A})_{f\in L})$, is given by some ...
user176332's user avatar
13 votes
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226 views

Is there a finite equational basis for the join of the commutative and associative equations?

I asked this on math stack exchange, but I was told to post it on mathoverflow. Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by the ...
user107952's user avatar
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When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...
Chris Schommer-Pries's user avatar
10 votes
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398 views

Equational theory in the signature (+,*,0,1) of sedenions and beyond

Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, ...
user107952's user avatar
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To what kind of generalized Lawvere theory does the "free cartesian closed category" 2-monad on $\mbox{Cat}_g$ correspond?

Thinking of Cat as a mere 1-category, there is a 1-monad $\Lambda$ for the free cartesian closed category on a category. To every category X it assigns the category $\Lambda(X)$ whose objects are ...
Mike Stay's user avatar
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General approaches to extension theorems as Caratheodory

I would like to know if there are some general studies about extension-like theorem, in the sense which i'm going to describe. This paragraph is not rigorous; I just would like the idea to be clear. I ...
Andrea Marino's user avatar
9 votes
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348 views

"Generalized theory of polynomials" for a given commutative Lawvere Theory

I am trying to understand Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing. Let $R$ be a commutative, associative ring with unit. We can ...
Georg Lehner's user avatar
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8 votes
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Group presentations where discarding generators always yields a subgroup

Consider a group presentation $ \left< G= \left\lbrace \text{generators}\right\rbrace \, \middle|\, R = \left\lbrace \text{relators}\right\rbrace \right>$ (no finiteness assumptions). Given $S\...
Arnaud Mortier's user avatar
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Is there any theorem achieving Conway's "Mathematician's Liberation Movement"

John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
Christopher King's user avatar
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224 views

Which semirings have enough injectives in their category of modules?

Let $R$ be a semiring and $Mod_R$ its category of modules. That is, $R$ is a monoid in the monoidal category of commutative monoids and $Mod_R$ is its category of modules in the usual sense. Question ...
Tim Campion's user avatar
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Varieties of groups with certain properties

Is there an example of a periodic variety $\mathbf{V}$ of groups that satisfies all of the following properties? $\mathbf{V}$ is finitely based $\mathbf{V}$ contains finitely many subvarieties $\...
E W H Lee's user avatar
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When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable? It seems that one can relatively easy ...
Dmitri Pavlov's user avatar
8 votes
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390 views

When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...
Theo Johnson-Freyd's user avatar
8 votes
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614 views

The name for a partial order

In a recent paper of mine, my co-authors found that a partial order that we were using was contained in a paper by Kundgen. In it, he called it "right-shifted partial order". I was curious, and found ...
Victor Miller's user avatar
8 votes
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618 views

Counting and understanging commuting functions.

Fix a positive integer $n$, and consider the functions from a set of size $n$ to itself. Let $cp(n)$ denote the number of ordered pairs $\langle f,g \rangle$ of these functions which commute, i.e., ...
Jeff Norden's user avatar
7 votes
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582 views

A new and subtle order-theoretic fixed point theorem

Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...
Paul Taylor's user avatar
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Relationship between measure theory and quantification

I was advised that this question might be better suited for mathoverflow, so I am reposting it here (original post). In a 1978 paper published by David P. Ellerman and Gian-Carlo Rota, the duo discuss ...
John Smith's user avatar
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325 views

What is going on in the field of algebraic logic these days?

I'm doing my masters in Mathematics and took a class in universal algebra and there I learned that for example: Boolean algebras have direct connection with classical logic, Heyting algebras with ...
Victor's user avatar
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142 views

Birkhoff's theorem with language expansion?

Let $\mathcal V$ be a variety (in the sense of universal algebra). Recall that Birkhoff's theorem characterizes when a class $\mathcal W \subseteq \mathcal V$ of $\mathcal V$-algebras forms a ...
Tim Campion's user avatar
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7 votes
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185 views

Were algebraic theories and abstract clones defined independently?

Algebraic theories (by which I mean the formalism based on bijective-on-objects functors) and abstract clones both capture universal algebraic structure, and are well-known to be equivalent. Algebraic ...
varkor's user avatar
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1 answer
586 views

Reversible varieties

We say that a variety $V$ is reversible if for each $n>0$ and $n$-ary fundamental operation $f$, there is some $m\geq n$ and $r$ along with terms $T_{2},\dotsc,T_{r},S_{1},\dotsc,S_{m}$ such that $...
Joseph Van Name's user avatar
7 votes
0 answers
341 views

Is there a theory of algebraic universal algebra?

An algebraic group is a group that is also an algebraic variety. There is also a theory of algebraic monoids. Is there are version of universal algebra that incorporates these examples, and other ...
arsmath's user avatar
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175 views

Universal identities on cubic surfaces or hypersurfaces

This question is inspired by this previous one. Generally speaking, I ask what algebraic identities are universally valid for the composition law on cubic surfaces (or hypersurfaces); since the law ...
Gro-Tsen's user avatar
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What Spec-like functors are there?

The real spectrum functor is an analog of Spec for partially ordered commutative rings and real closed fields in place of commutative reals and algebraically closed fields. I was hoping that there ...
Alex Mennen's user avatar
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Hemi-semi direct product of racks or quandles

In the category of racks (similarly quandles), instead of well-known semidirect product, we have the hemi-semi direct product construction as seen on Wagemann & Crans. As far as I know, semi ...
Kadir Emir's user avatar
7 votes
0 answers
129 views

Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...
Zhen Lin's user avatar
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7 votes
0 answers
390 views

Universal anti-Horn classes?

Is there published work about universal anti-Horn classes? Anti-Horn formulas are also sometimes known as dual Horn. See also related question Is there any research of universal algebras axiomatized ...
András Salamon's user avatar
7 votes
0 answers
346 views

Adjunction algebra - is there anything similar to this in algebra?

I call adjunction algebra a universal algebra with one binary operation denoted as the punctuation sign (;) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going ...
Ioachim Drugus's user avatar
7 votes
0 answers
269 views

Binary Operation on a Cubic Surface

If $P$ and $Q$ are two sufficiently general points on a cubic surface, the line between them intersects the cubic surface at a unique third point, $f(P,Q)$. This gives a binary operation on (generic) ...
Will Sawin's user avatar
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655 views

Invertible elements in generalized fields

Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For ...
Martin Brandenburg's user avatar
7 votes
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1k views

Undecidability degree of some elementary theories (two equivalence relations, ...)

I have a question about some results in the paper I. A. Lavrov. Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain theories (in Russian). ...
boumol's user avatar
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7 votes
0 answers
330 views

The graph of algebraic theories

Fix a logic $L$ and consider the category $\mathbf{AlgTh}_L$ with theories of $L$ as objects and theory interpretations as morphisms. For nice enough logics, this category has pushouts (which we will ...
Jacques Carette's user avatar
6 votes
0 answers
339 views

Cohomology without comonad?

TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be? For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple). ...
Student's user avatar
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6 votes
0 answers
297 views

An adjunction between monads on $\mathcal{C}$ and presentable categories under $\mathcal{C}$

Fix a regular cardinal $\kappa$ and let $\mathcal{C}$ be a $\kappa$-presentable $\infty$-category (comments about the 1-categorical case are welcome as well!). I'm looking for a reference for the ...
Saal Hardali's user avatar
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6 votes
0 answers
105 views

Permutative Yang-Baxter monoids

Suppose that $f,g:X^{2}\rightarrow X,T:X^{2}\rightarrow X^{2}$ are mappings such that $T(x,y)=(f(x,y),g(x,y))$. An element $1\in X$ is said to be an identity if $T(1,x)=(x,1),T(x,1)=(1,x)$. The ...
Joseph Van Name's user avatar
6 votes
1 answer
867 views

Does the likelihood of these tables exist?

Probably it does, and may be a number near $e^{-3/2}$ for 2-deficient tables. First some background. Early on in my studies of universal algebra, I encountered a result of Vadim Murskii, with the ...
Gerhard Paseman's user avatar
6 votes
0 answers
187 views

Can finite binary self-distributive algebras fit into small $n$-ary self-distributive algebras?

A binary operation $*$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$. An $n+1$-ary operation $t$ is said to be self-distributive if it satisfies the identity $$t(...
Joseph Van Name's user avatar
6 votes
0 answers
158 views

Why are there so few elements in the classical Laver tables with period 32?

Recall that the classical Laver table $A_{n}$ is the unique algebraic structure $(\{1,\ldots,2^{n}\},*_{n})$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$, and $x*_{n}1=x+1\mod n$ for all $x,y,z\in ...
Joseph Van Name's user avatar
6 votes
0 answers
127 views

What is the probability that a thread in the inverse limit of classical Laver tables is induced by a rank-into-rank embedding?

For this question, suppose that there exists a rank-into-rank cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Give $\mathcal{E}_{\...
Joseph Van Name's user avatar
5 votes
0 answers
75 views

Reciprocity for algebra objects in two algebraic categories

I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories. So, ...
Nik Bren's user avatar
  • 499
5 votes
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133 views

Classifying the algebraic structure on endomorphism sets

This is motivated by defining modules in a general sense, which is an appropriate homomorphism from $R$ to $\textrm{End}(X)$. If $X$ comes from different categories, the endomorphism set will have ...
Trebor's user avatar
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5 votes
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117 views

Can the equational theory of commutative rings be "unpacked" from the equational theory of exponentiation?

Below, I'll use "$\approx$" for the equality symbol in an equation, as opposed to "actual" equality. Suppose $\mathcal{V}$ is a variety (in the sense of universal algebra) in the ...
Noah Schweber's user avatar
5 votes
0 answers
156 views

Axiomatizability of image of functor

Let $\sigma$ and $\tau$ be first-order languages and let $S$ resp. $T$ be theories (sets of sentences) for $\sigma$ resp. for $\tau$ ($S$ and $T$ possibly empty). Let $\mathcal C$ resp. $\mathcal D$ ...
Daniel W.'s user avatar
  • 365
5 votes
0 answers
187 views

Algebraic/relational structures produced using evolutionary/machine learning algorithms?

Are there examples of algebraic structures which have been constructed using evolutionary algorithms and possibly machine learning algorithms? I am looking for algebraic structures like lattices ...
Joseph Van Name's user avatar
5 votes
0 answers
181 views

What are some examples of inner endomorphisms?

Let $(X,\mathcal{F})$ be an algebraic structure. If $t$ is an $n+1$-ary term, then let $L_{t,a_{1},...,a_{n}}:X\rightarrow X$ be the mapping defined by $L_{t,a_{1},...,a_{n}}(x)=t(a_{1},...,a_{n},x)$. ...
Joseph Van Name's user avatar