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).
141 questions with no upvoted or accepted answers
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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 ...
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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 ...
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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 ...
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If a map between unital rings preserves multiplication and successor, does it preserve addition?
Welcome to my first MathOverflow posting!
This is a question about rings, all of them assumed to be both unital and associative.
Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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"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 ...
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What is this quotient of the free product?
Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "...
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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\...
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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 ...
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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 ...
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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
$\...
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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 ...
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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 ...
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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 ...
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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., ...
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What happens if we add an initial object to a Lawvere theory?
Motivation
There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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, ...
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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). ...
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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 ...
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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 ...
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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 ...
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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(...
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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 ...
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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}_{\...
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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). ...
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Which pseudovarieties have (Eilenberg/Schutzenberger) minimal descriptions?
Suppose $\mathscr{V}$ is a pseudovariety in a countable language $\Sigma$. Say that a minimal description of $\mathscr{V}$ (in the sense of Eilenberg/Schutzenberger rather than Reiterman) is a pair $(...
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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 ...
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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 ...
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