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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160 views

Indecomposable weirdos (cnt.)

This post is a continuation of Weirdos but algebraic. Logically, the quoted post could follow the present one rather than precede it. Question Does there exist an indecomposable weirdo which is ...
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Topological connected eccentrics, not homeomorphic to commutative Lie groups

An eccentric is a universal algebra $\ (X\ \sigma\ \lambda\ \rho)\ $ such that operations $\ \sigma\ \lambda\ \rho\,:\,X\times X\to X\ $ satisfy: $\quad \forall_{x\ y\,\in X}\quad \lambda(\sigma(x\ y)...
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612 views

Weirdos but algebraic

Weirdos generalize Abelian groups as well as an algebra of arithmetic mean of reals (or geometric mean of positive reals). But first, I'll define eccentrics. (I will not ask about eccentrics here ...
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694 views

What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
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226 views

Comparing “axiomatized function spaces”

This was previously asked and bountied at math.stackexchange with no response. Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$ with the ...
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100 views

Explicit lifting characterization of complete lattices among posets?

It's well-known that the complete lattices are characterized among all posets as the regular-injectives. That is, a poset $L$ is a complete lattice if and only if $L$ has the right lifting property ...
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800 views

Why is the theory of small categories not algebraic?

In "Partial Horn logic and cartesian categories", E. Palmgren and S. J. Vickers state without proof that "The theory of categories is not algebraic." Is there a reference, or an elementary argument, ...
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66 views

When will a monad satisfy Moggi’s “equalizer” property?

I’m interested in finding when ($V$)-monads will satisfy the universal property that $\eta$ equalizes $\eta_T, T\eta$. I’m particularly interested in the case for monads on presheaf categories or that ...
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Varieties of non-distributive lattices with De Morgan negations and their ‘canonical’ members

After some revisions and checks, I have decided to repost the question. Recall that the variety $\mathcal{DM}$ of (distributive) De Morgan lattices has the lattice $\mathbf{4}$ as its canonical ...
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200 views

Birkhoff's completeness theorem put into practice

Birkhoff's completeness theorem (see here, Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in equational logic. Question. Does the proof of ...
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280 views

Does Higman's embedding theorem hold inside group varieties?

Suppose $\mathfrak{U}$ is a variety of groups. Let's define $F_n(\mathfrak{U})$ as relatively free groups in $\mathfrak{U}$. Suppose $G \in \mathfrak{U}$ is a finitely generated group. We call $G$ ...
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252 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 ...
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96 views

Is finite verbal subgroup equivalent to finite index of marginal subgroup?

There is a well known fact: If $G$ is a finitely generated group. Then $|G’| < \infty$ iff $[G:Z(G)]<\infty$. Suppose $\mathfrak{U}$ is a group variety. Let’s denote the corresponding verbal ...
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225 views

Universal property of the cocomplete category of models of a limit sketch

Let $\mathscr{S}$ be a limit sketch in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\...
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262 views

Non-abelian variety of groups in which finite groups are abelian

Is there a non-abelian variety of groups $V$ such that any finite group from $V$ is abelian? This was posed in a paper by Hanna Neumann (1967), but I cannot find the solution.
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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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1answer
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Can a compact object be a nontrivial self-retract?

Let $\mathcal C$ be a locally finitely-presentable category, and let $X$ be a finitely-presentable object of $\mathcal C$. Question: Can there exist a nontrivial idempotent on $X$ whose fixed points ...
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Bounds for Khukhro-Makarenko theorems

Let’s define the set of outer-commutator group words $OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$ using the following recurrence: $$\forall i \in \mathbb{N} \text{ } x_i \in OC$$ $$\forall u, v \...
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Does Levi operator always map one-word varieties to one-word varieties?

Suppose $\mathfrak{U}$ is a group variety. Now let’s define $L(\mathfrak{U})$ as the class of all such groups $G$, such that $\forall g \in G$ $\langle \langle g \rangle \rangle \in \mathfrak{U}$ (...
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Is the equational theory of groups axiomatized by the associative law?

Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory ...
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On group varieties and numbers

Suppose $\mathfrak{U}$ is a group variety. Let’s define $N_{\mathfrak{U}} \subset \mathbb{N}$ as a such set of numbers, that for any finite group $G$, $|G| \in N_{\mathfrak{U}}$ implies $G \in \...
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668 views

Simple book on model theory

I was expressed by how Mendelson describes models in his Introduction to mathematical logic. Now I am looking for a nice model theory guide. The book (video source, etc.) must: Include the concrete ...
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402 views

A survey on Universal Algebra

I've been searching for a good list of books and sources on Universal Algebra. Since the closest I could get from any site was this post, I decided to create a new post. For this, I would like ...
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Equivalence relations in arbitrary categories

Let $C$ be a category and $A\in\mathrm{ob}(C)$. A relation is a subobject $q:Q\hookrightarrow A^{\times 2}$ and the quotient is defined as the coequalizer $$A/Q:=\mathrm{coeq}\left(Q\stackrel{q}{\...
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Are $T_0$ topological quasigroups completely regular?

In 1957 H. Salzmann generalized to quasigroups but weakened the standard result that $T_0$ topological groups are completely regular. He was able to show that $T_0$ topological quasigroups are regular ...
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Is there a term for operations that are only defined for some members of a set? [closed]

For example, matrix multiplication is only defined for certain combinations of matrices in the set of all matrices. So, I can't call the combination of the set of all matrices and matrix ...
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Equivalence relations: Cosimplicial semilattice?

For $n\ge 0$, let $E_n$ be the set of all equivalence relations on $[n]:=\{0,\dotsc,n\}$. Now given two equivalence relations $R,R'\in E_n$, we build their join $$R\vee R' := \langle R\cup R'\rangle,$$...
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102 views

Does $\Sigma$ generate the variety of all groups?

Suppose $FT$ is the class of all isomorphism classes of finite T-groups (a T-group is a group, where all subnormal subgroups are normal). Define, $\Sigma$ as the set of all functions $f$ from $FT$ to $...
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613 views

Constructive proof of existence of free algebras for infinitary equational theories

Is it constructively true that all (not necessarily finitary) equational theories $T = (\Sigma, E)$ have an initial model? The usual proof for finitary equational theories I know constructs first ...
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How dangerous are set-size assumptions?

Many students' first introduction to the difference between classes and sets is in category theory, where we learn that some categories (such as the category of all sets) are class-sized but not set-...
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Braided lobsters

If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define ...
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Questions of the paper “PBW-pairs of varieties of linear algebras”

I am reading this paper "PBW-pairs of varieties of linear algebras", the link is here:https://www.tandfonline.com/doi/abs/10.1080/00927872.2012.720867. At page 672, there is a definition of PBW-pair. ...
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Faithful group actions on tuples from algebraic structures

So I am looking for examples of the following phenomenon. Suppose that $V$ is a variety with a computable equational theory which is not locally finite. Suppose that $G$ is an infinite finitely ...
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161 views

The question about elementary equivalence of free products

Let $A,B,C,D$ be algebraic systems and $A$ and $B$ be elementary equivalent as well as $C$ and $D$. Are free products of $A,C$ and $B,D$ elementary equivalent if $A,B,C,D$ are groups, or $A,B,C,D$ ...
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Lie structure over $R$-module

In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) the following definition is given: A Lie structure over the $R$-module ...
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Basic questions about varieties of uniformly partially permutative algebras

Define the Fibonacci terms $t_{n}(x,y)$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$. We say that an algebra $(X,*)$ is $N$-uniformly partially ...
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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 ...
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466 views

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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Which varieties are compatible with the classical Laver tables?

Let $$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$ denote the $n$-th classical Laver table. The operation $*_{n}$ is the unique binary operation on $\{1,\dots,2^{n}\}$ such that $$x*_{n}(y*_{n}z)=(x*_{n}...
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232 views

Is this condition sufficient for a variety to be reversible?

A variety $V$ is said to be reversible, if for each $n>0$ and fundamental operation $f$ there are $m\geq n$ and $r$ along with terms $T_{2},\dots,T_{r}$ and $S_{1},\dots,S_{m}$ such that if $G,H$ ...
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264 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 $T_{2},\dots,T_{r},S_{1},\dots,S_{m}$ such that $G,...
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Can one optimize the probability that an identity is satisfied until the probability is $1$?

I wonder how well one can obtain algebraic structures that satisfy interesting algebraic identities by simply slowly modifying those algebraic structures until they satisfy the required identities. I ...
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Does the period of the first row in the odd size bad Laver tables grow without bound?

Does the length of the period of the first row in the odd bad laver tables grow without bound? If $n$ is a natural number, then the $n$-th bad Laver table is the algebra $B_{n}=(\{1,...,n\},*)$ where ...
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What is the method to relax or weakening a structure “ripping off” from it all its identity elements?

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with ...
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A linear ordering on the quotient algebras of elementary embeddings?

We say that a finite self-distributive algebra $(A,*)$ is linear if there is some $1\in A$ where $a*1=1,1*a=a$ for all $a\in A$ and where if $\preceq$ is the relation where $x\preceq y$ if and only if ...
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What is the computational complexity of equivalence up to a critical point in the one generator free self-distributive algebra?

Suppose that there exists a rank-into-rank cardinal. Then let $R$ be the relation on the set of terms in the language of self-distributive algebras (or LD-monoids) on one generator where we set $R(s,t)...
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Arriving at the critical points in an algebra of elementary embeddings in a unique way

Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with a self-distributive operation $*$ defined ...
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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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246 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 ...
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Is the variety of algebras that satisfy the Yang-Baxter equation generated by its finite members?

Suppose that $f,g:X^{2}\rightarrow X$, and $T:X^{2}\rightarrow X^{2}$ is the function where $T(x,y)=(f(x,y),g(x,y))$. Then $(X,f,g)$ is said to satisfy the Yang-Baxter equation if $(T\times 1_{X})\...

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