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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71 views

Finitary endofunctors: “Support” of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

Consider the word a.k.a. list monad $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$: ...
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Finitely presentable grids and co-presentable topological spaces

In the 90s there has been some interest for the category $\mathsf{Top}^\circ,$ the dual of the category of spaces. Most of the relevant papers on the topic are co-authored by Pedicchio. Barr, ...
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value (element of an algebra), constant, variable, ground and non-ground terms, free algebras : there is a need for clarification

I have been developing an algorithm to compute the congruence defined by a finite set of "generators" and a finite set of equations (in the sense of equational theories). The algorithm ...
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1answer
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Finitely presented algebra [closed]

I found in the book "Universal Algebra for Computer Scientists", by W. Wechler, the following statement : "In general, even for finite presentations, the word problem is unsolvable"...
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343 views

Comparing the existing formulations of universal algebra and their levels of generality

I am a newcomer to universal algebra and I just read this (very good, IMO) book on the topic: Adámek, J., Rosický, J., & Vitale, E. M. (2010). Algebraic theories: a categorical introduction to ...
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162 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 ...
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101 views

Minimizing all aspects of the definition of Boolean algebra

There are many equivalent ways to describe Boolean algebras. There are a number of different ways to "minimize" the description. We can: Minimize the number of function symbols. Minimize ...
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801 views

Why does abelianization preserve finite products, really?

The abelianization functor $(-)^{ab} : \mathrm{Grp} \to \mathrm{Ab}$ is left adjoint to the inclusion of abelian groups into groups. As such, it preserves all colimits, but it doesn't generally ...
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72 views

Proving interpolation results through amalgamation

Notice: this is a cross-posting, I have asked essentially the same question on MSE (https://math.stackexchange.com/questions/4012960) but received no answers, and as this problem, although very basic, ...
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356 views

Varieties where every algebra is projective?

Is it possible to classify all varieties (in the sense of universal algebra) where every algebra is projective? Several years ago I asked a similar question, with "free" in place of "...
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1answer
163 views

On the tree-ishness of magmas and the stringiness of groups

Let me start off by saying that I suspect the answer to my question might fall under the domain of universal algebra, which is why I'm giving it that tag. However, I know only the very basics of ...
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143 views

Union star symbol in set theory

In the slides Provenance for Database Transformations, page 24, they provide a semiring for lineage, which include a $\cup^*$ symbol. However, I can not find any related materials about the meaning of ...
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1answer
199 views

Generalized cancelation properties ensuring a monoid embeds into a group

Context: an obvious necessary condition for a monoid to embed into a group (as submonoid) is to satisfy the left and right cancelation rules: $$xy=xz \quad\Longrightarrow y=z;$$ $$yx=zx \quad\...
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Ordering preserved by an inverse frame homomorphism

Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames): Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$. Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$. ...
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Equational identities of sedenions and beyond in the signature of multiplication

This is similar to my previous question about Cayley–Dickson algebras. However, now I am considering only multiplication. Consider a Cayley–Dickson algebra $(X,+,-,*,0,1)$, that is, an algebra ...
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276 views

Are gyrogroups useful for anything else other than the Einstein velocity addition rule?

Gyrogroups were discovered by Ungar in modelling the Einstein velocity addition rule in relativity. Have they been shown to be useful elsewhere in mathematics (or mathematical physics)?
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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. A ...
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155 views

Example of idempotent left quasigroups which are right-distributive but not left-distributive

I am looking for examples of the following algebraic structure: a set (X,.) which satisfy the axioms (idempotent) x.x = x (left quasigroup) the equation a.x = b has a unique solution denoted by x = ...
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Closedness of certain functional form; solvability of linear system; equationally compactness

To motivate my question, let $E \subseteq \mathbb{R} \times \mathbb{R}$ be an arbitrary set and suppose that a sequence of bivariate functions $h^{n}: E \to \mathbb{R}$ can be decomposed as $$ h^n (...
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1answer
112 views

What is the name for Boolean algebra's version of $\models$ between sets of identities and identities?

On p62 in Schaum's Outline of Theory and Problems of Boolean Algebra and Switching Circuits by Elliott Mendelson (1970), Part (b) of the corollary says that if an identity is satisfied by some ...
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197 views

Is equational logic in universal algebra a proof system not a logic system?

As far as I know a logic system defines its own semantics (e.g. $\models$), but not a proof calculus/system on its language. See p261 in Ebbinghaus et al's Mathematical Logic: In universal algebra, ...
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Varieties where any subalgebra is a regular sub-object

Following [LPAC] Chap.3, p.132 let $S$ a set of sorts and $\Sigma$ a $S$-sorted signature. From the [LPAC] treatment we have the category $Alg(\Sigma)$ with a $(\mathit{regular.Epi,Mono})$-...
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648 views

Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?

Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is ...
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79 views

Characterisation of presentations for varietal large equational theories

Let $T : \mathbf{Set}^\mathrm{op} \to \mathscr T$ be a large equational theory (i.e. a bijective-on-objects product-preserving functor). Following Linton in Some Aspects of Equational Categories, we ...
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“Tietze-like transformations” for defining interesting bijections between algebraic structures

Consider the following two definitions of the natural numbers: The natural numbers are the algebraic structure $\mathbb{N}_1$ generated by one constant, $0$ and one unary function, $S$ (and no ...
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Polynomial objects in any concrete category

EDIT: The original question had a trivial answer: it's just a coproduct. New question below New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...
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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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Invariant theory in universal algebra

Let $\mathcal{L}$ be a finite first-order language with no relation symbols, and $\mathcal{K}:=\mathcal{V}(\Theta)$ a variety in this language definited by a set of identities $\Theta$. My questions ...
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1answer
370 views

Jordan algebra identities

A Jordan algebra is a vector space with a commutative bilinear operation $\circ$ obeying an identity that's often written as $$ (x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) . $$ ...
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487 views

Is the class of power-associative binars finitely axiomatizable?

A binar is simply a set $S$ equipped with a single binary operation $*$. A power-associative binar is a binar where the subalgebra generated by a single element is associative. Equivalently, they can ...
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Morphism of sketches inducing a (co)reflective subcategory

Certain morphisms of theories gives rise to (co)reflective subcategories of their models. For the sake of simplicity, I'm going to fix my definition of sketch to be a category $\mathbb{C}$ with a ...
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158 views

Polynomial identities of supercommutative-gradable algebras

All algebras below are associative, and not assumed unital, and, to fix ideas, over the complex numbers. An algebra $A$ is supercommutative-gradable if it admits a grading $A=A_0\oplus A_1$ in $\...
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1answer
170 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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1answer
94 views

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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642 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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779 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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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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106 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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863 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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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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263 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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1answer
301 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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268 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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249 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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1answer
284 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
256 views

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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98 views

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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