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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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Does the specific condition on a normal subgroup of a finite group imply that it is a direct factor?

Suppose $G$ is a finite group, $H \triangleleft G$, such that $H$ is simple and $Var(H) = Var(G) = Var(\frac{G}{H})$ (Here $Var(G)$ stands for minimal group variety containing $G$). Does that imply ...
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Reflexive graph, meaning of the reflection

Here on the page 1 there is a definition of reflexive graph. I need an intuition how it works the morphism $e:X_0\to X_1.$ What is it and to what edge in $X_1$ it sends a vertex from $X_0$?
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Do we need the axiom of replacement (ZFC) to define a product of structures (Universal Algebra)?

Do we need the axiom of replacement (ZFC) to define a product of structures (Universal Algebra)? Here $\mathrm{V}$ is the class of all sets. From my perspective here is how the product of structures ...
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Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$?

Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$? Here $V_w(G)$ stands for the verbal subgroup of $H$, generated ...
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27 views

Weak Direct Powers

Given any non-empty set $X$, I want to consider the weak direct power $\mathbb{Z}_{n}^{(X)}$. But what is this set defined as? Is it $\mathbb{Z}_{n}^{(X)}=\left \{\mathbf{a}\in \prod_{x\in X}\mathbb{Z}...
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Showing $\Theta_{i}$ defined on $\prod _{i\in I}\mathcal{A}_{i}$ is a congruence relation.

I am trying to use the previously asked question to help me finish my question. But I have a simple question about their "case 1" (when $a = b$). Why is it that if $a=b$, then $f(a)=f(b)$? Is this by ...
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Counterexample for Converse of Surjective Homomorphisms

In universal algebra, I am trying to find a counterexample using groups for the converse of the following: If $\mathcal{A},\mathcal{B}$ are algebras, and $\phi:\mathcal{A}\to \mathcal{B}$ is a ...
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45 views

Give a subgroup with an identity such that this identity does not hold in the bigger group.

In universal algebra, I am trying to show a counterexample using groups for the converse of "If an identity $s=t$ holds in $\mathcal{L}$-algebra $\mathcal{A}$, and $\mathcal{B}$ is a subalgebra of $\...
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Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
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How are F-bialgebras defined?

First, does such a notion exist? If so, would it be as trivial as the triple $(S,\alpha, \beta)$, where $(S,\alpha)$ is an F-algebra and $(S,\beta)$ is an F-coalgebra? I'm assuming there would be ...
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Is a set together with an operation always a relational structure?

I'm reading Algebraic Methods in Philosophical Logic, an introductory book on Universal Algebra by by J. Michael Dunn, Gary Hardegree. This book start its presentation by introducing the notions of "...
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How to recognise an algebra form a ring?

Suppose $\mathcal{V}$ be a variety of algebras of some signature $\Sigma =\{f_1 ,f_2\dots f_n\}$. Let $\mathfrak{A}=\{A, f_1, f_2,\dots ,f_n\}$ be an algebra in $\mathcal{V}$. Sometimes we are able ...
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Troubles by understanding arity of terms in A Course in Universal Algebra.

In A Course in Universal Algebra in definition 10.1 terms are introduced. What puzzles me is the statement about arity: "A term $p$ is $n$-ary if the number of variables appearing explicitly in $...
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Is there a “low-powered” axiom system for producing all the $\infty$-groupoid axioms?

If I understand correctly, part of the buzz surrounding homotopy type theory is that a small system of axioms ends up producing all of the (weak) $\infty$-groupoid axioms, where by an "axiom" in this ...
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how can we define real number to form of a (first-order) structure of type $\mathcal L$?

We know that : Let $\mathbb{R}$ denote the set of all real numbers. Then: 1-) The set $\mathbb{R}$ is a field, meaning that addition and multiplication are defined and have the usual ...
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When can one atom be below the join of two distinct atoms?

Consider three distinct atoms of a lattice $a,b,c$. When can we rule out the possibility that $c\le a\lor b$? So, to be clear, the question is: What is the weakest natural property of a lattice that ...
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Definition of Tropical Hypersurface

Given the tropical semiring $(\mathbb{T},\oplus,\otimes)$, a tropical hypersurface associated to a tropical polynomial is the set of points where it is non-differentiable. I'm wondering how ...
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Is there a generalization of universal algebra in which inequalities are permitted?

In ordinary universal algebra, we consider only equational axioms like $$x + y = y + x, \qquad x + -x = 0.$$ There's a generalization of universal algebra in which quasi-equations are permitted, ...
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VAR, Algebra and local presentability

Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-\mathrm{lim}$, BUT $k-\textrm{colim}$ must be filtered. Where have we used that $k$ is ...
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Atomlessness of the Lindenbaum-Tarski $\sigma$-algebra of infinitary language $L_{\omega_1}$

Consider the Lindenbaum-Tarski $\sigma$-algebra $A$ of the infinitary language $L_{\omega_1}$ of propositional calculi. Since $L_{\omega_1}$ has $\omega_1$ propositional variables and allows ...
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Why is an isomorphism between (total) algebras required to have an inverse which is a homomorphism?

Let us consider homomorphisms between partial algebras as defined in https://planetmath.org/homomorphismbetweenpartialalgebras There, an isomorphism from $A$ to $B$ is a bijective homomorphism ...
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The Lattice of all subvarieties of a variety

Suppose that we have a variety $V$ and the corresponding lattice of all subvarieties. My question is: is that lattice an object of our ZFC theory? Especially, is it small? If we take and element of it,...
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The logic behind adding this sentence to the second condition in this definition of isomorphism

Below is the definition of isomorphism quoted from the textbook Introduction to Set Theory by Karel Hrbacek and Thomas Jech. First, we introduce relevant definitions: An $n$-ary relation $R$ in $A$ ...
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What is a presentation of a Lawvere theory formally, and how do you generate the associated Lawvere theory?

A Lawvere theory for a specific algebraic theory is usually given by some sort of presentation. You have generating functions and constants $f_i:A^n \to A$, and relations $(r_1,r_2) \in ``\mathrm{Hom}(...
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Implicit operations in finite semigroups.

what are some examples of implicit operations in finite semigroups other than expressions involving $\omega$? Like $x^\omega y^\omega$ or $x^{\omega+1}$. By Reiterman's theorem, pseudovarieties of ...
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Can the empty set serve as universe of a semigroup (i.e. set equipped with associative binary operation)?

The title of my question is more a representative of my more general question. In A course in universal algebra (nice material!) I encountered in definition 1.3 that the universe of an algebra is not ...
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Basic Misunderstanding of Commutative Diagram - Kernel

In A Categorical Introduction To Sheaves p. 3 (2), it says that (categorically) a kernel of a homomorphism is defined by the following diagram: I don't see how this characterizes the kernel. In ...
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Is a poset an algebra?

I encountered some nice material about so-called universal algebras. E.g. a lattice is an algebra (in the sense of universal algebra), and can be presented as a tuple $\langle L,\wedge,\vee\rangle$ ...
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Which functors between varieties of algebras are monadic?

I was wondering if there's any known result which classifies which functors $F : \mathcal{B} \to \mathcal{C}$, where $\mathcal{B}$ and $\mathcal{C}$ are both varieties of algebras, are monadic. I ...
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Saturated Addition Over Tri-state Pulser Input: What kind of algebraic structure is this?

I'm currently working with a "tri-state pulser" in an engineering context. To excite a response from this device, one provides a voltage sequence consisting of elements from $\{-1,0,1\}$. For example, ...
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Categories in which every monomorphism is regular / effectivie?

I'm interested in conditions on a category of modules of a monad on a topos satisfies the property that Every monomorphism is the equalizer of its cokernel pair. Every topos itself satisfies this ...
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What is the origin of the term `SQS skein';

What is the origin of the term `SQS skein'; it is a ternary operator that (in at least one avatar) arises in assigning algebras to (2,4) Steiner systems. But skein module evidently comes up in knot ...
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Rational rotational algebra (noncommutative torus) is not simple

I would like to show that the rational rotational algebra $A_\theta$ is not simple where $A_\theta= C^{*}(u,v : u,v$ are unitaries and $uv=e^{i2\pi\theta}vu$) and $\theta$ is a rational number. The ...
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What is so “direct” about the direct product?

Let $A_1, \ldots, A_n$ be objects in some category of universal algebra. Then we may form what is called their "direct product" $$ A_1 \times \cdots \times A_n $$ with pointwise operations. I've ...
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On the algebraic theory of Boolean algebras

I have a question which (I think) should be easy for the experts: Is the Lawvere theory of Boolean algebras commutative, i.e. are its operations "algebra homomorphisms under any interpretation"?
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Inclusion relations between equationally defined classes of finite semigroups

Let $S, T$ be two semigroups. In the following all semigroups are supposed to be finite. We write $S \prec T$ if there exists a surjective semigroup morphism from a subsemigroup of $T$ onto $S$. A ...
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Notation for many-sorted term algebras

I'm writing a computer science paper, and need to give a formal definition of many-sorted term algebras (which correspond to the abstract syntax of programming languages). I'm having trouble finding a ...
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1answer
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Is there a generalization of universal algebras that allows the use of multiple sets?

Algebras in Universal algebras are normally over only one set, which is enough to generalize over many algebraic structures. E.g. groups, rings or lattices. However, some algebraic structures ...
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Why does “$[\beta,\beta]\leq\alpha$” imply that “$\alpha$ has $\beta,\beta$-term condition” in general algebras?

Let $\mathbf A$ be an (universal) algebra and let us denote $\mathrm{Clo}(\mathbf A)$ the set of term operations of $\mathbf A$. Let $\alpha, \beta$ be congruences of $\mathbf A$. We say that $\alpha$ ...
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Morita theory for algebras for a monad $T$

There are convincing arguments that support the claim that universal algebra is essentially the theory of $\lambda$-accessible monads $T$ over Set. Now, given two equivalent categories of algebras ...
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Finite algebras can be given by finite presentations

Let $A$ be a finite algebra in variety $V$ with finite language $L$. Then let $X$ be the basis of $A$, which is obviously finite since $A$ is finite. I know that a finite presentation $\langle X|R \...
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Are these classes varieties?

a) Prove the class of all divisble groups isn't a variety of algebras in the language of groups $(\cdot , e, ^{-1})$. I can't use HSP for this one. I think that we need to show that there are some ...
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Prove this is a subalgebra

Let $\phi, \psi$ be two homomorphisms $A\rightarrow B$ of algebras $A$ and $B$. Prove $E = \{a\in A : \phi(a) = \psi(a)\}$ is a subalgebra of $A$. I'm not quite sure what I even have to show. In my ...
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Is a variety of algebras still a variety if you add in another operation or drop one?

Let $L$ be a language and let $t(x_1, x_2, . . . , x_n)$ be a term over $L$. Let the language $L_1$ be obtained from $L$ by adding a new n-ary functional symbol $g$. For any algebra $A$ over $L$ we ...
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What would an element of an algebra generated by a subalgebra look like?

If we have $A$ is an algebra and $X \subset A$ generates $A$, what does that look like? Also, if we have an algebra generated by a single element what would an element in that algebra look like?
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Prove the class of all algebras $G^ \backslash$ is a variety

I have no idea how to do this because in my notes there are no examples and HSP doesn't show up until the next section. How do I prove something is a variety without HSP? I think the only thing that ...
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Smallest variety containing $\mathbb{Z}$

I've been told that the answer is all abelian groups, but I don't see how. I know that the class of all nilpotent groups of degree 1 is a group variety and that a group being nilpotent of degree 1 ...
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If an algebra $A$ belongs to a variety and $\phi$ is a homomorphism then is $\phi(A)$ in the variety?

My attempt: Let $A$ be an algebra that belongs to variety $V$ and $\phi$ be a homomorphism $\phi: A \rightarrow B$. Then for any term $t(x_1,x_2,...,x_n)$ and any elements $a_1,a_2,...a_n \in A$, we ...
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$x^n = x$ implies commutativity, a universal algebraic proof?

I read in an answer on MO that Nathan Jacobson had given a universal algebraic proof that a ring satisfying the equation $x^n=x$ is commutative. The sketch given in the answer is very clear : wlog ...
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1answer
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Free objects as term algebras in universal algebra and free modules

In universal algebra a general algebra has a signature, that is a finite set of sort, which together form the universe, and relational symbols, functional symbols and symbols for constants. ...