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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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Generalising idea of projection of continuous laticces into any category

Definition: A continuous lattice $D$ is said to be a projection of a continuous lattice $D'$ if and only if there are a pair of continuous maps $$i:D\rightarrow D'$$ and $$j:D'\rightarrow D$$ ...
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Are upper quasiverbal and lower quasiverbal subgroups always the same subgroup?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a prevariety described by $Q$ as a class ...
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Do the quasiverbal subgroups always exist?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a quasivariety described by $Q$ as a class ...
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Is there a standard formulation of a 'null element'?

I want to define the set $F$ of functions on a set $X$ that can be generated from a finite number of selected operations defined on $X$ (the set from which these operations are selected need not be ...
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1answer
50 views

Is there a way to syntactically characterize homomorphic images and products?

Birkhoff's HSP theorem states that if a class of algebras (for a given type) is closed under products, subalgebras and homomorphic images ($\iff$ quotient), then it is actually defined by some ...
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38 views

Subdirectly irreducible algebra of language F with cardinality $>=2^\kappa$

If language F has cardinality $\kappa$ ($\kappa$ is some infinite cardinal) and arity of every operation in F is 1, then doesn't exist subdirectly irreducible algebra of language F with cardinality $&...
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49 views

General categorical notion of “structure induced by a map”?

In topology, we can speak of the "[topology induced by a function][1]", which is the (coarsest) topology on $X$ such that $f:X\to Y$ is continuous, given a topology on Y, or vice versa. I can imagine ...
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Algebra operations as natural transformations

Apologies in advance if the following makes little to no sense, but here goes .. Denote $m_G : G\times G\to G$ the multiplication of a group $G$. Does it make sense to think of the map $m_G$ as some ...
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If $(\alpha,\beta)$ is the factor pair congruences of algebra $\mathbb{A},$ ia $(\forall \gamma\in ConA)\alpha\circ\gamma=\gamma\circ\beta?$

Let $\mathbb{A}$ be an algebra such that $ConA$ is the distributive lattice. If $(\alpha,\beta)$ is the factor pair congruences of algebra $\mathbb{A},$ prove that $(\forall \gamma\in ConA)\alpha\circ\...
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1answer
45 views

CEP for (distributive) lattices and groups?

An algebra $A$ has the congruence extension property (CEP) if for every $B\le A$ and $\theta\in\operatorname{Con}(B)$ there is a $\varphi\in\operatorname{Con}(A)$ such that $\theta =\varphi\cap(B\...
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Cardinality of Sub(A) is smaller than ${\aleph_0}$ or equal $2^{\aleph_0}$?

Let $\mathbb{A}=(A,\mathcal{F}^\mathbb{A})$ be a countable algebra. I need to prove that $|Sub(\mathbb{A})|\leqslant\aleph_0$ or $|Sub(\mathbb{A})|=2^{\aleph_0},$ where $Sub(\mathbb{A})$ is the ...
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Equivalence of Frankl's Conjecture

Frankl’s conjecture is one of the most famous problems in combinatorics. Frankl's conjecture claims: For every finite non-empty set $A$ and for every Frankl's family $F$ on $A$ exists $a\in A$ such ...
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Query on algebraic systems, varieties and free objects

For many purposes one usually consider algebraic systems that are groups with respect one of their operations and $f^n(1,\dotsc, 1) = 1$ for all operations where $1$ is the neutral element with ...
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1answer
51 views

Trying to find a 4-nary bracket operator that is not a generalisation of commutator, associator and polynomial identities

Given a nonassociative algebra or ring $R$, the commutator and anticommutator are defined as: \begin{align} [a,b]=ab-ba, \{a,b\} = ab+ba\end{align} and the associator is defined as: \begin{align} (a,b,...
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Does the specific condition on a normal subgroup of a finite group imply that it is a direct factor? v2.0

Suppose $G$ is a finite group, $H \triangleleft G$, such that $\frac{G}{H}$ is simple and $Var(G) = Var(\frac{G}{H})$ (Here $Var(G)$ stands for minimal group variety containing $G$). Does that imply ...
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Can every β∈Eq(A) be represented by some terms using α1,α2,α3,α4∈Eq(A) and ∩,∨?

$A$ set $A$ is nonempty and finite. $Eq(A)$ is the set of all equivalence relations on a set $A.$ Then Eq(A)=$(Eq(A);\subset).$ How can I prove that we can choose $\alpha_1,\alpha_2,\alpha_3,\alpha_4\...
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Are the “compatibility axioms” of vector space axioms derivable from a general “compatibility principle”

I have seen a couple of times that concepts specific to particular structures that seem "sensible" but still somewhat arbitrary can be derived from a general canonical principle at a higher level of ...
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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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1answer
35 views

Why an alternative magma need not even be power-associative?

I don't understand well this situation Any associative magma (i.e., a semigroup) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be ...
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1answer
37 views

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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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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1answer
37 views

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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48 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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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3answers
250 views

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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1answer
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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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1answer
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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 ...