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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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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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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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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. ...
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Understanding The Equations Behind Dual Objects In Monoidal Categories

I think about the identities in monoidal categories $C$ (triangle and pentagon) the following way: Let $\Sigma=(\otimes,1)$ denote the monoid signature, $T(\Sigma,C_0)$ the set of terms over $\Sigma$ ...
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Comagma, Bimagma and their relation to Jónsson-Tarski algebra.

There doesn't seem to be any mention of something like a comagma: the categorical dual of a magma where the arrows are reversed for • : A × A → A, as in π : A → A × A. Only coalgebras and bialgebras ...
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Which algebraic structure is like a magma augmented with an operation which is an anti-function?

Is there any structure similar to a free magma, including an operation which splits, or undoes, the binary operation? ...
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Free Nilpotent Semigroup

Let a nilpotent semigroup be defined as in Neumann and Taylor's Subsemigroups of Nilpotent Groups. More precisely, we call a semigroup nilpotent of class n if it satisfies the following identity: $q_n(...
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Scalar multiplication well-defined using tensor product universal property

I have a question concerning the proof, that the scalar multiplication of a module is well-defined. I guess, that the universal property of the tensor product will help, but I am not quite sure in ...
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Graphs in a regular category

Let $\mathcal{C}$ be a regular category, and let $X,Y,Z\in Ob(\mathcal{C})$. Let $g:Z\to Y$ be a regular epi, and let $R\in Sub(X\times Y)$ (subobjects of $X\times Y$). Define $S:=(id_X\times g)^\ast(...
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Characterizing the algebras on $\mathbb(Z)/2\mathbb(Z)$

Is there a characterization or discussion of which algebraic structures may be equipped to the set $\{0,1\}$? For example, $\{0,1\}$ admits a unique group structure, a unique ring structure, $2!=2$ ...
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$K$-free lattice on two generators where $K=\{$two element lattice$\}$

This is an example from [A Course in Universal Algebra][1]. Sorry, I can't copy all necessary definitions here; there are a lot of them. Let $\mathscr{F}$ be a type of algebras and $K$ be a class of ...
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What is non-algebraic structure

If an algebraic structure is a set of operations on a set of elements, what is a non-algebraic structure? https://en.wikipedia.org/wiki/Outline_of_algebraic_structures#...
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Can we axiomatize a field starting with the binary operations and only “equational” axioms?

The usual field axioms include the existence of (additive and multiplicative) identities and inverses. Is there a set of field axioms where all axioms are purely equational (see below for what I mean)?...
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Subdirect products

My question actually regards to slightly degenerate case of a subdirect product: Is every algebra a subdirect product of itself alone I.e. of the product only involving itself once?
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Subdirect product of algebras

Consider two classes of algebras(not necessarily of the same structure), say $\mathbf{A}$ and $\mathbf{B}$ and suppose the following claim: An algebra is an $\mathbf{A}$-algebra if and only if it is ...
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Algebraic theories, the category Set, and natural transformations

An algebraic theory is a small category $\cal T$ with finite products.An algebra for the theory $\cal T$ is a functor $A:\cal T\to Set$ preserving finite products. We denote by $Alg\ \cal T$ the ...
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Question on the composition of homomorphisms

It is known that if $f:A\to B$ is a homomorphism and $g:B\to C$ is another homomorphism, then $g\circ f:A\to C$ is a homomorphism. In other words, the composition of two homomorphisms is a ...
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What does it mean - “to derive” operation from some existing one on a particular set?

On the set $S = \{0, 1 \}$ let $f : S^3 \mapsto S$. Ternary operator $f(a, b, c)$ evaluates to $0$ when at least two arguments equal $0$; otherwise it returns $1$. One can form various terms in a ...
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What is the most general algebraic structure that a finite set has?

For an object $X$ in a category with finite products, define its endomorphism Lawvere theory to be the Lawvere theory generated by $X$: its $n$-ary operations are given by $\text{Hom}(X^n, X)$, and so ...