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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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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 is an example of a groupoid which is not a semigroup?

I know that groupoid refers to an algebraic structure with a binary operation. The only necessary condition is closure. However, I couldn't find any easy-to-understand example of a groupoid which is ...
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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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2answers
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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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1answer
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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. ...
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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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1answer
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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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1answer
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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 ...
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Minimal completion of a Boolean algebra

According to Halmos (Lectures on Boolean Algebras, p. 92), the completion of Boolean algebra $A$ is a complete Boolean algebra $B$ together with a monomorphism $h$ from $A$ into $B$ such that (1) $h$ ...
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Making a quantifier-free type finitary

Consider the theory of Heyting algebras. I have a model $L$ and a tuple $a$ in $L$. The quantifier-free type $p(x)$ of $a$ in $L$ is not generally equivalent to a single finitary formula. But ...
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Why is $\mathbb{Z}$ a free algebra in the class of abelian groups?

Following "Term rewriting and all that" page 47, I find that $\mathbb{Z}$ is free in the class of all Abelian groups being generated by $X = \{1\}$. The definition of free is: Let $\Sigma$ be a ...
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1answer
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Looking for a reference of quasi inverse in rings

This question is related to exercise 1.51 from Rotman's "Introduction to the Theory of Groups". down vote favorite 2 My question is related to exercise 1.51 from Rotman's "Introduction to the ...
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People keep describing algebras as though evaluation takes expressions back to the set of generators and I don't get it

Let $F$ be the free monoid functor, and $G$ its right adjoint. Then $G \circ F$ is a monad on $\mathbf{Set}$; and its unit $\eta$ is a natural transformation taking every set $X$ to the set $(G \circ ...
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Finitely generated substructures and model completions

Finitely generated substructures and/or local finiteness seem to play a great role in the study of model completions. For instance, I am told the following: A universal theory $T$ has a model ...
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Congruence lattice of direct power of Steiner seven element quasigroup

The Steiner seven element quasigroup is the algebra $\mathbf{S}_7 = \langle S_7, \cdot \rangle$, where $S_7 = \{ 1,2,3,4,5,6,7 \}$, and, up to isomorphism, its multiplication table is the one given ...
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The direct product of a family of algebras

I have a confusion about the concept of a direct product of a family of algebras in Universal algebra. I shall use the definition in Blackburn $\textit{et al}$ (2001) Modal Logic, p:498-9. (A similar ...
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1answer
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If a homomorphism of Boolean algebras is injective on a set of generators, is it injective?

Situation: Given two Boolean algebras, $A$ and $B$, where $A$ is defined using a set of generators subject to some relations: \begin{align*} A = \operatorname{BA}\langle \Box a, \Diamond a \; (a \in I)...
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1answer
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If a pointed variety is a subtractive category, then it is a subtractive variety.

Let $\mathcal{V}$ be a variety of algebras. Every class of algebras is a category. In particular, assume that $\mathcal{V}$ is a subtractive category. This means that in $\mathcal{V}$ every reflexive ...
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Categories as varieties of universal algebras.

I was wondering if it is possible to consider a category $\mathcal{C}$ as a variety of algebras. My idea was to consider pairs of composable morphisms of $\mathcal{C}$ as algebras, compositions and ...
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free algebras and K-free algebras

Given a set $X$ and a class $K$ of universal algebras of the same type $F$, an $F$-algebra $U$, generated by $X$, is said to be free over $X$ for the class $K$ if, for every $F$-algebra $A$ in $K$ and ...