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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Free algebras from model theory perspective

Let $\mathbb{V}$ be a non-trivial variety of algebras, and let $F_S\in\mathbb{V}$ be a free algebra on a set $S$. I want to know what is known about the model theory of $F_S$; I know these objects are ...
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8 votes
0 answers
105 views

Group presentations where discarding generators always yields a subgroup

Consider a group presentation $ \left< G= \left\lbrace \text{generators}\right\rbrace \, \middle|\, R = \left\lbrace \text{relators}\right\rbrace \right>$ (no finiteness assumptions). Given $S\...
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17 votes
1 answer
547 views

How hard is it to say "not exactly $p$" with a Horn sentence?

EDIT: immediately after bountying the question (whoops ...) I found, while looking for something else entirely, that Sauro Tulipani gave an explicit algorithm for producing a Horn sentence $\varphi_p$ ...
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2 votes
1 answer
213 views

Possible symmetry groups of power terms

Previously asked and bountied at MSE: Let $\mathfrak{E}=(\mathbb{N};\mathit{exp})$ be the algebra in the sense of universal algebra consisting of the natural numbers with just exponentiation. To each ...
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9 votes
1 answer
691 views

What classes of groups can arise as "symmetry groups of terms"?

Let $\mathfrak{A}$ be an algebra (in the sense of universal algebra). To each term $t(x_1,...,x_n)$ in the language of $\mathfrak{A}$ in which each variable actually appears we can assign a group $G_\...
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3 votes
0 answers
118 views

Analogue of Kolmogorov/Arnold superposition for general manifolds?

Previously asked and bountied at MSE with slightly different language: Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
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2 votes
2 answers
172 views

Are free functors usually injective up to isomorphism? [duplicate]

Let $U$ be the forgetful functor from categories to quivers. Then the left adjoint $F$ of $U$ is the functor sending a quiver to its path category. It's a fact that $F$ is injective up to isomorphism, ...
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2 votes
1 answer
159 views

What does it mean for the surreal numbers/partizan games to be "universally embedding"?

In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan ...
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-3 votes
1 answer
130 views

Is the underlying set of every renormalization group countable and finite? [closed]

Is the underlying set of every renormalization group countable and finite? Suppose A is a renormalization group, and the elements of it compose of the set B. Is B the set countable and finite?
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4 votes
1 answer
138 views

A question regarding equational bases of the theory of the commutative and associative properties

Suppose we are working in the language of a binary operation symbol $*$. Let $S$ be a set of equations which generate precisely the same equational theory generated by the set containing the ...
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Is Burgess' ST theory an algebraic set theory? If it is, what kind of algebras are described by this theory?

I found the mention of Burgess' ST set theory in the article https://en.wikipedia.org/wiki/General_set_theory about George Boolos' generalized set theory (GST). It sounds like this: "ST is GST ...
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4 votes
1 answer
713 views

How do finite door spaces work?

Recall that a door space is a topological space where every set is either open or closed (or both). A topological space is finite if it has finitely many points. I'm interested in learning about ...
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6 votes
1 answer
359 views

How many sublattices are contained in the powerset lattice of a finite set?

How many sublattices does the powerset lattice $2^n$ contain for $n$ finite? (up to equality, not isomorphism) I thought for sure this would be easy to find on OEIS, but so far I am coming up empty. I ...
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4 votes
0 answers
89 views

Minimization of second-order unifiers

We know that first-order unification is decidable. More generally, if there exists a unifier for a first-order unification problem, then there exists a most general unifier. I'm interested in the ...
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13 votes
0 answers
333 views

The free complete lattice on three generators, beyond ZF

This was originally asked at MSE; although it is still under bounty it seems unlikely to be answered there. $\mathsf{ZF}$ proves that there is no free complete lattice on three generators since any ...
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7 votes
1 answer
534 views

First isomorphism theorem for sets?

Let $f\colon S\to T$ be any function. There is the obvious refinement of $f$, by replacing the codomain $T$ with the image. Thus, every function factors into a surjection followed by an injection (...
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5 votes
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134 views

Axiomatizability of image of functor

Let $\sigma$ and $\tau$ be first-order languages and let $S$ resp. $T$ be theories (sets of sentences) for $\sigma$ resp. for $\tau$ ($S$ and $T$ possibly empty). Let $\mathcal C$ resp. $\mathcal D$ ...
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4 votes
1 answer
107 views

Do almost-point-transitive algebras generate almost-point-transitive varieties?

Say that an algebra $\mathfrak{A}$ (in the sense of universal algebra) is point-transitive iff for every $a,b\in\mathfrak{A}$ there is a $\pi\in Aut(\mathfrak{A})$ with $\pi(a)=b$. While genuinely ...
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1 vote
0 answers
49 views

First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?

In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
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2 votes
0 answers
84 views

Function and algebraic system

Let $f$ be a surjective function from $X$ to $\{1,2,3\}$. Let $* :X^2 \to X$, such that $$f(x)\neq f(y) \implies f(x) \neq f(xy)\neq f(y), $$ and $$f(x)=f(y)\implies f(x)=f(xy).$$ Let's call the ...
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8 votes
0 answers
290 views

What is going on in the field of algebraic logic these days?

I'm doing my masters in Mathematics and took a class in universal algebra and there I learned that for example: Boolean algebras have direct connection with classical logic, Heyting algebras with ...
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7 votes
0 answers
130 views

Birkhoff's theorem with language expansion?

Let $\mathcal V$ be a variety (in the sense of universal algebra). Recall that Birkhoff's theorem characterizes when a class $\mathcal W \subseteq \mathcal V$ of $\mathcal V$-algebras forms a ...
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1 vote
1 answer
223 views

Lawvere theory of Lawvere theories

There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a ...
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7 votes
0 answers
170 views

Were algebraic theories and abstract clones defined independently?

Algebraic theories (by which I mean the formalism based on bijective-on-objects functors) and abstract clones both capture universal algebraic structure, and are well-known to be equivalent. Algebraic ...
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20 votes
1 answer
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Does sine interact equationally with addition alone?

$\DeclareMathOperator\Eq{Eq}\DeclareMathOperator\Th{Th}$Originally asked at MSE without success: For a structure $\mathcal{A}$ whose signature only contains function and constant symbols, let $\Eq(\...
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1 vote
1 answer
82 views

Algebras for general transfors

Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more ...
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2 votes
2 answers
171 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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0 votes
3 answers
133 views

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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0 votes
1 answer
126 views

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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11 votes
3 answers
411 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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12 votes
0 answers
190 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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1 vote
0 answers
111 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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19 votes
2 answers
934 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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3 votes
0 answers
122 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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9 votes
2 answers
397 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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3 votes
1 answer
260 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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2 votes
0 answers
199 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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10 votes
1 answer
229 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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0 votes
1 answer
89 views

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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3 votes
2 answers
294 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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5 votes
0 answers
279 views

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). ...
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1 vote
2 answers
186 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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3 votes
1 answer
116 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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-2 votes
1 answer
226 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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2 votes
0 answers
57 views

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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23 votes
1 answer
734 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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3 votes
1 answer
83 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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5 votes
1 answer
194 views

"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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4 votes
0 answers
272 views

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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13 votes
0 answers
218 views

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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