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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3 votes
0 answers
127 views

Is there an ordered algebra analogue of the HSP theorem?

For an algebraic signature (= set of function symbols) $\Sigma$, say that an ordered $\Sigma$-algebra is a pair $\mathfrak{A}=(\mathcal{A};\le)$ where $\mathcal{A}$ is a $\Sigma$-algebra in the sense ...
9 votes
3 answers
664 views

Algebraic axiomatization for AB+BA^T operation on matrices

Let us consider a matrix algebra $\operatorname{Mat}_{n\times n}(K)$, where $K$ is a field, $\operatorname{char} K \neq 2$. It is well-known that the axiomatization of commutator operation $[A,B]=AB-...
1 vote
0 answers
92 views

Eventual stabilization for repeatedly adding multiplayer games

This question is an outgrowth of a couple previous questions of mine. In order: 1,2,3. This should be fully self-contained, but those questions may help motivate this one. To keep things readable, I'...
4 votes
2 answers
207 views

Presentationally finite group "extensions"

Fix a group $G$ and fix a presentation of $G$ as $\langle X\mid R\rangle$. A presentationally finite extension of $G$ is any group that can be presented as $H=\langle X\cup X'\mid R\cup R'\rangle$, ...
5 votes
1 answer
238 views

Monoid associated to $>2$-player Hackenbush

There is some literature on multiplayer combinatorial game theory, but as far as I can tell none of it follows the line of attack below. I'd love a pointer to a similar approach taken in the ...
3 votes
0 answers
71 views

Relation algebras and quantifier rank

I had originally posted the following questions on math stack exchange but feel they are better suited for mathoverflow (original post$-$updated & reorganized for clarity). On the Wikipedia page ...
7 votes
0 answers
154 views

Relationship between measure theory and quantification

I was advised that this question might be better suited for mathoverflow, so I am reposting it here (original post). In a 1978 paper published by David P. Ellerman and Gian-Carlo Rota, the duo discuss ...
2 votes
0 answers
191 views

The trigonometric $C^*$-algebra

The trigonometric $C^*$-algebra is the universal $C^*$-algebra generated by $\mathcal{G}=\{x,y,z\}$ subject to relations \begin{align}x^2=x=x^*, &\quad y^2=y=y^*\\ [x, z]=y, &\quad [y,z]=...
3 votes
0 answers
89 views

Algebraic logical structure

Let $M=(W,R)$ be a Kripke frame, $A=(f_1,...,f_m)$ is a tuple of operations $f_i:W^{n_i}\to W$, and $\Phi=(\varphi_1,...,\varphi_m )$ is a tuple of first-order logic formulas in vocabulary $\sigma=\{=...
4 votes
1 answer
355 views

Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?

Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic ...
7 votes
1 answer
171 views

Category of multisorted Lawvere's theories

Multisorted Lawvere's theory consists of sets of sorts $S$ small category $T$ a preserving-product essentially bijective functor $(\mathrm{finSet}/S)^{\mathrm{op}} \to T$ How is category of ...
7 votes
2 answers
569 views

Deriving consequences of identities

Suppose we are given a variety in the universal algebra sense. For concreteness, suppose that we have two binary operations $+,\cdot$, three unary operations $-,\ast,'$, and two zeroary operations $0,...
15 votes
2 answers
518 views

Does every Lawvere theory arise in this way?

By a Lawvere theory, I mean a finite-product category $\mathsf{T}$ equipped with a distinguished object, such that every object of $\mathsf{T}$ can be expressed as a finite product of the ...
8 votes
1 answer
278 views

How many categories $C$ are there such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$?

In this question, bimonadic category is a category $C$ such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$. How many bimonadic categories are there? Can we classify them all? ...
8 votes
1 answer
441 views

Can a Shelah semigroup be commutative?

A semigroup $S$ is called $\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$; $\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N$...
7 votes
1 answer
190 views

Free median algebras and maximal linked systems

$\DeclareMathOperator\MLS{MLS}$Recall that the median operation, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements ...
5 votes
0 answers
117 views

Can the equational theory of commutative rings be "unpacked" from the equational theory of exponentiation?

Below, I'll use "$\approx$" for the equality symbol in an equation, as opposed to "actual" equality. Suppose $\mathcal{V}$ is a variety (in the sense of universal algebra) in the ...
2 votes
1 answer
108 views

Algebras determined by their globals

If $A= (A, f_1, f_2, ...f_n)$ is an algebra, then its global (sometimes referred to as complex algebras) $\mathcal{U}(A)$ is defined on the power set $\wp(A)$ in the usual way. It is known that $\...
3 votes
1 answer
63 views

Reference request for a proof of the Mal'cev condition for congruence $n$-permutability

By a theorem of Hagemann and Mitschke, a condition (A) that a variety $\mathcal{V}$ is congruence $n$-permutable, is equivalent to a condition (B) that there exist ternary terms $p_1,\dots,p_{n-1}$ ...
2 votes
2 answers
243 views

Of what kind of complemented bounded poset are the structures in my quasi-variety?

I feel that my question is very basic, but, somewhat suprisingly, nobody was able to give me an answer so far: Let $\mathbf{M} := \langle \{ 0,1 \}, 0, 1, \leq, \neg \rangle$ be the structure with ...
7 votes
1 answer
406 views

Is the equational theory of this "orthocentrish" algebra finitely based?

Let $\mathcal{A}$ be the algebra (in the sense of universal algebra) whose underlying set is the four-element set $\{a,b,c,d\}$ and whose structure consists just of the ternary operation $F$ defined ...
19 votes
0 answers
554 views

What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?

Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
20 votes
2 answers
717 views

Why is every variety of bands determined by a single identity?

A band is a semigroup where every element is idempotent: $a a = a$. A collection of bands is a variety if it is closed under taking subobjects (subsemigroups), quotient objects (images of ...
3 votes
1 answer
114 views

When does a clone on a two-element set have almost abelian symmetry groups?

Say that a clone (in the sense of universal algebra) $\mathfrak{C}$ has almost abelian symmetry groups (= aasg) iff for each function $f(x_1,...,x_n)\in\mathfrak{C}$ there is an abelian subgroup $A\...
4 votes
0 answers
143 views

Is the orthocenter "(roughly) equationally finitely-based"?

Let $T$ be the "almost everywhere" equational theory of the orthocenter function, "tweaked appropriately" to avoid partiality issues (see this earlier question of mine for details)....
9 votes
1 answer
290 views

Equational theory of the orthocenter

Previously asked at MSE: Briefly speaking, I'm looking for a description of the equational theory of the orthocenter function, $\mathsf{orth}$. By $\mathsf{orth}$ I mean the (partial) function sending ...
13 votes
1 answer
427 views

Looking for a half-remembered reference on 'magnitude algebras'

I've been trying and failing to find a paper/article/blog post (I think it was a paper) on a particular algebraic structure. The paper describes a structure consisting of something like a constant $0$,...
33 votes
7 answers
3k views

Do non-associative objects have a natural notion of representation?

A magma is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional ...
5 votes
1 answer
239 views

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 ...
14 votes
1 answer
1k views

When are epimorphisms of algebraic objects surjective?

Let $C$ be the category of $\tau$-algebras for some type $\tau$. Consider the statements: Every monomorphism is regular. Every epimorphism in $C$ is surjective. It is easy to see that 1. implies 2. ...
8 votes
0 answers
123 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\...
29 votes
3 answers
4k views

Lawvere theories versus classical universal algebra

A Lawvere theory is a small category with finite products such that every object is isomorphic to a finite product of copies of a distinguished object x. A model of the theory in a category with ...
62 votes
5 answers
9k views

Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)

Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated? Basically I want to believe I can ...
17 votes
1 answer
605 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 $\varphi_p$ ...
9 votes
1 answer
747 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_\...
2 votes
1 answer
220 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 ...
7 votes
1 answer
713 views

Does ⬦ generate all De Morgan algebras?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...
37 votes
19 answers
5k views

Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
16 votes
2 answers
1k views

Comparing "axiomatized function spaces"

This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing ...
3 votes
0 answers
138 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 $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
2 votes
2 answers
209 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, ...
4 votes
1 answer
129 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 ...
3 votes
1 answer
292 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 ...
6 votes
0 answers
339 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). ...
-4 votes
1 answer
173 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?
4 votes
1 answer
171 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 ...
0 votes
0 answers
116 views

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 ...
4 votes
1 answer
791 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 ...
8 votes
1 answer
493 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 ...
4 votes
0 answers
97 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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