# 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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### Is there any good methods for writing down basis for laws of groups?

I am wondering if there is a good method to write down a finite equational basis for a finite group.
Especially I am wondering if there is a good method in following situations:
We can write a group ...

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### Are "very conservative" connectives already definable?

I'm sadly an outsider to nonclassical propositional logics. All terminology below comes from Humberstone's book The Connectives, specifically section 4.2.
A new connective - a bit more precisely, a ...

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### When does HSP reduce to SPH?

This is actually a poorly camouflaged attempt to use the answers to When is the opposite of the category of algebras of a Lawvere theory extensive? (all very interesting) for the purposes of my ...

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### How large must algebras with a given congruence lattice be?

This is a follow-up to a recent question of mine:
For $n\in\mathbb{N}$ let $C(n)$ be the smallest $k$ such that every bounded lattice with cardinality $\le n$ which is isomorphic to the congruence ...

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### Example of trickiness of finite lattice representation problem?

I'm trying to come up with a good explanation for my students of why the finite lattice representation problem is difficult. I've already shown that the "greedy approach" to representing the ...

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### Significance of subdirect irreducibility outside universal algebra, particularly in category theory?

Subdirect irreducibility is a central notion in universal algebra. Trying to formulate it as abstractly as possible, your object is subdirectly irreducible if, in the opposite category, it does not ...

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### 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 ...

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### A new and subtle order-theoretic fixed point theorem

Sometimes a very simple argument appears out of the blue and overturns a subject. It is not based on pre-existing theory and heavy involvement in such a theory is actually a handicap in finding such ...

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### 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'...

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### 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 $H=\langle X\cup X'\mid R\cup R'\rangle$, ...

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### 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 ...

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### 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 ...

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### 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]=...

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### 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 ...

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### 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=\{=...

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### 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,...

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### 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 ...

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### Enriched categories from Lawvere theories

Given a Lawvere theory $\mathbb{T}$, is there a convenient way (e.g. a functorial construction) that freely enriches a category $\mathcal{C}$ to a $\mathbb{T}$-enriched category? For example, given ...

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### 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 ...

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### 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?
...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 $p_1,\dots,p_{n-1}$ ...

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### Ultrafilter subtraction and "zero"

This is related to a couple recent MO/MSE questions of mine, namely 1,2. Belatedly, I've tweaked this post to remove an overly-ambitious secondary question; see the edit history if interested.
Let $\...

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### 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 ...

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### 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 ...

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### 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\...

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### 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)....

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### 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 $0$,...

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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...