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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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### 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 ...
151 views

### 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 ...
159 views

### 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 ...
135 views

### 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 ...
306 views

### 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 ...
162 views

### 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 ...
113 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 ...
456 views

### 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 ...
1 vote
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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'...
165 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 $H=\langle X\cup X'\mid R\cup R'\rangle$, ...
60 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 ...
138 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 ...
171 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]=...
200 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 ...
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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 ...
94 views

### 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 ...
340 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 ...
256 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? ...
170 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 ...
113 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 ...
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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 ...
399 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 ...
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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 $\varphi_p$ ...
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 ...