# 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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### How much choice is needed to prove the completeness of equational logic?

All the proofs of the completeness of (Birkhoff's) equational logic I have read seem to pick representatives for equivalence classes of terms and hence require the axiom of choice. Is AC (or a weak ...

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### Classifying the algebraic structure on endomorphism sets

This is motivated by defining modules in a general sense, which is an appropriate homomorphism from $R$ to $\textrm{End}(X)$. If $X$ comes from different categories, the endomorphism set will have ...

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### Can the set of parafinite congruences be descriptive-set-theoretically complicated?

Fix an algebra $\mathfrak{A}$ with underlying set $\mathbb{N}$ and finite language $\Sigma$. The set of congruences on $\mathfrak{A}$ is a closed subset $C_\mathfrak{A}$ of $2^\mathbb{N}$ (with the ...

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### Congruences that aren't "finite from above," take 2: semigroups

This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...

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### Congruences that aren't "finite from above"

Let $\mathfrak{A}=(A;...)$ be an algebra in the sense of universal algebra. Say that a congruence $\sim$ on $\mathfrak{A}$ is parafinite iff there is an equivalence relation $E\subseteq A^2$ with ...

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### Posets of equational theories of "bad quotients"

This is a follow-up to an older question of mine:
Suppose $\mathfrak{A}=(A;...)$ is an algebra (in the sense of universal algebra) and $E$ is an equivalence relation - not necessarily a congruence - ...

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### Classes of algebras axiomatizable by special formulas; and free objects

Let $\mathcal{L}$ be a first-order language without relation symbols, and let $\mathcal{K}$ be a class of $\mathcal{L}$-algebras. $\mathcal{K}$ is axiomatizable if there is a set $T$ of first-order ...

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### Sizes of linearly ordered subalgebras of powers

On the grounds that I'm currently teaching a linear algebra class and I enjoy making my students furious, let a linear algebra be an algebra $\mathcal{A}$ in the sense of universal algebra equipped ...

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### Two notions of generalized quotient/substructure

Given a language $\Sigma$ and a $\Sigma$-algebra (in the sense of universal algebra) $\mathcal{A}=(A;\dotsc)$ and a function $f:A\rightarrow A$, let $\mathcal{A}_f$ be the $\Sigma$-algebra whose ...

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### What algebraic structure controls endomorphisms of algebras over a Lawvere theory

Given a Lawvere theory $T,$ is it possible to describe a Lawvere theory $\textrm{End}(T)$ such that $\textrm{End}(T)$-algebras describe "endomorphisms of $T$-algebras"?
In other words, what ...

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### Definition of term functions, in universal algebra

According to the definitions in Sankappanavar's universal algebra :
Assume $p$ is a term, then $p(x_1,x_2,...,x_n)$ indicates that the variables occurring in $p$ are among $x_1,...,x_n$. But there is ...

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### Is there an infinite chain of endofunctors of finite sets?

We consider the category of endofunctors of finite sets with natural transformations.
Is there an infinite chain $F_1, F_2, \ldots$ of endofunctors such that there is a natural transformation from $...

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### Counting permutations of $X^2$ that induce 4 quasigroup operations up to isotopy

Let $X$ be a finite set. Recall that a binary operation $\ast$ on $X$ is said to be a quasigroup operation if there are binary operations $/,\backslash$ where $(x/y)\ast y=(x\ast y)/y=x$ and $x\ast(x\...

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### Sequences that don't count algebraic structures on finite sets

People count $n$-element groups, $n$-element monoids, $n$-element commutative monoids, etcetera - always up to isomorphism. The algebraic structures I've listed, and many more, are studied ...

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### Cantor-Bernstein phenomena for structures (and a "moderate zigzag" property)

My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as ...

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### Product types: algebraic structure for modeling product types with commutative and associative product operation

Is there a known algebraic structure over set of Types (however they are defined) which is equipped with:
commutative and associative product operation for building product types from simpler types, ...

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### Equational identities of unordered $n$-tuples

I asked this question on math stack exchange, but I did not get an answer, so I am asking it here. This question combines both universal algebra and set theory. Let $V$ be a model of ZFC set theory, ...

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### Follow up to a question on equational bases of the theory of the commutative and associative properties

This is a natural follow-up to my previous question, here: A question regarding equational bases of the theory of the commutative and associative properties. As before, suppose we are working in the ...

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### Equational theory for the cyclic groups

Is there a finite, first-order equational theory whose models are exactly the finite cyclic groups? (An equational theory that uses $\mathbb{N}$ clearly would be 'cheating').
For example of a similar ...

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### Nomenclature help: action vs. module and “pointed” monadic algebras are module objects?

I'm confused about some nomenclature in a paper by J. Goguen from 1975. I'm happy to use definitions however they appear. But I aim to summarise the paper for a non-expert audience and I want them to ...

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### Parametrization of topological algebraic objects

There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...

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### Proving a property in De Morgan residuated lattices

A residuated lattice is an algebra $(L, \wedge, \vee,\odot, \rightarrow, 0, 1)$
of type $(2, 2, 2, 2, 0, 0)$ satisfying the following axioms:
(RL1) $(L, \wedge, \vee)$ is a bounded lattice (the ...

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### Is there an identity between the associative identity and the constant identity?

This is a follow-up to my previous question, here: Is there an identity between the commutative identity and the constant identity?. Let our signature be that of a single binary operation $+$. I ...

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### Is there an identity between the commutative identity and the constant identity?

I asked this on Math Stack Exchange, but it didn't get a single answer. So, I am now asking it here. Let our signature be that of a single binary operation $+$. I define the constant identity to be $x+...

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### Theorem constructing a mathematical structure from a set of internal isomorphisms

I am searching for information about a specific theorem mentioned in the book "Discriminator-algebras: algebraic representation and model theoretic properties" by Heinrich Werner. The ...

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### Lattices of clones: is 4 worse than 3?

Let $\mathscr{C}_n$ be the lattice of clones on the $n$-element set $\{1,...,n\}$. $\mathscr{C}_2$ is complicated but countable, but $\mathscr{C}_3$ (and all higher lattices) is of size continuum.
...

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### Are modular lattices shallow?

Let $A$ be a universal algebra with finitely many finitary operations. Write $F_n$ for the $n$-ary operations.
We define the affine maps on $A$ inductively: $\eta \mapsto \eta$ and $\eta \mapsto c$ ...

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### How does $R \equiv 1\ (\text {mod}\ h)\ $?

Definition $:$ Let $H$ be a Hopf algebra. An invertible element $R \in H \otimes H$ is called a coboundary structure on $H$ if
$(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$
$(2)$ $R_{21} = R^{-1},$
$(...

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### Is a principal filter in a free Heyting algebra a projective Heyting algebra?

A Heyting algebra is a bounded distributive lattice $(L,\vee,\wedge,0,1)$ together with a binary operation $\rightarrow$ called implication or relative pseudocomplementation with the property that, ...

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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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### 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 ...