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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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A syntactic characterisation of morphisms of algebraic theories whose induced algebraic functors admit right adjoints

Let $f : S \to T$ be a morphism of algebraic theories. Such a morphism induces a monadic functor $f^* : \mathrm{Mod}(T) \to \mathrm{Mod}(S)$ (hence $f^*$ has a left adjoint). We may view $f$ ...
varkor's user avatar
  • 10.6k
6 votes
1 answer
243 views

Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?

For a single-sorted algebraic theory $\mathcal{T}$ denote by $t_n$ the number of $\mathcal{T}$-algebras with $n$ elements (up to isomorphism). Is there an example for $\mathcal{T}$ such that ...
Martin Brandenburg's user avatar
4 votes
2 answers
366 views

Notion of prime congruences

We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime ...
Lave Cave's user avatar
  • 293
1 vote
0 answers
100 views

Lawvere theory and presentations of groups

In his dissertation on "Functorial semantics of algebraic theories", Lawvere says in his introduction that "from the category (or more precisely from an underlying-set functor) we can ...
Sylvain Cabanacq's user avatar
7 votes
1 answer
330 views

Is the partial order of all equations in the signature of magmas a lattice?

$\newcommand\Eq{\mathrm{Eq}}$I asked this question on math stack exchange, here, but there were no comments or answers. So, I am asking it here on mathoverflow. Consider the signature of a single ...
user107952's user avatar
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9 votes
1 answer
1k views

Is the number of varieties of groups still unknown?

A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
Martin Brandenburg's user avatar
7 votes
0 answers
160 views

What happens if we add an initial object to a Lawvere theory?

Motivation There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects ...
andres's user avatar
  • 355
8 votes
2 answers
596 views

If a semigroup embeds into a group, then is it a subdirect product of groups?

The title has it all: Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups? If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
Salvo Tringali's user avatar
3 votes
0 answers
152 views

What is known about the word problem on free algebraic models?

Consider the fragment of first-order logic with equality and universal quantification as the only logical symbols; we call this logic the logic of universal algebra. I am interested in languages $\...
Hernán Ibarra Mejia's user avatar
7 votes
2 answers
488 views

Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?

By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
Salvo Tringali's user avatar
2 votes
0 answers
178 views

Are there two equations with no equation strictly between them, but at least one quasi-equation between them?

I asked this question on Math Stack Exchange a while ago, but no one has responded yet. So, I am asking it here. Consider the signature of a single binary operation $+$, and consider the set $Eq$ of ...
user107952's user avatar
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11 votes
1 answer
415 views

Examples of natural algebraic irreflexive relations

To motivate the question, consider the theory of rings. Define $x \parallel y$ to mean $\exists w \exists z .((x - y) z = w (x - y) = 1)$, or in words, "$x - y$ is a unit". Then $\parallel$ ...
Zhen Lin's user avatar
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1 vote
0 answers
73 views

Why can we not convert GATs / EATs / limit sketches to sites?

I think I'm in the process of understanding something very subtle here, and I could use an expert's double check. So basically, my question is whether what I write is correct. (Non-finitary) GATs, ...
anuyts's user avatar
  • 511
16 votes
0 answers
218 views

If a map between unital rings preserves multiplication and successor, does it preserve addition?

Welcome to my first MathOverflow posting! This is a question about rings, all of them assumed to be both unital and associative. Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
Fred Wehrung's user avatar
2 votes
1 answer
118 views

Reference request for a proof of the fact that every congruence-permutable variety is semidegenerate

Given an algebra $\mathbf{A}$, a pair of congruences $ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when $ \alpha \circ \beta =\beta \circ \alpha$, and an algebra $\mathbf{A}$ is called ...
Arena's user avatar
  • 21
14 votes
2 answers
725 views

Are there any non-conjugation "extendible automorphisms" in the category of finite groups?

Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment ...
Noah Schweber's user avatar
5 votes
0 answers
81 views

Which pseudovarieties have (Eilenberg/Schutzenberger) minimal descriptions?

Suppose $\mathscr{V}$ is a pseudovariety in a countable language $\Sigma$. Say that a minimal description of $\mathscr{V}$ (in the sense of Eilenberg/Schutzenberger rather than Reiterman) is a pair $(...
Noah Schweber's user avatar
3 votes
0 answers
99 views

Counting Eilenberg/Schutzenberger-type definitions of pseudovarieties

See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that ...
Noah Schweber's user avatar
4 votes
1 answer
426 views

Is $\mathbb Z$ prime in the class of abelian groups?

Let $B$, $C$, and $D$ be abelian groups such that $\mathbb Z\times B$ is isomorphic to $C\times D$. Is there a group $E$ such that $C$ or $D$ is isomorphic to $\mathbb Z\times E$? Reference: page 263 ...
Tri's user avatar
  • 1,644
4 votes
1 answer
150 views

Comparing semiring of formulas and Lindenbaum algebra

This is motivationally related to an earlier question of mine. Given a first-order theory $T$, let $\widehat{D}(T)$ be the semiring defined as follows: Elements of $\widehat{D}(T)$ are equivalence ...
Noah Schweber's user avatar
1 vote
1 answer
215 views

Reference about cancellation property for semigroups

Have the semigroups with the following cancellation property been studied? Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
Hector Pinedo's user avatar
2 votes
0 answers
196 views

On "necessary connectives" in a structure

Given a clone $\mathcal{C}$ over $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with connectives from $\mathcal{C}$ in place of the usual Booleans. Given a clone $...
Noah Schweber's user avatar
3 votes
0 answers
115 views

Are "equi-expressivity" relations always congruences on Post's lattice?

Given a clone $\mathcal{C}$ over the set $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with (symbols corresponding to) elements of $\mathcal{C}$ replacing the ...
Noah Schweber's user avatar
13 votes
4 answers
843 views

What is a "general" relation algebra?

I'm trying to understand why (or if) the axioms of relation algebras are "the right ones." For example, we can back up the idea that the group axioms exactly capture the notion of "...
Noah Schweber's user avatar
2 votes
0 answers
128 views

What is the name of the largest subobject where a map is equivariant?

Suppose we have two objects $X,Y$ with a $G$-operation and a non-equivariant map between them. In this situation, we can look at the largest subobject $X'$ of $X$ on which $f$ is $G$-equivariant. Is ...
HenrikRüping's user avatar
9 votes
2 answers
377 views

Reference for an old result of P. M. Cohn

As it was shown by Malcev, unlike the commutative case, in which every domain can be embedded in a field, there are noncommutative domains that can't be embedded in a division ring. For noncommutative ...
jg1896's user avatar
  • 3,318
8 votes
1 answer
282 views

About the characterization of categories of model of algebraic theories

So, in his Handbook of categorical algebra Vol 2, Borceux states a theorem (the 3.9.1, page 158) that says that: Given a category $C$ with a functor $U:C \to Set$, $C$ is the category of models of a (...
Simon Henry's user avatar
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2 votes
0 answers
81 views

The involutive structure on a division ring

This question is motivated by foundations of geometry, namely, by studying scalars in affine spaces. Let $F$ be a field (or better a division ring). It has the operations of addition and ...
Taras Banakh's user avatar
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2 votes
1 answer
480 views

homotopic to a constant map

Let $X$ and $Y$ be topological spaces and more precisely connected finite CW complexes. Let $f\colon X \to Y$ be a continuous map such that there exist a second continuous map $F\colon X^3 \to Y$ and $...
Sebastian Meyer's user avatar
8 votes
0 answers
148 views

What is this quotient of the free product?

Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "...
Noah Schweber's user avatar
4 votes
1 answer
254 views

Are there atoms in the lattice of intermediate logics?

A few days ago I stumbled upon this question on MS. The question is: Does the lattice of intermediate logics have an atom, i.e. an element that is strictly stronger than IPC but not strictly stronger ...
Navid's user avatar
  • 143
3 votes
1 answer
142 views

Is there any (specially Algebraic Geometrical) exposition of Koike Terada's Young-diagrammatic methods for the representation theory paper?

I am talking about the paper by Koike, Kazuhiko and Terada, Itaru, Young-diagrammatic methods for the representation theory of the classical groups of type ($B_n$), ($C_n$), ($D_n$), J. Algebra 107, ...
Dibyendu's user avatar
6 votes
0 answers
86 views

Reciprocity for algebra objects in two algebraic categories

I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories. So, ...
Nik Bren's user avatar
  • 519
5 votes
1 answer
376 views

Kuratowski's 14 theorem and universal algebra

For a tuple of functions $\overline{p}$ on a set $Y$, let $cl_{\overline{p}}$ be the associated closure operation: $cl_{\overline{p}}(Z)$ is the smallest subset of $Y$ containing $Z$ and closed under ...
Noah Schweber's user avatar
7 votes
1 answer
433 views

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 ...
ralphS16's user avatar
5 votes
0 answers
163 views

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 ...
Trebor's user avatar
  • 1,263
3 votes
0 answers
176 views

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 ...
Noah Schweber's user avatar
1 vote
1 answer
142 views

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 ...
Noah Schweber's user avatar
5 votes
3 answers
542 views

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 ...
Noah Schweber's user avatar
3 votes
1 answer
152 views

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 - ...
Noah Schweber's user avatar
4 votes
1 answer
183 views

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 ...
jg1896's user avatar
  • 3,318
1 vote
1 answer
83 views

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 ...
Noah Schweber's user avatar
9 votes
1 answer
306 views

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 ...
Noah Schweber's user avatar
9 votes
2 answers
601 views

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 ...
Grisha Taroyan's user avatar
0 votes
0 answers
105 views

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 ...
BAD MAN's user avatar
  • 11
5 votes
1 answer
713 views

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 $...
Sebastian Meyer's user avatar
2 votes
0 answers
50 views

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\...
Joseph Van Name's user avatar
15 votes
2 answers
910 views

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 ...
John Baez's user avatar
  • 22.3k
4 votes
0 answers
247 views

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 ...
Noah Schweber's user avatar
0 votes
0 answers
56 views

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, ...
000 000's user avatar

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