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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58
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5answers
8k 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 ...
39
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1answer
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Identities of commutators

Let $G$ be a group and set $[x,y]:= x^{-1}y^{-1}xy$ as usual and consider it as a binary operation. Question: Is there a description of the identities that the operation $[.,.]$ satisfies for all ...
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19answers
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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 ...
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6answers
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Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions $f$ and $g$ on a (suitable) abelian group $G$, we have two bilinear operations: multiplication - $$(f\cdot g)(x) = f(x)g(x),$$ and convolution - $$(f*...
34
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21answers
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Why are so few operations with arity bigger than 2?

In usual algebraic structures, like groups, rings, monoids, etc, or in algebras coming from logics like Boolean algebras, Heyting algebras and the like the operations are usually of arity 0 (constants)...
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7answers
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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 ...
28
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4answers
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How dangerous are set-size assumptions?

Many students' first introduction to the difference between classes and sets is in category theory, where we learn that some categories (such as the category of all sets) are class-sized but not set-...
27
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3answers
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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 ...
25
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1answer
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A preprint of Sela concerning the work of Kharlampovich-Miyasnikov

Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...
22
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1answer
695 views

Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?

Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is ...
22
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0answers
958 views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
21
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4answers
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Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
20
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3answers
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Relation between monads, operads and algebraic theories

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any ...
20
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1answer
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Does sine interact equationally with addition alone?

$\DeclareMathOperator\Eq{Eq}\DeclareMathOperator\Th{Th}$Originally asked at MSE without success: For a structure $\mathcal{A}$ whose signature only contains function and constant symbols, let $\Eq(\...
19
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3answers
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Why are ring actions much harder to find than group actions?

I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia: A module is a ring action on an abelian group. ...
19
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2answers
657 views

Do all subtraction-free identities tropicalize?

If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+...
19
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2answers
854 views

Why does abelianization preserve finite products, really?

The abelianization functor $(-)^{ab} : \mathrm{Grp} \to \mathrm{Ab}$ is left adjoint to the inclusion of abelian groups into groups. As such, it preserves all colimits, but it doesn't generally ...
18
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4answers
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Complete Boolean algebra not isomorphic to a $\sigma$-algebra

Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
18
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4answers
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Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
18
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2answers
690 views

Constructive proof of existence of free algebras for infinitary equational theories

Is it constructively true that all (not necessarily finitary) equational theories $T = (\Sigma, E)$ have an initial model? The usual proof for finitary equational theories I know constructs first ...
17
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2answers
1k views

Relation between monads, operads and algebraic theories (Again)

This question (as the title obviously suggests) is similar to, or a continuation of, this question that was asked years ago on MO by a different user. The present question, though, is different from ...
17
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2answers
639 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 ...
17
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1answer
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What are the relations between conjugates and commutators?

The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operations are: $a^b= b^{-...
17
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1answer
684 views

Topological universal algebra: what is a variety?

Very roughly, universal algebra is the study of those classes of algebraic structures which can be defined via a set of equations; such a class is called a variety. Of course there is far more to the ...
17
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0answers
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Why did Bourbaki not use universal algebra?

I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra? Well, universal algebra is not much older than category ...
16
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3answers
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What is the theory of polynomials?

We all know what polynomials are, along with their elementary properties and many effective algorithms for different representations of polynomials. The question here is more of a universal algebra ...
16
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1answer
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So, did Poincaré prove PBW or not?

This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer? It is known that Poincaré, at least, invented something that can be called Poincaré-...
16
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1answer
465 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
15
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3answers
696 views

Is the Amitsur-Levitzki identity essentially unique?

Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...
15
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1answer
817 views

What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
15
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2answers
552 views

Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...
14
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4answers
741 views

Are norms intrinsically $\mathbb{R}$-valued?

Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$? I have searched a ...
14
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2answers
501 views

Contexts and notations for composing asymmetric simplices

Imagine the elements of a group-like structure as puzzle pieces with essential two sides, an IN-side and an OUT-side. You can compose two such pieces in two obvious ways: Now consider triangular ...
13
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2answers
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Free division rings?

Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?
13
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3answers
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$fgf = f$, $gfg = g$, $fg$ not necessarily identity, what is this called?

A very simple question, I just totally forgot how it was called, and Google is not helping. There's a pair of functions $f:X\to Y$, $g:Y\to X$. $fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to ...
13
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4answers
813 views

The groupoid of algebraic expressions and proofs

Fix a set of variables $V$, and suppose we're given a presentation of a monosorted algebraic theory, with variable symbols taken from $V$. For the sake of example, suppose the presentation consists of ...
13
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1answer
314 views

characterization of subalgebras of universal enveloping algebra coming from Lie subalgebras

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}'$ its subalgebra. Then the universal enveloping algebra $U(\mathfrak{g}')$ can be canonically embedded into $U(\mathfrak{g})$, that of $\mathfrak{...
13
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1answer
826 views

Which categories are the categories of models of a Lawvere theory?

Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...
13
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2answers
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What is the status of (universal) algebra in type theory?

With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...
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3answers
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About a construction of Borel $\sigma$-algebra associated to a lattice

Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$). Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ i....
13
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0answers
293 views

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 ...
13
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0answers
215 views

Is there a finite equational basis for the join of the commutative and associative equations?

I asked this on math stack exchange, but I was told to post it on mathoverflow. Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by the ...
12
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5answers
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Jonsson Boolean algebras?

Let us say that a mathematical structure of cardinality $\omega_1$ is Jonsson whenever every one of its proper substructures is countable. There are examples of Jonsson groups due to Shelah or ...
12
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0answers
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Birkhoff's HSP theorem in categories other than $\mathbf{Set}$

Fix a category $C$ with finite products and a set $L$ of function symbols (each equipped with an arity in $\mathbb N$). An $L$-algebra in $C$, $\mathbf A=(A,(f^\mathbf{A})_{f\in L})$, is given by some ...
11
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3answers
391 views

Comparing the existing formulations of universal algebra and their levels of generality

I am a newcomer to universal algebra and I just read this (very good, IMO) book on the topic: Adámek, J., Rosický, J., & Vitale, E. M. (2010). Algebraic theories: a categorical introduction to ...
11
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2answers
667 views

What is known about ideal and divisibility lattices of GCD domains and their generalizations?

The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}...
11
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3answers
577 views

IBN for algebraic theories

Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
11
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5answers
877 views

What are the invariant Pseudo-differential operators on a Lie group?

It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group. On the other ...
11
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2answers
987 views

Uses of Zorn's Lemma when the thing is actually unique

Are there uses of the sledgehammer Zorn's Lemma that are embedded in arguments similar to the more delicate one below, showing that the thing being constructed is actually unique? Many uses of Zorn's ...
11
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0answers
404 views

When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...

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