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29
votes
15answers
2k views

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 ...
24
votes
1answer
2k views

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
votes
20answers
2k views

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 ...
21
votes
3answers
2k views

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 ...
18
votes
6answers
1k views

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 ...
18
votes
0answers
679 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 ...
17
votes
1answer
660 views

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= ...
13
votes
3answers
2k views

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 ...
13
votes
3answers
1k views

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 ...
13
votes
3answers
450 views

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 ...
12
votes
2answers
367 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 ...
12
votes
2answers
725 views

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 ...
12
votes
0answers
515 views

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 ...
11
votes
1answer
789 views

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 ...
11
votes
5answers
504 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
votes
0answers
342 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 ...
10
votes
1answer
312 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 ...
10
votes
0answers
183 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 ...
9
votes
2answers
748 views

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?
9
votes
3answers
701 views

When finitely generated free algebras are finite

The variety (in the sense of universal algebra) of Boolean algebras, for example, has the property that finitely generated free algebras have finite cardinality; in that case specifically ...
9
votes
3answers
487 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 ...
9
votes
3answers
601 views

Algebraic axiomatization for AB+BA^T operation on matrices

Let us consider a matrix algebra $Mat_{n\times n}(K)$, where $K$ is a field, $char K \neq 2.$ It is well-known that the axiomatization of commutator operation $[A,B]=AB-BA$ on matrix algebra leads us ...
8
votes
3answers
582 views

What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
8
votes
2answers
614 views

Right actions of operads and monads

Given an operad $A$, there is an associated monad $M_A$ given by $M_A(X) = \coprod_n A(n) \otimes X^{\otimes n}$ such that being an $A$-algebra and being an algebra over the monad $M_A$ is the same ...
8
votes
1answer
338 views

Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
7
votes
4answers
328 views

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?
7
votes
2answers
296 views

Is there a general result that theorems about finite structures proved in ZFC can be proved in ZF?

The title question is too vague so let me be specific. Much of modern finite semigroup theory uses profinite semigroups and properties of profinite semigroups that depend on the existence of prime ...
7
votes
1answer
242 views

Varieties of groups with infinite relatively free group of rank 2 finite, infinite in rank 3

Does there exist a variety of groups $\mathfrak{V}$ such that the relatively $\mathfrak{V}$-free group of rank 2 is finite, but the relatively $\mathfrak{V}$-group of rank 3 is infinite? (In other ...
7
votes
1answer
598 views

Lattice-ordered commutative monoids

By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...
7
votes
1answer
499 views

Does ⋄ generate all De Morgan algebras?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...
7
votes
0answers
270 views

When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...
7
votes
0answers
405 views

The name for a partial order

In a recent paper of mine, my co-authors found that a partial order that we were using was contained in a paper by Kundgen. In it, he called it "right-shifted partial order". I was curious, and found ...
7
votes
0answers
520 views

Counting and understanging commuting functions.

Fix a positive integer $n$, and consider the functions from a set of size $n$ to itself. Let $cp(n)$ denote the number of ordered pairs $\langle f,g \rangle$ of these functions which commute, i.e., ...
6
votes
5answers
482 views

Computing colimits in a Lawvere theory

Let Prod(C, D) be the set of finite-product preserving functors from C to D. Is it true that for any Lawvere theory L, Prod(L, Set) has small colimits? This seems to be the case, as it is invoked ...
6
votes
3answers
839 views

Does “finitely presented” mean “always finitely presented”, considered in general

I'm wondering about the question, "If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?" I know this is true for groups and for ...
6
votes
3answers
433 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 ...
6
votes
2answers
241 views

Is HSP(A) = ISP(A) decidable?

Let $A$ be a finite algebra for some finitary signature. Is it decidable whether $\mathbb{H}\mathbb{S}\mathbb{P}(A) = \mathbb{I}\mathbb{S}\mathbb{P}(A)$? That is, whether the variety ...
6
votes
3answers
202 views

Axiomatizing orientation in the complex plane

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the ...
6
votes
1answer
227 views

Generalizations of Birkhoff's HSP Theorem

Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and ...
6
votes
1answer
371 views

Two questions about commutative theories

Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...
6
votes
5answers
529 views

Binary operations compatible with the usual order on the reals

An officemate passes along the following natural-seeming question: Say that a binary operation $\oplus$ is compatible with the usual order $\leq$ on $\mathbb{R}$ if for any $w, x, y, z$ in ...
6
votes
0answers
179 views

Binary Operation on a Cubic Surface

If $P$ and $Q$ are two sufficiently general points on a cubic surface, the line between them intersects the cubic surface at a unique third point, $f(P,Q)$. This gives a binary operation on (generic) ...
6
votes
0answers
527 views

Invertible elements in generalized fields

Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For ...
5
votes
3answers
860 views

Monad arising from operad

It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...
5
votes
3answers
170 views

Locally finite varieties which are not finitely generated

Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...
5
votes
3answers
622 views

Presenting Lawvere theories?

(Re)Reading Lawvere theories versus classical universal algebra, I was reminded of a question I have had for quite some time: What is the best way to present Lawvere theories? By present, I mean ...
5
votes
1answer
221 views

Do plain functors out of monoidal categories factor into a nontrivial monoidal part and a plain part?

Given symmetric monoidal closed categories $C, D$, a symmetric monoidal (not closed!) functor $F:C\to D$ factors into two parts: the first is a symmetric monoidal closed functor from $C$ to a ...
5
votes
1answer
159 views

What do algebraic theories with strictly terminal trivial models look like?

By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...
5
votes
1answer
303 views

Is there a general theory of “representation theorems”?

Let $V$ and $W$ be classes of algebraic structures, and suppose we have some canonical way of constructing objects of $W$ from objects of $V$. Let's call this construction $C$, so that for all $A\in ...
5
votes
1answer
280 views

What are the essential properties of algebraic closure on an arbitrary structure?

Define the "model theoretic" notion of a closure function as follows: Definition (1): Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the ...