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2
votes
1answer
461 views

Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.) Feel free to re-tag if you can think of ...
1
vote
1answer
164 views

Bodnarchuk, Kaluzhnin, Kotov, Romov’s Theorem on inclusion of Polymorphism ($Pol \rho \subseteq Pol \sigma$)

Bodnarchuk, Kaluzhnin, Kotov, Romov’s paper [1] is well-known. Anne Fearnley [2] infered from it the following theroem and used it to prove the inclusion of polymorphisms. Theorem (Bodnarchuk, ...
2
votes
1answer
140 views

How to prove that a binary relation is a strongly rigid relation? i.e. Polρ only contains projections

I first quote a definition from Clone theory in Universal Algebra: A binary relation $\rho$ on a set U is strongly rigid if every universal algebra on U such that $\rho$ is a subuniverse of its square ...
3
votes
2answers
184 views

Coequalizers in an Eilenberg-Moore category

Last month I proved that some category $\mathbf C$ that I happen to care about is isomorphic to the Eilenberg-Moore category for a monad on the category of bounded posets $\mathbf{BPos}$. I know from ...
3
votes
1answer
397 views
3
votes
2answers
482 views

Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...
3
votes
1answer
216 views

Iterating Monad-Comonads structures

Let $(T, \mu , \eta )$ a monad on the category $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle: \mathscr{C}^T \to \mathscr{C}$ where ...
5
votes
1answer
298 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 ...
10
votes
1answer
298 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 ...
11
votes
0answers
505 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 ...
6
votes
0answers
178 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) ...
5
votes
2answers
174 views

Algebras with finite essential arity

We are talking about algebras in the universal algebraic sense, that is, a set that $A$ is equipped with a set $F$ of finitary operations on $A$. Definition: An algebra $(A,F)$ is said to have ...
11
votes
5answers
496 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 ...
12
votes
2answers
362 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 ...
6
votes
3answers
430 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 ...
2
votes
1answer
200 views

Methods to tell if a magma has idempotents

(Disclaimer: below, when I say "compact" I mean "compact Hausdorff.") I asked a version of this question on math stackexchange ...
4
votes
4answers
279 views

What is an ideal-supporting algebra?

I'm sorry if this question is too elementary, but I asked it at MathStackExchange and it received no responses. On the Wikipedia page for congruence relation it mentions how for groups and rings, ...
1
vote
0answers
49 views

Looking for a uniform explanation of algebras with canonical generators.

Let $\mathcal{V}$ be a finitary variety i.e. the algebras for a signature whose operations have finite arity and for some arbitrary set of equations. Then any algebra $A \in \mathcal{V}$ has a ...
2
votes
1answer
118 views

Distributive lattice embedding into a finite lattice.

Suppose one has an inclusion $\iota : D \hookrightarrow S$ where $D$ is a finite distributive lattice and $S$ is a finite join-semilattice. If $\iota$ preserves all meets and joins one can show that ...
17
votes
1answer
638 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= ...
3
votes
1answer
263 views

Classification of plethories over $\mathbb{Q}$

Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). ...
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 ...
4
votes
3answers
561 views

Survey of finite axiomatizability for relational theories?

An $L$-theory $T$ is finitely axiomatizable if there is a finite set $A$ of $L$-sentences with the same consequences as $T$, i.e. such that $M \models T$ iff $M \models A$ for every $L$-structure $M$. ...
3
votes
2answers
180 views

Are the categories of bialgebras and weak biaglebras cocomplete/algebraic?

I am trying to verify whether the category of bialgebras and then the category of weak bialgebras are cocomplete. We know that algebraic categories are cocomplete (Thm. 4.5 of this book), so I have ...
1
vote
1answer
186 views

Cyclic Distribution on the reals?

Do there exist binary operators *, **, and *** on the real numbers, such that ...
3
votes
1answer
301 views

How strong is the condition that an operad splits, i.e. O(n)=O(s)xO(n-s)?

I recently proved something for the operad $Comm$ valued in a model category $M$ and am trying to generalize it to other symmetric operads. I'm very new to operads, so please forgive me if there are ...
2
votes
1answer
208 views

What to call substructures in universal algebra in which we restrict the signature?

Suppose $\Sigma$ is a signature in the sense of universal algebra and $\Sigma' \subseteq \Sigma$ a sub-signature. Every $\Sigma$-algebra is also a $\Sigma'$-algebra in a forgetful way. Suppose $A$ is ...
7
votes
2answers
294 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 ...
8
votes
3answers
579 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 ...
4
votes
0answers
205 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
2
votes
0answers
228 views

Algebras with supremum-founded subalgebra lattice

I am interested in algebras whose subalgebra lattice is supremum-founded. Let us call those algebras small. A complete lattice $(L, \leq)$ is called supremum-founded, if for any two elements $x < ...
9
votes
3answers
700 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 ...
6
votes
0answers
526 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 ...
7
votes
0answers
269 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 ...
18
votes
0answers
677 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 ...
5
votes
3answers
618 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 ...
2
votes
1answer
278 views

Prove that algebraic structure is not semiring?

I have a [finite] lattice enriched with additional operations. I would like either: find a pair of binary operations (and constants) satisfying semiring laws, or prove that no such operations exist ...
4
votes
0answers
252 views

Embedding of relatively free groups of bigger rank into ones of smaller rank

This question is prompted by this one by Arturo Magidin: whether there exist varieties of groups in which the relatively free group of rank 2 is finite, and the relatively free group of rank 3 is ...
7
votes
1answer
239 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 ...
11
votes
1answer
755 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
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 ...
12
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 ...
7
votes
0answers
400 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 ...
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 ...
3
votes
1answer
179 views

Can ordinary schemes be described as sheves of algebras/models for a certain Lawvere theory?

Consider the definition of a $\mathcal{C}^{\infty}$-scheme given in Dominique Joyce's "Algebraic geometry over C-infty rings". As far as I uderstand (not being an expert either in C-infty rings nor in ...
3
votes
2answers
1k views

Why Lawvere theories have finite products? and more

According to Wikipedia, a Lawvere theory consists of a small category $L$ with (strictly associative) finite products and a strict identity-on-objects functor $I:\aleph_0^\text{op}\rightarrow L$ ...
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 ...
4
votes
1answer
374 views

Higher-order preservation theorems?

The Łos-Tarski preservation theorem states that a set of formulas $F$ of first-order language $L$ is preserved under substructures for models of theory $T$ in $L$ precisely when $F$ is equivalent ...
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 ...
5
votes
3answers
850 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 ...