**2**

votes

**1**answer

596 views

### Suggestions on the best introductory Model Theory texts [closed]

Any recommendations on the best texts for introducing Model Theory?

**4**

votes

**2**answers

576 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

**1**answer

277 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 ...

**6**

votes

**1**answer

390 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 $A\in V$...

**4**

votes

**2**answers

267 views

### Theorems (from clone theory) that can be stated only by using operations and their composition.

To state my question, I first have to explain what I mean by "using operations and their compositions only". In short, it means that I am looking for theorems from clone theory (or on operations and ...

**12**

votes

**1**answer

464 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**

votes

**0**answers

793 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 ...

**7**

votes

**0**answers

193 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) ...

**11**

votes

**5**answers

684 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 ...

**1**

vote

**1**answer

206 views

### What are sources for pathological and non-so-pathological Gabriel filters on commutative rings?

The heavy lifting in the theory of Gabriel filters is for noncommutative rings, and discussions I've been able to find all focus there.
I am trying to develop a theory of Gabriel-filter localization ...

**8**

votes

**0**answers

319 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 ...

**2**

votes

**1**answer

222 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 (http://math.stackexchange.com/questions/305186/left-continuous-magmas-...

**1**

vote

**0**answers

67 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

**1**answer

142 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

**1**answer

931 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= b^{-...

**6**

votes

**1**answer

405 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 ...

**3**

votes

**1**answer

318 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). ...

**4**

votes

**3**answers

837 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

**2**answers

202 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 ...

**6**

votes

**5**answers

531 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 here:...

**3**

votes

**1**answer

363 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

**1**answer

218 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 ...

**24**

votes

**3**answers

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 ...

**7**

votes

**2**answers

330 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 ...

**2**

votes

**1**answer

331 views

### Essential arities in quasi-varieties.

For my question, let us consider the following scenario.
We have a quasi-variety $\mathcal{A}$ generated by a finite algebra $\mathbf{M}$ (i.e. $\mathcal{A} = \mathbb{ISP}(\mathbf{M})$). Now, let $\...

**9**

votes

**3**answers

705 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 ...

**20**

votes

**0**answers

802 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 ...

**9**

votes

**3**answers

741 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 $|F_n|=2^{2^...

**4**

votes

**0**answers

219 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 ...

**24**

votes

**20**answers

3k 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 (constants)...

**2**

votes

**0**answers

236 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 < ...

**6**

votes

**0**answers

547 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 ...

**3**

votes

**1**answer

185 views

### Equational definition of Residuated Lattices

The usual axiomatization of residuated lattices involve using ≤. I know I can expand away the use of ≤ using a definition such as (x ≤ y) := (x ∧ y = x), but I fear I will get a set of messy axioms.
...

**4**

votes

**0**answers

269 views

### The graph of algebraic theories

Fix a logic $L$ and consider the category $\mathbf{AlgTh}_L$ with theories of $L$ as objects and theory interpretations as morphisms. For nice enough logics, this category has pushouts (which we will ...

**7**

votes

**0**answers

547 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., ...

**2**

votes

**1**answer

304 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

**0**answers

261 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 ...

**9**

votes

**3**answers

616 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 ...

**7**

votes

**1**answer

258 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

**0**answers

370 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

**1**answer

926 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 Poincaré-...

**14**

votes

**3**answers

3k 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

**0**answers

490 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 ...

**3**

votes

**1**answer

206 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 ...

**16**

votes

**3**answers

2k 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 ...

**3**

votes

**2**answers

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 $I:\aleph_0^\text{op}\rightarrow L$ ...

**3**

votes

**5**answers

1k views

### Defining 'free monoid' without Nat?

Is there a definition of what is a 'free monoid' which does not pre-suppose that the natural numbers has already been defined? The definitions that I have been able to track down all use the natural ...

**4**

votes

**1**answer

486 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 ...

**7**

votes

**3**answers

1k 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

**1**answer

241 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 "...