**33**

votes

**16**answers

3k 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

**20**answers

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 ...

**24**

votes

**1**answer

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 ...

**21**

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 ...

**18**

votes

**6**answers

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

**2**answers

487 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 ...

**18**

votes

**0**answers

746 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

**1**answer

781 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= ...

**15**

votes

**4**answers

637 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 ...

**14**

votes

**3**answers

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

**2**answers

399 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 ...

**13**

votes

**3**answers

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

**2**answers

934 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

**3**answers

410 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}} ...

**12**

votes

**1**answer

372 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 ...

**12**

votes

**1**answer

262 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 ...

**11**

votes

**1**answer

832 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

**5**answers

606 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

**0**answers

673 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

**0**answers

359 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

**3**answers

531 views

### Natural associative law for a ternary “group”?

Suppose one were to define a group-like structure based on a set $G$
with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$.
One possible definition for the ...

**10**

votes

**4**answers

527 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 ...

**9**

votes

**2**answers

785 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

**3**answers

646 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 ...

**9**

votes

**3**answers

721 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

**3**answers

504 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

**3**answers

609 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 ...

**9**

votes

**1**answer

155 views

### Does every Lawvere theory arise in this way?

By a Lawvere theory, I mean a finite-product category $\mathsf{T}$ equipped with a distinguished object, such that every object of $\mathsf{T}$ can be expressed as a finite product of the ...

**9**

votes

**0**answers

62 views

### Has any work been done on the concept intermediate between symmetric operads and Lawvere theories?

There ought to be a variant of universal algebra in which we replace $\mathbf{Set}$ with the monoidal category $\bot\mathbf{Set}$, by which I mean the monoidal category whose underlying category ...

**8**

votes

**4**answers

485 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?

**8**

votes

**2**answers

628 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

**1**answer

373 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 ...

**8**

votes

**0**answers

298 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

**2**answers

310 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

**3**answers

455 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 ...

**7**

votes

**1**answer

251 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

**1**answer

90 views

### A decision problem for clones

E. Post proved that there are only countably many clones on a two-element set (classes of operations closed under superposition and containing all projections). All these clones are finitely ...

**7**

votes

**1**answer

684 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

**1**answer

518 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

**0**answers

187 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) ...

**7**

votes

**0**answers

441 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

**0**answers

538 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

**5**answers

512 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

**3**answers

856 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

**2**answers

462 views

### Primitive Recursive Arithmetic via Universal Algebra

From the wikipedia article on Primitive Recursive Arithmetic (see http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic):
"Primitive recursive arithmetic, or PRA, is a quantifier-free ...

**6**

votes

**2**answers

269 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

**3**answers

222 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

**1**answer

285 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

**1**answer

390 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

**5**answers

536 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 ...