# 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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### 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 ...
1k 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 ...
294 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 ...
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 ...
173 views

### 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 ...
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 ...
304 views

### Equational theory in the signature (+,*,0,1) of sedenions and beyond

Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, ...
299 views

### "Generalized theory of polynomials" for a given commutative Lawvere Theory

I am trying to understand Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing. Let $R$ be a commutative, associative ring with unit. We can ...
279 views

### What is going on in the field of algebraic logic these days?

I'm doing my masters in Mathematics and took a class in universal algebra and there I learned that for example: Boolean algebras have direct connection with classical logic, Heyting algebras with ...
531 views

### Is there any theorem achieving Conway's "Mathematician's Liberation Movement"

John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
204 views

### Which semirings have enough injectives in their category of modules?

Let $R$ be a semiring and $Mod_R$ its category of modules. That is, $R$ is a monoid in the monoidal category of commutative monoids and $Mod_R$ is its category of modules in the usual sense. Question ...
187 views