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7
votes
0answers
514 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., ...
4
votes
2answers
241 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 ...
2
votes
1answer
320 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 ...
6
votes
5answers
469 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 ...
4
votes
2answers
549 views

Is there a notion of congruence relation for essentially algebraic structures?

In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations. A congruence ...
1
vote
4answers
744 views

Is monomorphism going in both directions sufficient for isomorphism?

In category theory, it seems that a monomorphism from $A$ to $B$ and one from $B$ to $A$ should be enough to guarantee isomorphy, but it doesn't seem to be so. (If I'm right then there's something ...
4
votes
0answers
259 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 ...
1
vote
1answer
194 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 ...
9
votes
2answers
722 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?
16
votes
6answers
995 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 ...
1
vote
2answers
824 views

What is a semigroup or, what do I do with that associativity proof?

Mathematically, I know what a semigroup is: It is a set S along with an associative binary operation $* : S \times S \rightarrow S$. So far, so good. From a computational perspective, one can ...
8
votes
2answers
602 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 ...
2
votes
1answer
448 views

Higher-order, multi-sorted, non purely equational version of universal algebra ?

I have been looking around, unsuccessfully, for generalizations of universal algebra based on higher-order logic (rather than first order) and where the relations are not purely equational. ...
3
votes
5answers
866 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 ...
3
votes
1answer
174 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. ...
3
votes
2answers
264 views

Terminology: Name for a homomorphism from the free object?

Is there a standard name for taking a homomorphism from the free object over an algebraic structure? Roughly speaking, this should amount to evaluation of any element of the free object under the ...
1
vote
1answer
480 views

Coequalizer in the category of primitive recursive functions

What does a coequalizer in the category of primitive recursive functions look like? I know that in Set, a coequalizer is a minimum congruence but...what is it in particular in the category of ...
7
votes
1answer
482 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 ...
4
votes
2answers
419 views

Is every lattice the fixed-point set of an order endomorphism of ⋄^n?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) Let ⋄ be the 4 element lattice τ / \ i j \ / f Is every lattice isomorphic to the fixed point lattice of some ...