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Results tagged with universal-algebra
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user 290
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).
33
votes
7
answers
3k
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 ax …
- 118k
15
votes
Why are so few operations with arity bigger than 2?
Here's one I learned from Todd Trimble. Giving a set $X$ the structure of a compact Hausdorff space is the same as equipping $X$ with $J$-ary operations $X^J \to X$ for every set $J$, one for each ul …
17
votes
In what sense are operads "better" than PROPs?
One thing you can do with an operad that you cannot do with a prop is write down a monad such that algebras over the monad correspond to algebras over the operad. For example, Hopf algebras have a pro …
- 118k
16
votes
1
answer
961
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 …
- 118k
17
votes
Relation between monads, operads and algebraic theories (Again)
Lawvere theories can be thought of as "cartesian operads." That is, we have an analogy
$$\text{Lawvere theories} : \text{cartesian monoidal categories} :: \text{operads} : \text{symmetric monoidal c …
- 118k
42
votes
Accepted
Why are ring actions much harder to find than group actions?
First, I am not really sure what you mean by "it is hard to come across a general theory of ring actions." This is precisely module theory! If $f : R \to S$ is any ring homomorphism whatsoever, then c …
- 118k
7
votes
On the tree-ishness of magmas and the stringiness of groups
People have done lots of interesting works along these lines. This is a discussion that would be best had at a blackboard to facilitate easy drawing, but here is one version of the story among many. F …
- 118k
4
votes
What is a "general" relation algebra?
I don't know anything about the history and context of this idea of a relation algebra but the definition doesn't smell like "the right one" to me, and for a simple reason: why restrict attention to b …
- 118k
11
votes
Are there any non-conjugation "extendible automorphisms" in the category of finite groups?
Not a complete answer. Your definition of an extendible map says that $\beta$ is an endomorphism of the forgetful functor $U$ from the under category $G \downarrow \mathrm{FinGrp}$ to $\mathrm{Set}$ s …
- 118k