11
votes
4answers
201 views

Why are particular combinations of algebraic properties “better” (richer and more pervasive) than others?

I am a second-year graduate student of pure mathematics. Like most graduate students, I have been exposed to many types of algebraic structures, and it seems standard for the major emphasis, if not ...
6
votes
1answer
78 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
1
vote
0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
3
votes
2answers
84 views

For what categories do category algebras exist?

The monoid algebra $R[M]$, for a commutative ring $R$ and a monoid $M$, can be described as the free $R$-algebra on $M$. We think of $R[M]$ as the set of finite formal sums of elements of $M$ with ...
3
votes
2answers
124 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
17
votes
2answers
425 views

A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking ...