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Stream: theory: category theory

Topic: Is there a name for a monoid with extra operations?

Davi Sales Barreira (Aug 21 2022 at 11:55):

This question is a bit odd, but I'm quite juvenile in abstract algebra, so perhaps someone can understand what I mean and help out. I've asked this in Math Exchange, but I got no answers. I have a hunch that this might be related to $\mathbf{Poly}$, but I'm not sure.

Suppose that I can define an operation $+$ and $\times$, such that $\forall m,n \in M$ we have $m+n \in M$ and $m\times n \in M$, and also, $m+n = n +m$,

$$m \times (n + l) = m\times n + m \times l \\ m \times n \neq n \times m \\ m* (n_1\times n_2) \neq (m *n_1) \times (m * n_2)$$

Hence, I have $(M,e,*,+,\times)$. My question then is what is this thing called? I'm trying to learn more about such structures, cause I actually have something like this in a program that I'm coding.

Just to give more perspective. The $*$ operator is like a composition (nesting), where $m*n$ means something like put $n$ inside of $m$ (like the $\vartriangleleft$ in $\mathbf{Poly}$).
The $+$ is like an "append", and the $\times$ creates a new object.

Here is an actual example. Consider the category of all sets. Hence, $M$ is the class of all sets. The operator $*$ is the "nesting" operator. For example:

$$A := \{1,2\} \quad,B := \{a,b\} \implies A * B = \{1,2,\{a,b\}\} \\ A \times B = \{\{1,a\},\{1,b\},\{2,a\},\{2,b\}\} \\ A + B = \{1,2,a,b\}$$

John Baez (Aug 21 2022 at 13:47):

Davi Sales Barreira said:

This question is a bit odd, but I'm quite juvenile in abstract algebra, so perhaps someone can understand what I mean and help out. I've asked this in Math Exchange, but I got no answers. I have a hunch that this might be related to $\mathbf{Poly}$, but I'm not sure.

Suppose that I can define an operation $+$ and $\times$, such that $\forall m,n \in M$ we have $m+n \in M$ and $m\times n \in M$, and also, $m+n = n +m$,

$$m \times (n + l) = m\times n + m \times l \\ m \times n \neq n \times m \\ m* (n_1\times n_2) \neq (m *n_1) \times (m * n_2)$$

Your question is a bit odd because you start by mentioning two operations and then you give a (non-)equation involving a third, $\ast$. But I guess you meant to start by saying there's also a third operation $\ast: M \times M \to M$.

John Baez (Aug 21 2022 at 13:49):

Anyway, there's no simple general name for this structure because your third operation because you're not asserting any equations relating this operation to the other two.

John Baez (Aug 21 2022 at 13:52):

A binary operation $\ast : M \times M \to M$ not necessarily obeying any equation is called a [[magma]], so what you have is some structure involving $+$ and $\times$, something similar to a [[near-ring]], together with an unrelated magma structure involving $\ast$.

John Baez (Aug 21 2022 at 13:55):

(I said "similar to", because there are some differences.)