Defining Simplicity in Magmas

[Stephen Tashi]

"simple" magmas?

I gather the modern term for a set with a closed binary operation is "magma" and that the old term "groupoid" now applies to something in category theory.

Is there a standard definition for a "simple" magma? For example, I think its straighforward to define the direct product of two finite magmas. So one thought is that a finite magma could be called "simple" if it is not isomorphic to the direct product of two smaller magmas.

(An amusing obstacle to looking up facts about magmas on the web is that the computer algebra software called Magma is the subject of so many links.)

 

[rasmhop]

There is a standard definition of a simple object in a finitely complete category with an initial object 0. See:
http://ncatlab.org/nlab/show/simple+object
for the general definition.

A congruence on a magma M is an equivalence relation ~ on M such that a ~ b and c ~ d implies ac ~ bd. For any such congruence we get a quotient magma M/~ in the usual manner by identifying x,y in M if x ~ y.

We say that M is simple if M has precisely 2 quotient magmas: 0 and M.

I don't know of any applications of this, or whether there is any use for this notion. However this notion agrees with the general notion of a simple object which generalize the notion of a simple object in an abelian category or a simple group.

EDIT:
To see why your proposed definition is no good we can take an example from group theory where a similar issue arises. Let C4 be the cyclic group {0,1,2,3} and let C2 be the cyclic group {0,1}. We then have a short exact sequence
$$0 \to C_2 \xrightarrow{\times 2} C_4 \xrightarrow{\times 2} C_2 \to 0$$
$C_4$ is not a simple magma (or group) because it has a quotient object $C_2$. However it can't be written as the product of two non-trivial magmas (or groups). One may interpret the existence of these non-simple magmas (or groups) that are not products as a result of the fact that there exists non-split exact sequences in the category of magmas (or groups).

Actually now that I think a bit about it the explanation may be a bit more complicated for magmas than for groups. In group theory there is a 1-1 correspondence between normal subgroups and congruences, but I am not sure if we can define something similar for magmas. Perhaps if we restricted to magmas with a cancellation law (ab=ac implies b=c). I'm not really sure and in any case know next to nothing of magmas so don't take anything I say on faith.

 

[Stephen Tashi]

My desire for "simplicity" must at odds with the idea of simplicity for groups. To use the example of $C_4$. Let's write the group operation table in the unusual order of:$$\begin{array}{c|cccc} * & 0 & 2 & 1 & 3\\ \hline 0 & 0 & 2 & 1 & 3 \\ 2 & 2 & 0 & 3 & 1 \\ 1 & 1 & 3 & 2 & 0 \\ 3 & 3 & 1 & 0 & 2 \end{array}$$

Define the sets $A = \{0,2\}, B = \{1,3\}$ and define the product $(X)(Y)$ of two sets as the set of those results formed by a product of an element from $X$ times an element of $Y$.

Then one can see "within" the table for $C_4$ the pattern:

$$\begin{array}{c|cc} * & A & B \\ \hline A & A & B \\ B & B & A \\ \end{array}$$To me, the natural extension of that idea to a magma can be illustrated by a magma with the table

$$\begin{array}{c|cccc} * & p & q & r & s\\ \hline p & p & p & s & r \\ q & q & q & s & s \\ r & r & r & p & q \\ s & r & s & q & p \end{array}$$

where we can define the sets $A = \{p,q\}, B= \{r,s\}$ and see the same pattern within the table.

On the one hand this idea of "non-simplicity" conveys the thought that there is a simpler pattern inside an operation table. However, it doesn't (to me) convey a notion that something is "not simple" if it can be systematically constructed from less complicated pieces. The ability to visualize a pattern within a table doesn't guarantee that you can take that pattern and combine it with some other component to construct the table from simple pieces. (That's just a commentary on the type of "simplicity" I want to know about - not a proof that the conventional definition of simplicity should be changed!)