Semigroups: Exploring the Logic Behind the Name

[airpocket]

This is a stupid question, but perhaps somebody else has had the same stupid question before and found an answer.

Why is a <b>semi</b>group so named? If a group were a set and a binary operation satisfying 2 additional properties, then semigroup would be the perfect name, since it satisfies only 1 additional property, but that's not the case.

Is there some logic to the name? Is it because $$\frac{3}{2} = 1$$ in integer arithmetic ;-)? Wikipedia and other sources are no help, and I'm hoping there is a logic to the name, as mathematical terminology is usually extremely logical.

 

[masudr]

It's called that because it's kind of half way to being a group. Think of it like this:

1. You have a set S with a binary operation S x S -> S, then we call that a magma.

2. Make that operation associative, then we call it a semigroup.

3. Include an identity element of the operation within the set, then it becomes a monoid.

4. Include an inverse element for each element under the operation within the set, then it becomes a group.

And so on...

 

[airpocket]

Thanks for the reply. You labeled your points 1 to 4 but you could just as easily split set and binary operation on the set into 2 separate points and then you'd have 1 through 5 (and semigroup=3, group=5).

And that doesn't change the fact that the standard formulation is 1 property for a semigroup, 2 for a monoid, and 3 for a group.

If it were the case that originally the binary operation on the set was explicitly counted as a property and there really were 2 for a semigroup and 4 for a group, then your explanation would be perfect, but I haven't seen it explained in that way before, so it's unconvincing.

It doesn't really matter, but often I've found mathematical terminology to be perfectly precise, and this seems to fall short of that standard.

Any other thoughts?

p.s. Apologies for screwing up the formatting in the first post. I realize now that HTML doesn't work, but I'm not sure why the LaTeX got escaped instead of showing up. It doesn't seem that I can edit it to fix it.

 

[masudr]

I think closure was originally accepted as a proper axiom, but it is kinda self-evident in the definition of the binary operation and its domain and range. The reason for specifying closure is because it is an axiom that can be overlooked when checking to see if an algebraic system is a group or not. (e.g. closure of integers under division).

If we take closure as a meaningful axiom, then a semigroup has 2 properties, and a group has 4. What do you think? It doesn't quite fit the "perfectly precise" (and I agree, maths terminology often is exactly that), but I'd say it's good enough. I am a physicist, however...

 

[airpocket]

If we take closure as a meaningful axiom, then a semigroup has 2 properties, and a group has 4. What do you think? It doesn't quite fit the "perfectly precise" (and I agree, maths terminology often is exactly that), but I'd say it's good enough. I am a physicist, however...

Yeah, closure as an explicit axiom does make it sound like a much better match to me, so that's probably it if you've seen it mentioned like that as an axiom. You've convinced me.

Thanks again for your help.