Excellent.

So welcome Zoomies.

We just basically came back to the old room where the stream is.

So I'll restart with the structures.

We basically came to somehow here.

We looked at the structure of groups.

Can you also post on the matrix channel

that we're in H19 again

back to the original one?

And so the idea is we have a collection of things,

in this case a base set, an operation

that doesn't drop you outside of the base set.

That is also associative.

And a unit element with some properties

namely composing with it under the operation,

doesn't change an object.

And we have an inverse function that kind of undoes composition.

OK?

That's something we've seen quite a lot.

For instance

the integers with addition form a group.

They don't form a group with multiplication

because we don't have an inverse.

Actually, we might have an inverse,

but the inverse function that we all know and love

dumps you outside.

It's the one over function, and one over two

is not a natural number.

So there's a very natural thing to do.

We might actually be interested in groups

that don't have inverses.

In a way, castrated groups.

But of course, we are positive people,

so we give it a nice sounding scientific name.

Indeed, we call that thing a monoid.

A monoid is something that doesn't have inverses,

but it still has a unit.

And indeed

the integers together with times

forms a monoid because we still have that one times x

is the same as x times one

and that is x.

OK?

We don't have an inverse.

We cannot undo.

And there are other interesting monoids around.

One of them is the strings with concatenation.

You can run two strings together.

You get another string.

If you run a string together with the empty string,