So what about the analog to the Cartesian product operation?

Well, it's much more complicated for labelled classes.

When we take the product of two classes, so we're going to say,

this first example 1, we call it the star product.

1 star product of 1, 2, 3, what we're going to get is

object of size 4, but they have to be numbered from 1 to 4.

And what the combinatorics requires is that we do that

numbering 1 to 4 but we do it in all consistent ways.

So in this case, the second argument is a sequence that's in increasing order.

So when we renumber,

we have to put that in increasing order on each one of the possibilities.

So we can assign 1, 2, 3, and 4 to the first one, and then the reaming labels,

we assign to the remaining 3 atoms, but they have to be in increasing order.

And here's a more complicated example,

where we take the star product of a two-cycle and a three-cycle.

Again, when we do that, we get 5.

We have objects consist of five atoms that have this structure,

a set of a two-cycle and a three-cycle.

The atoms have to all be numbered with 1 through 5, and

we have to do it in all possible ways.

So in this case, you can choose any label.

So there's only way to label the two-cycle.

And then you can choose 5, choose 2, or 10 different ways to label the two- cycle.

So, the first column is,

with the top one being 1, you can have 2, 3, 4, 5 for the bottom one.

Second column's top one being 2, 3 and 4.

Then, after you've labelled the two-cycle then

you take the remaining labels and assign them to the three-cycle,

but you have to be consistent and maintain the order.

So, for example, with 3, 4, in this case with 3, 4's, the labelled two-cycle,

the remaining labels are 1, 2, and 5, and they have to go in that order.

So that's the star product operation relabeled in all consistent ways.

And when we get to applications, we'll see why that's

not just intuitive, that's fundamental to working with labelled objects.

So with labelled objects, since you can tell the difference in the ordering,

we have more basic constructions and

it's a much richer set of operations that we're going to work with.

And actually, the ones that I'm giving work for labelled and

unlabeled are only the beginning.

Research is ongoing and people have developed many,

many more constructions than I'm going to present here.