Partitions (of some underlying set) can be thought of as variables like this:
The number of values the variable can take on is the number of parts in the partition.
Every element of the underlying set has some value for the variable, namely, the part that that element is in.
Another way of looking at it: say we’re thinking of a variable v:S→D as a function from the underlying set S to v’s domain D. Then we can equivalently think of v as the partition {{s∈S∣v(s)=d}∣d∈D}∖∅ of S with (up to) |D| parts.
In what you quoted, we construct the underlying set by taking all possible combinations of values for the “original” variables. Then we take all partitions of that to produce all “possible” variables on that set, which will include the original ones and many more.
Partitions (of some underlying set) can be thought of as variables like this:
The number of values the variable can take on is the number of parts in the partition.
Every element of the underlying set has some value for the variable, namely, the part that that element is in.
Another way of looking at it: say we’re thinking of a variable v:S→D as a function from the underlying set S to v’s domain D. Then we can equivalently think of v as the partition {{s∈S∣v(s)=d}∣d∈D}∖∅ of S with (up to) |D| parts.
In what you quoted, we construct the underlying set by taking all possible combinations of values for the “original” variables. Then we take all partitions of that to produce all “possible” variables on that set, which will include the original ones and many more.
Makes perfect sense, thanks!