What's the term for making a new partition from two existing partitions of a space? For example, if you have a partition $X = \{A, B\}$ and $Y = \{C, D\}$, and you form the partition $\{AC, AD, BC, BD\}$, what do you call it? (Example: on the interval $[0, 4]$, $X = \{[0, 2), [2, 3]\}$ and $Y = \{[0, 1), [1, 3]\}$, so the new partition is $\{[0, 1), [1, 2), [2, 3]\}$.)
For example, do you call it "the partition formed through intersecting $X$ and $Y$"? Or "the partition formed through crossing $X$ and $Y$"?
Thank you!