> Is that right?
It's certainly a monoid, but I don't think it respects the partial order structure. Since \\(c \le a\\) and \\(d \le d\\), we expect \\(c \otimes d \le a \otimes d\\). But \\(c \otimes d = c \not\le d = a \otimes d\\).
Here's a multiplication table I built out of your rule: