> 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:

|abcd
-+----
a|abcd
b|bccc
c|cccc
d|dccc