Here are examples of multiplications in family F+ that, although valid, would probably be deemed trivial. In each case, a × b = a × c, so they are certainly not cancellative.
With commutativity and rotativity, a × b = a × c more broadly implies that all products of two distinct atoms are equal.
Script ⟨ 0; 0 ⟩:
a × a = 0 | b × a = 0 | c × a = 0 | ||
a × b = 0 | b × b = 0 | c × b = 0 | ||
a × c = 0 | b × c = 0 | c × c = 0 |
Script ⟨ +1, 0, 0; 0, 0, 0 ⟩:
a × a = a | b × a = 0 | c × a = 0 | ||
a × b = 0 | b × b = b | c × b = 0 | ||
a × c = 0 | b × c = 0 | c × c = c |
Script ⟨ +1, +1, +1; +1, +1, +1 ⟩:
a × a = ⟨ +1, +1 +1 ⟩ | b × a = ⟨ +1, +1 +1 ⟩ | c × a = ⟨ +1, +1 +1 ⟩ | ||
a × b = ⟨ +1, +1 +1 ⟩ | b × b = ⟨ +1, +1 +1 ⟩ | c × b = ⟨ +1, +1 +1 ⟩ | ||
a × c = ⟨ +1, +1 +1 ⟩ | b × c = ⟨ +1, +1 +1 ⟩ | c × c = ⟨ +1, +1 +1 ⟩ |
Script ⟨ +2, +2, +2; −1, −1, −1 ⟩:
a × a = ⟨ +2, +2, +2 ⟩ | b × a = ⟨ −1, −1, −1 ⟩ | c × a = ⟨ −1, −1, −1 ⟩ | ||
a × b = ⟨ −1, −1, −1 ⟩ | b × b = ⟨ +2, +2, +2 ⟩ | c × b = ⟨ −1, −1, −1 ⟩ | ||
a × c = ⟨ −1, −1, −1 ⟩ | b × c = ⟨ −1, −1, −1 ⟩ | c × c = ⟨ +2, +2, +2 ⟩ |