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 〉 |