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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 ⟩