Michael wrote:

>Define \\((a, b, c) \leq (a', b', c')\\) if and only if \\(a \gt a', b \gt b' \, and \, c \lt c'\\).

I assume you are using the opposite convention I was using above where \\(x \le y \\) means "If we have \\(x\\) we can get \\(y\\)". I'm also assuming that you meant \\((a, b, c) \leq (a', b', c')\\) if and only if \\(a \ge a', b \ge b' \, and \, c \le c'\\), since under your original definition \\((a,b,c) \not\le (a,b,c)\\) since \\(a \not\gt a\\), \\( b \not\gt b\\) and \\(c \not\lt c \\).

Using this modified definition it is indeed the case that \\((2,1,0) \le (0,0,1) \\) as desired. But it is also the case that \\((2,1,0) \le (0,0,8) \\) which violates conservation of mass (probably undesired) as well as \\((1,1,0) \le (0,0,1) \\) which does not take the stoichiometry of the reaction into account (probably also undesired).

As far as I understood the question we need to take the set of all possible complexes into account, not just the ones resulting from valid applications of the Petri net.

>Define \\((a, b, c) \leq (a', b', c')\\) if and only if \\(a \gt a', b \gt b' \, and \, c \lt c'\\).

I assume you are using the opposite convention I was using above where \\(x \le y \\) means "If we have \\(x\\) we can get \\(y\\)". I'm also assuming that you meant \\((a, b, c) \leq (a', b', c')\\) if and only if \\(a \ge a', b \ge b' \, and \, c \le c'\\), since under your original definition \\((a,b,c) \not\le (a,b,c)\\) since \\(a \not\gt a\\), \\( b \not\gt b\\) and \\(c \not\lt c \\).

Using this modified definition it is indeed the case that \\((2,1,0) \le (0,0,1) \\) as desired. But it is also the case that \\((2,1,0) \le (0,0,8) \\) which violates conservation of mass (probably undesired) as well as \\((1,1,0) \le (0,0,1) \\) which does not take the stoichiometry of the reaction into account (probably also undesired).

As far as I understood the question we need to take the set of all possible complexes into account, not just the ones resulting from valid applications of the Petri net.