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Sep 29, 2011 at 17:21 comment added user16974 I appologize for not being clear. My intention is $A \subset \bigcup_{x \in C} {\rm Past}(x) \cup \bigcup_{x \in C} {\rm Future}(x)$.
Sep 29, 2011 at 15:48 comment added Emil Jeřábek David, while the OP’s terminology is not quite clear, I think you are a bit overdoing it. Poset and maximal chain are standard terms. It seems that the intended reading of 1 is $A\subseteq\bigcup_{x\in C}\mathrm{Past}(x)$ and $A\subseteq\bigcup_{x\in C}\mathrm{Future}(x)$.
Sep 29, 2011 at 15:30 comment added Colin Reid Judging by previous questions, I think the OP is feeling around for axioms that will ensure a reasonably tame automorphism group. My suggestion would be to change 1. to 'every finite set lies in an interval' or something like that.
Sep 29, 2011 at 15:18 history answered David CC BY-SA 3.0