Hey mastermind2007.

I'm wondering if there are two different kinds of orders here of if the partial order used refers to the same order.

Also do you know the partial order properties of x with respect to y?

The z <= x and z <=y implies that x <= y if we are talking about the same partial order. This means that there is a point somewhere where everything is "less than" some particular value of y which means that if a <= y then everything gets filtered on this condition.

I get the feeling this is needed because of the partial ordering axiom, but I'm not completely sure since this rule is part of the ordering axioms themselves.

When I read the definition, it seems that all this thing is doing is classifying one subset to be <=, it's complement to be not part of the filter and then keeping the subset that satisfies the partial order with respect to some particular reference point, where the ordering can be whatever the hell it wants to be as long as it divides the set into two groups (filtered and non-filtered) and carries some additional ordering within the filtered subset characterized by the partial order itself.