Modeling $(x > 0 \wedge y > 0) \Leftrightarrow z > 0$ in a linear program: impossible?
A linear program consists of a finite collection of inequalities of the form $\sum_i a_ix_i \leq b_i$. Each such inequality defines a closed convex set, and so their intersection defines a closed convex set. Any domain which is not closed and convex thus cannot be represented using linear programming constraints.