Decreasing event

An event $A$ in a partially ordered set is a decreasing event if $x\in A$ and $y\le x$ imply $y\in A$. Its complement is an increasing event. In bond percolation, with an open edge coded by $1$, closing more edges cannot destroy a decreasing event.

For example, the event that no open path connects two fixed vertices is decreasing; the event that such a path exists is increasing. Two decreasing events are positively associated under a product measure, since both of their complements are increasing events. An increasing event and a decreasing event instead satisfy the negative correlation of increasing and decreasing events. The order convention must be stated when the coordinates encode something other than open edges.