Definition:Binary Mess

Definition
Let $S$ be a set.

Let $I$ be the set of all finite subsets of $S$.

Let $M \subseteq \Bbb B^I$ be a set of mappings from finite subsets of $S$ to a Boolean domain.

Suppose that $M$ satisfies the following:
 * For every $P \in I$, there exists some $t \in M$ such that:
 * $\Dom t = P$


 * For every $P \in I$ and $t \in M$:
 * $t \restriction_P \in M$
 * where $t \restriction_P$ is the restriction of $t$ to $P$.

Then $M$ is a binary mess on $S$.