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:
 * $\Dom t = P$
 * That is, every finite subset of $S$ is the domain of some mapping in $M$


 * For every $P \in I$ and $t \in M$:
 * $t {\restriction_P} \in M$
 * where $t {\restriction_P}$ is the restriction of $t$ to $P$.
 * That is, $M$ is closed under subsets.

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