Definition:Binary Mess

Definition

Let $S$ be a set.

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

Let $\Bbb B$ be a Boolean domain.

Let $M$ be a set defined as:

$\ds M \subseteq \bigcup_{P \mathop \in I} \Bbb B^P$

where $\Bbb B^P$ denotes the set of all mappings from $P$ to $\Bbb B$.

That is, such that $M$ is a set of mappings from finite subsets of $S$ to $\Bbb B$.

Suppose that $M$ satisfies the binary mess axioms:

$(\text M 1)$	$:$			$\ds \forall P \in I: \exists t \in M:$	$\ds \Dom t = P $				Every finite subset of $S$ is the domain of some mapping in $M$
$(\text M 2)$	$:$			$\ds \forall P \in I: \forall t \in M:$	$\ds t {\restriction_P} \in M $				The restriction of every mapping in $M$ to every finite subset of $S$ is in $M$

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

Let $S$ be a set.

Let $M$ be a binary mess on $S$.

Let $f : S \to \Bbb B$ be a mapping from $S$ to a Boolean domain.

Then, $f$ is consistent with $M$ if and only if, for every finite subset $P \subseteq S$:

$f {\restriction_P} \in M$

where $f {\restriction_P}$ denotes the restriction of $f$ to $P$.

\((\text M 1)\)	$:$			\(\ds \forall P \in I: \exists t \in M:\)	\(\ds \Dom t = P \)				Every finite subset of $S$ is the domain of some mapping in $M$
\((\text M 2)\)	$:$			\(\ds \forall P \in I: \forall t \in M:\)	\(\ds t {\restriction_P} \in M \)				The restriction of every mapping in $M$ to every finite subset of $S$ is in $M$