Definition of the n-Trace

Suppose $\Sigma$ is a set of inputs $\Sigma = \{\alpha_1,\alpha_2,\alpha_3...,\alpha_n\}$. Then the set $\wp(\Sigma)$is the power set, the set of all subsets of $\Sigma$.Each sentence can be viewed as an arbitrary source of information but here is considered as a Boolean value. We define traces as $T\subseteq\wp(\Sigma)$ is an n-trace over $\Sigma$ iff no n-partition of $\Sigma$ partitions every element of T.

We define various categories of n-traces as:

• $\forall$ n, $\forall \Sigma$, $\{\emptyset\}$ is an n-trace over $\Sigma$.
• If $x\in\Sigma$, then $\forall{\it n}, \{\{x\}\}$ is an n-trace over $\Sigma$.
• if T is an n-trace over $\Sigma$, and $\Sigma\subseteq\Sigma'$,then T is an n-trace over $\Sigma'$
• An n-trace T over $\Sigma$ is non-trivial iff it has neither $\emptyset$ nor any unit set as a member.
• An n-trace T is a minimal n-trace over $\Sigma$ iff no proper subset of T is an n-trace over $\Sigma$.
• An n-trace T is a full n-trace over $\Sigma$ iff any system B of supersets of T is an n-trace over $\Sigma$ iff T=B.
• An n-trace T is a proper n-trace over $\Sigma$ iff $\forall\Sigma'\subset\Sigma; T$ is not an n-trace over $\Sigma'$.
• An n-trace is said to be fundamental iff it is non-trivial, minimal, full and proper.