We now want to extend our notion of orthogonality to conditional orthogonality. This will take a bit of work. In particular, we will have to first extend our notions of partition generation and history to be defined on partitions of subsets of S.
4.1 Generating a Subpartition
Definition 20 (subpartition). A subpartition of a set S is a partition of a subset of S. Let SubPart(S)=⋃E⊆SPart(E) denote the set of all subpartitions of S.
Definition 21 (domain). The domain of a subpartition X of S, written dom(X), is the unique E⊆S such that X∈Part(E).
Definition 22 (restricted partitions). Given sets S and E and a partition X of S, let X|E denote the partition of S∩E given by X|E={[e]X∩E∣e∈E}.
Definition 23 (generating a subpartition). Given a finite factored set F=(S,B), and X∈SubPart(S)</
We now want to extend our notion of orthogonality to conditional orthogonality. This will take a bit of work. In particular, we will have to first extend our notions of partition generation and history to be defined on partitions of subsets of S.
4.1 Generating a Subpartition
Definition 20 (subpartition). A subpartition of a set S is a partition of a subset of S. Let SubPart(S)=⋃E⊆SPart(E) denote the set of all subpartitions of S.
Definition 21 (domain). The domain of a subpartition X of S, written dom(X), is the unique E⊆S such that X∈Part(E).
Definition 22 (restricted partitions). Given sets S and E and a partition X of S, let X|E denote the partition of S∩E given by X|E={[e]X∩E∣e∈E}.
Definition 23 (generating a subpartition). Given a finite factored set F=(S,B), and X∈SubPart(S)</