Implication-matrix Model

IMPLICATION-MATRIX MODEL An implication Matrix model includes a reflexive, binary matrix Ψ and an association … where Bl represents "implies." More specifically, if Ψij = 1, the truth of hj follows from the truth of vi; also the falsity of v1 necessarily follows from the falsity of hj. If truth of hj is represented by the equation hj = 1, then falsity is represented both by hj = 0 and hj = 1.

From a structural point of view, Ψij = 1 means that there is a digraph path oriented from v1 to hj in an implication digraph.

It follows that, if Ψ(V, H) represents an implication matrix indexed by the ordered sets V and H, the transpose matrix Ψ(H,V) will be an imlplication matrix indexed by H and V.

If the index pair of an implication matrix is (V, V), the matrix is a self-implication matrix.

If V and H have no elements in common, the matrix is a cross-implication matrix.

All other implication matrices are hybrid.

An Implication Matrix Model is not necessarily complete. If it is complete, then the Implication Matrix Ψ is also a reachability matrix, because of the transitivity of the implication relation, i.e., Ψ2 = Ψ and Ψ + I = Ψ, where I is the Identity Matrix. The Boolean sum of any two Implication Matrices with the same index pairs is clearly an Implication Matrix.

Likewise, because of the transitivity of the Implication relation, any power of an Implication Matrix is an Implication Matrix.