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An '''Implication Matrix Model''' includes a reflexive, binary matrix Ψ and an association … where Bl represents | |||
An | "implies." | ||
matrix Ψ and an association … where Bl represents | |||
"implies." More specifically, if Ψij = 1, the truth of hj | More specifically, if Ψij = 1, the truth of hj follows from the truth of vi. | ||
follows from the truth of vi | |||
follows from the falsity of hj. If truth of hj is represented | Also, the falsity of v1 necessarily follows from the falsity of hj. | ||
by the equation hj = 1, then falsity is represented both by | |||
If truth of hj is represented by the equation hj = 1, then falsity is represented both by | |||
hj = 0 and hj = 1. | hj = 0 and hj = 1. | ||
From a structural point of view, | From a structural point of view, | ||
Ψij = 1 means that there is a digraph path oriented from | Ψij = 1 means that there is a digraph path oriented from v1 to hj in an implication digraph. | ||
v1 to hj in an implication digraph. | |||
It follows that, if Ψ(V, H) represents an | 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. | ||
matrix Ψ(H,V) will be an | |||
by H and V. | |||
If the index pair of an | If the index pair of an [[Imlplication Matrix]] is (V, V), the matrix is a [[Self-implication Matrix]]. | ||
matrix is a | |||
If V and H have no elements in common, the matrix is a | If V and H have no elements in common, the matrix is a [[Cross-implication Matrix]]. | ||
All other implication matrices are hybrid. | All other implication matrices are hybrid. | ||
An Implication Matrix Model is not necessarily complete. | An Implication Matrix Model is not necessarily complete. | ||
Likewise, because of the transitivity of the Implication | 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]]. | ||
relation, any power of an Implication Matrix is an Implication 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]]. | |||