Implication-matrix Model - Revision history

Laouris at 11:27, 10 January 2022

2022-01-10T11:27:00Z

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

			[[Category: ISM Terminology]]

Laouris at 16:53, 6 January 2022

2022-01-06T16:53:53Z

← Older revision		Revision as of 16:53, 6 January 2022
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	~~IMPLICATION-MATRIX MODEL~~		An '''Implication Matrix Model''' includes a reflexive, binary matrix Ψ and an association … where Bl represents
	An ~~implication~~ Matrix ~~model~~ includes a reflexive, binary		"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~~; also~~ the falsity of v1 necessarily
	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 ~~implication~~		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~~ 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		If the index pair of an [[Imlplication Matrix]] is (V, V), the matrix is a [[Self-implication Matrix]].
	matrix is a ~~self~~-implication ~~matrix~~.

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

Laouris: Created page with "IMPLICATION-MATRIX MODEL An implication Matrix model includes a reflexive, binary matrix Ψ and an association … where Bl represents "implies." More specifically, if Ψij =..."

2022-01-06T16:44:37Z

Created page with "IMPLICATION-MATRIX MODEL An implication Matrix model includes a reflexive, binary matrix Ψ and an association … where Bl represents "implies." More specifically, if Ψij =..."

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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.