Three Dimensional IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 97

The non-strict relation “inclusion about value” is

A ⊆ v B iff ( K = P ) & ( L = Q ) & ( H = R ) & ( ∀ k ∈ K )( ∀ l ∈ L )( ∀ h ∈ H )

( a k , l , h ≤ b k , l , h ).

The strict relation “inclusion” is

A ⊂ B iff ((( K ⊂ P ) & ( L ⊂ Q ) & ( H ⊂ R )) ∨ (( K ⊆ P ) & ( L ⊂ Q ) & ( H ⊂ R ))

∨ (( K ⊂ P ) & ( L ⊆ Q ) & ( H ⊂ R )) ∨ (( K ⊂ P ) & ( L ⊂ Q ) & ( H ⊆ R )))

& ( ∀ k ∈ K )( ∀ l ∈ L )( ∀ h ∈ H )( a k , l , h < b k , l , h ).

The non-strict relation “inclusion” is

A ⊆ B iff ( K ⊆ P ) & ( L ⊆ Q ) & ( H ⊆ R ) & ( ∀ k ∈ K )( ∀ l ∈ L )( ∀ h ∈ H )

( a k , l , h ≤ b k , l , h ).

6.4 Operations “Reduction” Over an 3D-IM

First, we introduce operations ( k , ⊥ , ⊥ ) -, ( ⊥ , l , ⊥ ) - and ( ⊥ , ⊥ , h ) -reduction of a

given 3D-IM A =[ K , L , H , { a k i , l j , h g }] :

A

, ⊥ , ⊥ ) =[

K −{ k } , L , H , { c t u ,v w , h g }]

( k

where

c t u ,v w , x y = a k i , l j , h g for t u = k i ∈ K −{ k } ,v w = l j ∈ L and x y = h g ∈ H ,

A

, ⊥ ) =[

K , L −{ l } , H , { c t u ,v w , x y }] ,

( ⊥ , l

where

c t u ,v w , x y = a k i , l j , h g for t u = k i ∈ K ,v w = l j ∈ L −{ l } and x y = h g ∈ H

and

A ( ⊥ , ⊥ , h ) =[ K , L , H −{ h } , { c t u ,v w , x y }] ,

where

=

a k i , l j , h g for t u =

∈ ,v w = ∈

L and x y = h g ∈

−{ } .

c t u ,v w , x y k i K l j H h