Extended Index Matrices - Index Matrices: Towards an Augmented Matrix Calculus - page 52

The strict relation “ inclusion about value ”is

A ⊂ v B iff ( K = P ) & ( L = Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( R s ( a k , l , b k , l )).

The non-strict relation “ inclusion about value ”is

A ⊆ v B iff ( K = P ) & ( L = Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( R n ( a k , l , b k , l )).

The strict relation “ inclusion ”is

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

∨ (( K ⊂ P ) & ( L ⊆ Q ))) & ( ∀ k ∈ K )( ∀ l ∈ L )( R s ( a k , l , b k , l )).

The non-strict relation “ inclusion ”is

A ⊆ ∗ B iff ( K ⊆ P ) & ( L ⊆ Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( R n ( a k , l , b k , l )).

3.4 Hierarchical Operators Over EIMs

In [10, 14], two hierarchical operators are defined. They are applicable on EIM, when
their elements are not only numbers, variables, etc, but when they also can be whole
(new) IMs.
Let A be an ordinary IM and let its element a k f , e g be an IM by itself:

=[ , , {

b p r , q s }] ,

a k f , l g P Q

where

K ∩ P = L ∩ Q =∅ .

Here, we will introduce the first hierarchical operator:

A | ( a k f , l g ) =[ ( K −{ k f } ) ∪ P ,( L −{ l g } ) ∪ Q , { c t u ,v w }] ,

where

⎧
⎨

a k i , l j , if t u = k i ∈ K −{ k f } and v w = l j ∈ L −{ l g }

c t u ,v w = b p r , q s , if t u = p r ∈ P and v w = q s ∈ Q .

⎩

,

0 otherwise

Let us assume that in the case when a k f , l g is not an element of IM A , then