Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 72

Termwise subtraction

A ∗ (T ) − ( ◦ , ∗ ) B ∗ (T ) = A ∗ (T ) ⊕ ( ◦ , ∗ ) ¬ B ∗ (T ).

Operations “reduction”, “projection” and “substitution” coincide with the res-
pective operations defined over IMs in [10], while the hierarchical operations over
IMs are not applicable here.

4.2 Relations Over ETIFIMs

Let the two ETIFIMs

A ∗ (T ) =[ K ∗ (T ), L ∗ (T ), { μ k i , l j ,τ , ν k i , l j ,τ }]

and

B ∗ (T ) =[ P ∗ (T ), Q ∗ (T ), { ρ p r , q s ,τ , σ p r , q s ,τ }]

be given. We introduce the following definitions where ⊂ and ⊆ denote the relations

“ strong inclusion ” and “ weak inclusion ”.
The strict relation “inclusion about matrix-dimension and elements” is

A ∗ (T ) ⊂

e
d

B ∗ (T ) K ∗ (T ) ⊂ P ∗ (T )) L ∗ (T ) ⊂ Q ∗ (T )))

((( (

iff &

K ∗ (T ) ⊆ P ∗ (T )) L ∗ (T ) ⊂ Q ∗ (T )))

∨ (( & (

K ∗ (T ) ⊂ P ∗ (T )) L ∗ (T ) ⊆ Q ∗ (T ))) ∨ ( & (

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )( μ k i , l j ,τ , ν k i , l j ,τ = ρ k i , l j ,τ , σ k i , l j ,τ ).

The non-strict relation “inclusion about matrix-dimension and elements” is

A ∗ (T ) ⊆

e
d

B ∗ (T ) K ∗ (T ) ⊆ P ∗ (T )) L ∗ (T ) ⊆ Q ∗ (T )) iff ( & (

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )( μ k i , l j ,τ , ν k i , l j ,τ = ρ k i , l j ,τ , σ k i , l j ,τ ).

The strict relation “inclusion about element values” is

A ∗ (T ) ⊂

e
v

B ∗ (T ) K ∗ (T ) = P ∗ (T )) L ∗ (T ) = Q ∗ (T )) iff ( & (

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )( μ k i , l j ,τ , ν k i , l j ,τ < ρ k i , l j ,τ , σ k i , l j ,τ ).