Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 11

⎧
⎨

a k i , l j , if t u = k i ∈ K and v w = l j ∈ L − Q

or t u = k i ∈ K − P and v w = l j ∈ L ;

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

or t u = p r ∈ P − K and v w = q s ∈ Q ;

c t u ,v w = ,

⎩

a k i , l j ◦ b p r , q s , if t u = k i = p r ∈ K ∩ P

and v w = l j = q s ∈ L ∩ Q

false , otherwise

where here and below ◦∈{∧ , ∨ , → , ≡} .

A ⊗ ( ◦ ) B =[ K ∩ P , L ∩ Q , { c t u ,v w }] , where

c t u ,v w = a k i , l j ◦ b p r , q s , for t u =

= ∈ ∩

k i p r K P and

v w = l j = q s ∈ L ∩ Q ;

A ( ◦ , ∗ ) B =[ K ∪ ( P − L ), Q ∪ ( L − P ), { c t u ,v w }] , where

⎧
⎨

a k i , l j , if t u = k i ∈ K

and v w = l j ∈ L − P − Q

b p r , q s , if t u =

∈ − −

p r P L K

v w =

q s ∈ and Q

c t u ,v w = ,

⎩

◦

P (

a k i , l j ∗ b p r , q s ), if t u = k i = p r ∈ K and v w = q s ∈ Q

l j = p r ∈ L ∩

false , otherwise

where .

Operation (1.4) from Sect. 1.2 preserves its form, while operations (1.5) and (1.6)
from the same section are impossible in general. They are possible only in the case,
when

( ◦ , ∗ ) ∈{ ( ∨ , ∧ ), ( ∧ , ∨ ) }

α is the operation negation. In this case, operations (1.5) and (1.6) obtain the

forms

¬ A =[ K , L , {¬ a k i , l j }] ,

A − ( ◦ ) B =[ K ∪ P , L ∪ Q , { c t u ,v w }] ,

where