Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 12

⎧
⎨

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

and “ ” means operation “negation”. This notation is used, because of the variety

of different possible forms of the operation “negation”, as discussed in Sect. 2.1 .
Only operations Max-row-, Min-row-, Max-column- and Min-column aggrega-
tion from Sect. 1.2 exist for L-IMs, but in the following forms.

¬

∨ -row-aggregation

l 1 l 2 ... l n

ρ ∨ ( A , k 0 ) =

a k i , l n ) , k 0 ∨

m (

a k i , l 1 ) ∨

m (

a k i , l 2 ) ... ∨

m ( 1 ≤ i ≤ 1 ≤ i ≤ 1 ≤ i ≤

∧ -row-aggregation

l 1 l 2 ... l n

ρ ∧ ( A , k 0 ) =

a k i , l n ) , k 0 ∧

m (

a k i , l 1 ) ∧

m (

a k i , l 2 ) ... ∧

m ( 1 ≤ i ≤ 1 ≤ i ≤ 1 ≤ i ≤

∨ -column-aggregation

l 0

∨

k 1 a k 1 , l j

1 ≤ j ≤ n

σ ∨ ( A , l 0 ) = ,

. .

k m ∨ a k m , l j

1 ≤ j ≤ n