Extended Index Matrices - Index Matrices: Towards an Augmented Matrix Calculus - page 50

={

k 1 , k 2 ,..., k m } ,

={

l 1 , l 2 ,..., l n }

≤ ≤ , ≤ ≤ :

where K L ,for1 i m and 1 j n

∈ X

a k i , l j .

3.2 Operations Over EIMs

Let in this section, the sets X, Y, Z, U be fixed. Let operations “ ∗ ” and “ ◦ ” be defined

so that .

The first six operations from Sect. 1.2 remain valid here without changes.
Now, we see that for operations “addition” and “termwise multiplication”,

∗: X × Y → Z and ◦: Z × Z → U

• in the case of standard, i.e., R -IM, X = Y = R , where here and below, R is the

set of the real numbers, operation “ ∗ ” is the standard operation “ + ”or“ × ” and

obviously, Z = R ;

• when X = Y ={ 0 , 1 } , operation “ ∗ ” is “max” or “min”, and Z = X ;

• when X = Y is a set of logical variables, sentences or predicates, then “ ∗ ”is“ ∨ ”

or “ ∧ ” and Z = X ;

• when

X = Y = L ∗ ≡{ a , b | a , b , a + b ∈[ 0 , 1 ]} ,

then Z = X and operation “ ∗ ” is defined for the intuitionistic fuzzy pairs a , b

and c , d ,by

, ∗ , = ( , ), ( , )

a b c d max a c min b d

or

a , b ∗ c , d = min ( a , c ), max ( b , d ) .

In the case of operation “multiplication”,

• in the case of standard IM, X = Y = R , operation “ ∗ ” is the standard operation

“ + ” and operation “ ◦ ”—standard operation “ . ”, obviously, Z = R ;

• when X = Y ={ 0 , 1 } , operation “ ∗ ” is “max” and “ ◦ ”—“min”, or opposite, “ ∗ ”

is “min” and “ ◦ ”—“max”, and Z = X ;

• when X = Y are a set of logical variables, sentences or predicates, then “ ∗ ”is“ ∨ ”

and “ ◦ ”—“ ∧ ”, or vice versa, “ ∗ ”is“ ∧ ” and “ ◦ ”—“ ∨ ”, and Z = X ;

X = Y = L ∗ , then

• when Z = X and operation ∗ is defined for the intuitionistic

fuzzy pairs a , b and c , d , as above.

In the case of operation “termwise subtraction”,

• if X = R , then the constant α ∈ R ;

• if X ={ 0 , 1 } , then α ∈{ 0 , 1 } ;

• when X = Y is a set of logical variables, propositions or predicates, then α has

sence only when it is an operation “negation”.

• when the set contains IFPs, then for each one of the above discussed operations

over IMs, the operation “

X

∗ ◦ ∗

” is “max” and “ ”is“min”,orviceversa,“ ”is“min”