T-Preorders and T-Indistinguishabilities - Fuzzy Logic: An Introductory Course for Engineering Students - page 130

Finally,

T ( E ( a , b ), E ( b , c )) = T ( Inf

i ∈ I

E i ( a , b ), Inf

i ∈ I

E i ( b , c ))

≤ Inf

i

T ( E i ( a , b ), E i ( b , c ))

∈ I

≤

E i (

, ) = ( , )

Inf
i ∈ I

a c E a c

since T is a continuous t-norm.

5.2 The Characterization of T-Preorders

X , define J T (

. Obviously J T

If ˃ ∈[ 0 , 1 ] x , y ) = J T (˃( x ), ˃( y )) is a T-Preorder in

X

X × X , and if F ↂ[ 0 , 1 ] it is also

J T Inf

˃ ∈F

a T-Preorder.
Given a fuzzy relation μ : X × X ₒ[ 0 , 1 ] , consider the set T (μ) of its T-states,

that is the fuzzy sets ˃ : X ₒ[ 0 , 1 ] , such that

T (˃( x ), μ( x , y )) ≤ ˃( y ) , for all x , y ∈ X .

As it is known this last inequality is equivalent to

J T (

μ( x , y ) ≤ J T (˃( x ), ˃( y )) = x , y ),

thus,

J T ˃ ∈ (μ) ⃔ μ ≤

T

Hence,

J T , ˃ ∈ T (μ) ⃔ μ ≤ Inf

˃ ∈ T (μ)

J T , μ and it is clear that if is a T-Preorder.

Avoiding some technical difficulties in the proof of the converse statement, let us
state

μ = Inf

˃ ∈ T (μ)

J T ,
a result characterizing the structure of all T-preorders.

• μ is a T-Preorder if and only if μ = Inf

˃ ∈ T (μ)