On Random Topological Structures
نویسندگان
چکیده
and Applied Analysis 3 As a result of 2.2 we obtain, for each x, y ∈ 0, 1 , T ( x, y ) ≤ T x, 1 x, T ( x, y ) ≤ T ( 1, y ) y. 2.5 Since, for all x, y ∈ 0, 1 , trivially T x, y ≥ 0 TD x, y , we get for an arbitrary t-norm T , TD ≤ T ≤ TM. 2.6 That is, TD is weaker and TM is stronger than any other t-norms. Also since TL < TP , we obtain the following ordering for four basic t-norms TD < TL < TP < TM. 2.7 Proposition 2.2 see 2 . i The minimum TM is the only t-norm satisfying T x, x x for all x ∈ 0, 1 . ii The weakest t-norm TD is the only t-norm satisfying T x, x 0 for all x ∈ 0, 1 . Proposition 2.3 see 2 . A t-norm T is continuous if and only if it is continuous in its first component, that is, if for each y ∈ 0, 1 the one-place function T ( ·, y ) : 0, 1 −→ 0, 1 , x −→ T ( x, y ) , 2.8 is continuous. For example, the minimum TM and Łukasiewicz t-norm TL are continuous but the t-norm TΔ defined by TΔ ( x, y ) : ⎧ ⎨ ⎩ xy 2 , if max ( x, y ) < 1, xy, otherwise, 2.9 for x, y ∈ 0, 1 , is not continuous. Definition 2.4. i A t-norm T is said to be strictly monotone if T ( x, y ) < T x, z whenever x ∈ 0, 1 , y < z. 2.10 ii A t-norm T is said to be strict if it is continuous and strictly monotone. For example, the t-norm TΔ is strictly monotone but the minimum TM and Łukasiewicz t-norm TL are not. Proposition 2.5 see 2 . A t-norm T is strictly monotone if and only if T ( x, y ) T x, z , x > 0 ⇒ y z. 2.11 4 Abstract and Applied Analysis If T is a t-norm, then x n T is defined for every x ∈ 0, 1 and n ∈ N ∪ {0} by 1, if n 0 and T x n−1 T , x , if n ≥ 1. Definition 2.6. A t-norm T is said to be Archimedean if for all x, y ∈ 0, 1 2 there is an integer n ∈ N such that
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