Arithmetic operators in interval-valued fuzzy set theory

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

“…For general x and y inL I we extend the subtraction using the connection between ⊕ L I and ⊖ L I (see Example 8) that was obtained in [6]: for all x, y in…”

“…It was proven in [6] that these operators satisfy (add-1)-(add-5), (mul-1)-(mul-5), (add-5'), (mul-5'), (1), (2), (3) and (4). In [5] these operators are used to define additive and multiplicative generators on L I and it is shown that the only t-norms that can have a continuous additive generator based on this addition are t-representable t-norms.…”

“…It was proven in [11] that these operators satisfy (add-1)-(add-5), (mul-1)-(mul-5), (add-5'), (mul-5'), (1) and (2). We prove (2) and (4).…”

“…In [5] we introduced additive and multiplicative generators on L I based on a special kind of addition introduced in [6]. It was shown that the only t-norms on L I which have a continuous additive generator on L I that is a natural extension of an additive generator on the unit interval are pseudo-t-representable, and this is due to the typical construction of the addition operator introduced in [6].…”

“…It was shown that the only t-norms on L I which have a continuous additive generator on L I that is a natural extension of an additive generator on the unit interval are pseudo-t-representable, and this is due to the typical construction of the addition operator introduced in [6]. Because of this limitation, in [11] another addition was introduced. In fact many more additions can be introduced.…”