This paper proves that a binary operation ⋆ on [0, 1], ensuring that the binary operation ⋏ is a t-norm or ⋎ is a t-conorm, is a t-norm, where ⋏ and ⋎ are special convolution operations defined by
(
f
⋏
g
)
(
x
)
=
sup
{
f
(
y
)
★
g
(
z
)
:
y
▵
z
=
x
}
,
(
f
⋎
g
)
(
x
)
=
sup
{
f
(
y
)
★
g
(
z
)
:
y
▿
z
=
x
}
,
for any f, g ∈ Map([0, 1], [0, 1]), where △ and ▽ are a continuous t-norm and a continuous t-conorm on [0, 1], answering negatively an open problem posed in [8]. Besides, some characteristics of t-norm and t-conorm are obtained in terms of the binary operations ⋏ and ⋎.
Discussion(0)
No comments yet. Be the first to comment.