On a family of t-norms. (English) Zbl 0739.39006

Mayor, Gaspar; Torrens, Joan

Fuzzy Sets Syst. 41, No. 2, 161-166 (1991).

The authors consider the functional equation (1) \(T(x,y)+| x- y|=T(\hbox{Max}(x,y)\), \(\hbox{Max}(x,y))\), where \(T\) is a \(t\)-norm. (A \(t\)-norm is a binary operation on \([0,1]\) that is associative, commutative, non-decreasing in each place, and has identity 1.) The authors prove that if \(T\) is a continuous \(t\)-norm which satisfies (1), then, for some \(\alpha\in[0,1]\), \(T\) is of the form \(T(x,y)=\hbox{Max}(0,x+y-\alpha)\), whenever \(x,y\in[0,\alpha]\), and \(=\hbox{Min}(x,y)\), otherwise.

Reviewer: R.Tardiff (Salisbury)

MSC:

39B22	Functional equations for real functions
03E72	Theory of fuzzy sets, etc.

Keywords:

strong negation; functional equation; continuous \(t\)-norm

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