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Let x and y be the unique real numbers such that L((m – x) / α) = ω = L((n – y) / γ), where ω is a fixed value in [0, 1]. This is equivalent to x = m – α L – 1(ω), which implies y = n – γ L – 1(ω), z = x + y = m + n – (α + γ ) L–1(ω) and  m + n − z L  = ω.  α+γ  The same reasoning holds on decreasing parts of M and N and  z − ( m + n)  R  = ω.  β +δ  Using Theorem 1, we prove (m, α, ß)LR  (n, γ, δ)LR = (m + n, α + γ, ß + δ)LR. (12) More generally, (m, α, ß)L′R′  (n, γ, δ)L′R′ = (m + n, 1, 1,)L′′R′′ with L′′ = (αL – 1 + γL′ – 1) – 1, R′′ = (ßL – 1 + δL′ – 1) – 1.

Xr) such that y = f (x1, . . , xr). This extension principle sounds more probabilistic than fuzzy, particularly if we also replace min by product. 2. Extension Principle 38 general f (A1, . . , Ar) is a classical subset of y when X = ‫( ޒ‬with min or product) and continuous membership functions are considered. So the result depends only on the supports of the Ai, which invalidates this principle as one of fuzzy extension. Another extension principle can be obtained by just replacing min by product in (1).

Sambuc, R. (1975). “Fonctions ⌽-flous. ” Thèse Univ. de Marseille, Marseille. , and Sambuc, R. (1976). Relations floues. Fonctions ⌽-floues. Application à l’aide au diagnostic en pathologie thyroidienne. Med. Data Process. , Toulouse, Fr. , and Riera, T. (1978). Entopies of finite fuzzy sets. Inf. Sci. 15, No. 2, 159–168. , and Malvache, N. (1976). Utilisation d’un référentiel de sous-ensembles flous. Application à un algorithme flou. Inf. Conf. Syst. , Wroclaw, Poland. Zadeh, L. A. (1965). Fuzzy sets.

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