By Michał Baczyński (auth.), Humberto Bustince, Javier Fernandez, Radko Mesiar, Tomasas Calvo (eds.)

This quantity collects the prolonged abstracts of forty five contributions of contributors to the 7th overseas summer season college on Aggregation Operators (AGOP 2013), held at Pamplona in July, 16-20, 2013. those contributions hide a really wide variety, from the in basic terms theoretical ones to these with a extra utilized concentration. in addition, the summaries of the plenary talks and tutorials given on the related workshop are included.

Together they supply an outstanding evaluate of modern tendencies in examine in aggregation services which are of curiosity to either researchers in Physics or arithmetic engaged on the theoretical foundation of aggregation capabilities, and to engineers who require them for applications.

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**Sample text**

When Cr = CH . Consider an arbitrary copula C : [0, 1]2 → [0, 1] and its diagonal section δ : [0, 1] → [0, 1] given by δ (x) = C(x, x). Recall that δ is non–decreasing, 2–Lipschitz, δ (0) = 0, δ (1) = 1 and δ (x) ≤ x for all x ∈ [0, 1]. Then the function f : [0, 1] → [0, 1] given by f (x) = x − δ (x) is 1–Lipschitz, f (0) = f (1) = 0. For p = (M, M, −1, f , f ), it holds (7) Cp (x, y) = max (0, M(x, y) − M(x − δ (x), y − δ (y))). Applying formula (5) considering N = f , one gets C f (x, y) = max (0, x ∧ y − f (x ∨ y)) = max (0, δ (x ∨ y) − |x − y|) = CδMT (x, y), where CδMT is a Mayor–Torrens copula [14] derived from the diagonal section δ .

The function Cu,f ,gδ : [0, 1]2 → [0, 1] defined by f ,g ˆ gˆ f, (1 − x, 1 − y) , l,δˆ Cu,δ (x, y) = x + y − 1 + C (3) Semiquadratic Copulas 51 is a copula with diagonal section δ if and only if (i) f (0) = g(0) = 0 , (ii) max ( f (t) + (1 − t) | f (t)| , g(t) + (1 − t) |g (t)|) ≤ (iii) f (t) + g(t) ≥ (1 − t) 2t−1−δ (t) (1−t)2 t−δ (t) 1−t , , for all t ∈ [0, 1[ where the derivatives exist. The function Cu,f ,gδ defined by (3) is called an upper semiquadratic function with diagonal section δ . 2 Horizontal and Vertical Semiquadratic Copulas with a Given Diagonal Section Horizontal (resp.

34 A. Pradera • Characterizations of implications by means of aggregation and negation functions. Another important issue when dealing with fuzzy implications is to characterize them, in the sense of obtaining the minimal set of properties required to a fuzzy implication in such a way that it belongs to a given family of implications. Such characterizations usually involve properties of the underlying aggregation and/or negation functions. For example (see [4]), the class of (A,N)-implications generated from triangular conorms and continuous negations has been characterized, as well as the family of R-implications built from left-continuous triangular norms.