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As a particular case of a more general result of Aglianò and Montagna in [1], we recall a slightly different notion of ordinal sum for finite linearly-ordered Wajsberg hoops. Actually, a hoop is an algebra A = A, ∗, →, 1 such that A, ∗, 1 is a commutative monoid and for all x, y, z ∈ A the following equations hold: x → x = 1, x ∗ (x → y) = y ∗ (y → x), x → (y → z) = (x ∗ y) → z. A Wajsberg hoop is a hoop satisfying the equation: (x → y) → y = (y → x) → x. A bounded hoop is an algebra A = (A, ∗, →, 1, 0) such that A, ∗, →, 1 is a hoop and 0 ≤ x for all x ∈ A.

J. Frank, B. Schweizer, Problems on associative functions. Aequationes Math. 66, 128–140 (2003) 42. W. Trutschnig, On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384, 690–705 (2011) 43. J. Fernández-Sánchez, W. Trutschnig, Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems. J. Theor. Probab. (2015). 1007/s10959-014-0541-4 44. P. D. Taylor, A remark on associative copulas.

Many of the definitions and theorems for bivariate copulas have analogous multivariate versions (see [1, 12]). 3 Associative Copulas: A Survey 35 The main problem in the theory of copulas is to determine which sets of (possible different dimensions) copulas are margins of a higher-dimensional copula. The associativity of n-copulas in the sense of Post (see [70]) is studied in [71]— solving an open problem posed in [72]. Specifically, if n ≥ 2 is a natural number and S is a nonempty set, an n-ary operation f : S n −→ S is associative on S if, for any 1 ≤ i < j ≤ n, the equality j−1 n+j−1 2n−1 = f x1 , f xj f x1i−1 , f xin+i−1 , xn+i 2n−1 , xn+j q holds for all x1 , .

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IBM SY22-2798-2 Logic Blocks - Automated Logic Diagrams (SLT, SLD, ASLT, MST) - Maintenance


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