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Ordinal sums and idempotents of copulas. (English) Zbl 1205.62063
Summary: We prove that the ordinal sum of $$n$$-copulas is always an $$n$$-copula and show that every copula may be represented as an ordinal sum, once the set of its idempotents is known. In particular, it will be shown that every copula can be expressed as the ordinal sum of copulas having only trivial idempotents. As a by-product, we also characterize all associative copulas whose $$n$$-ary forms are $$n$$-copulas for all $$n$$.

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 28E99 Miscellaneous topics in measure theory 26E99 Miscellaneous topics in real functions 28E10 Fuzzy measure theory 26E50 Fuzzy real analysis
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