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Let $f_0,\dots,f_{n-1}$ are relations, where $f_i$ is a relation of arity $m_i$ for every $i=0,\dots,n-1$.

How to construct an $(m_0+\dots+m_{n-1})$-ary relation from them?

For my previous erroneous attempt to do this see here: Name for product of relations

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If all on same set, can define new relation as holding if all the component relations hold, or if at least one holds, or any other desired Boolean combination. – André Nicolas Jun 9 '12 at 15:09
So, you want to combine several sets of n-tuples into a single set? There is, of course, no standard way to do this. Do you have an application in mind? – Dan Christensen Jun 15 '12 at 15:24

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Oh, I found:

$$\left\{ \operatorname{uncurry}z \hspace{0.5em} | \hspace{0.5em} z \in \prod f \right\}$$

where uncurrying is defined in Wikipedia: http://en.wikipedia.org/wiki/Currying by the formula $\operatorname{uncurry}(f)(x;y)=(fx)y$.

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This would not seem to be the answer to your question. This is "a technique of transforming a function that takes multiple arguments (or an n-tuple of arguments) in such a way that it can be called as a chain of functions each with a single argument." – Dan Christensen Jun 15 '12 at 15:41

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