Functions with the domain having greater cardinality than $\aleph_1$

Are there any examples of an algebra on elements belonging to a set of greater cardinality than that of real numbers. Can a reference to their use be given?

• continuous functions on stone-cech compactification of $\mathbb{N}$. – Norbert Nov 11 '13 at 7:22
• @Norbert Thanks. This would be an example of domain having the cardinality $\aleph_2 / 2^\mathfrak c$ are their examples of even greater cardinalities. – ARi Nov 11 '13 at 7:40

It is consistent that the real numbers have cardinality $\aleph_2$, in which case a function $f\colon\Bbb{R\to R}$ satisfies your requirements.
If you want a set whose cardinality is explicitly $\aleph_2$ then you will have to resort, in one way or another, to $\omega_2$ which is the second uncountable ordinal (i.e. the cardinal $\aleph_2$). The function can be anything, from the successor ordinal, to anything crazy.
However, I feel that you're not talking about $\aleph_2$, but rather $\beth_2$ i.e. $|\mathcal{P(P(}\Bbb R))|$. In that case you can consider the Lebesgue measure as a function whose domain has cardinality $\beth_2$, and its range is $\Bbb R$.
• "It is consistent that the real numbers have cardinality $\aleph_2$...isnt $\aleph_1$ the cardinality of the set of real numbers? – ARi Nov 11 '13 at 8:37
• math.stackexchange.com/a/490420/622 There are probably many other questions and answers about this question. But no. The cardinality of the real numbers is not necessarily $\aleph_1$. – Asaf Karagila Nov 11 '13 at 8:41