## Algebraic structures of greater cardinality than the continuum?

Are there interesting algebraic structures whose cardinality is greater than the continuum? Obviously, you could just build a product group of $\beth_2$ many groups of whole numbers to get to such a structure, but the properties of this group seem to be not very different from those you get when you just put together $\beth_1$ many whole number groups here.

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 I think this is a duplicate but can't find the other question... it was about groups, maybe? – Qiaochu Yuan Jun 2 2011 at 13:44 Dear Qiaochu Yuan: even if not an exact duplicate, the OP could read the answers to the similar question mathoverflow.net/questions/32370/…. Was you refering to it? – Giuseppe Jun 2 2011 at 18:06

The MO question Martin is referring has several good examples of algebraic structures larger than the continuum; one that I did not see talked about there is Conway's field of surreal numbers.

The surreal numbers form a proper class, so their cardinality dwarfs the cardinality of any set.

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This MO question "Cardinalities larger than the continuum in areas besides set theory" has recieved a couple of interesting answers, also algebraic ones (e.g. Zariski cotangent space of a manifold which is not smooth, automorphism tower length of a group)

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