It is a very standard fact that commutative semigroups admit an invariant mean and the proof basically relies on Markov-Kakutani fixed point theorem. Now, it seems to me that the proof of this theorem does not need the associativity (see for instance http://www.springerlink.com/content/g777570166731376/) and then the answer to the following question should be affirmative.

Question: Recall that a magma is a set equipped with a binary operation. Is it true that a commutative magma admits an invariant mean?

So, is that trivially true or am I missing some detail?

Thanks in advance,

Valerio

It is a very standard fact that commutative semigroups admit an invariant mean and the proof basically relies on Markov-Kakutani fixed point theorem. Now, it seems to me that the proof of this theorem does not need the associativity (see for instance http://www.springerlink.com/content/g777570166731376/) and then the answer to the following question should be affirmative.

Question: Recall that a magma is a set equipped with a binary operation. Is it true that a commutative countable magma admits an invariant mean?

So, is that trivially true or am I missing some detail?

Thanks in advance,

Valerio