Proposition. Let X denote an infinite set. Then for each partitioning Π of X, there exists a function f:X→X whose coimage is Π.
I'd like to know whether the analogous statement holds for associative binary operations.
Question. Let X denote an infinite set. Is it true that for each partitioning Π of X2, there exists an associative operation f:X2→X whose coimage is Π?
This question occurred to me as a result of Behrouz Maleki's answer here.
Answer
Consider the discrete partition of X2, and assume it is the coimage of an associative binary operation. Then for all a,b,c∈X we have a(bc)=(ab)c, and thus a=ab, and this determines the operation as projection to the left argument. Similarly, bc=c, and this determines the operation as projection to the right argument. This can only happen if X is empty or a singleton.
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