Tuesday, 4 September 2018

combinatorics - Let X denote an infinite set. Is every partitioning of X2 induced by an associative operation f:X2rightarrowX?





Proposition. Let X denote an infinite set. Then for each partitioning Π of X, there exists a function f:XX 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:X2X 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,cX 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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