Noncommutative algebraic operation.

Question

Can we always find a non-commutative algebraic operation in a non-empty set?

Answer 1

composition is a non-commutative binary operation on the set of bijections with domain and codomain $A$ whenever $|A|>2$

Answer 2

If the set has just one element, there's only one binary operation, which is obviously commutative.

If the set has at least two elements, let $x$ and $y$ be distinct elements and define $$ x\cdot y=x,\quad y\cdot x=y, $$ and $z\cdot x=x\cdot z=z\cdot y=y\cdot z=x$ for all other elements $z$

Answer 3

If I understand your question, the answer is yes - given any set $X$ with more than one element, let $*$ be left projection: $x*y=x$.

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If we ask for a noncommutative operation satisfying some other restrictions, of course, then the answer can be quite surprising. My favorite example: there are no noncommutative division rings of finite cardinality! That is, given any finite abelian group $(G, +)$, there is no binary operation $*$ on $G$ such that $(G, *)$ is a division ring and $*$ is noncommutative. This is Wedderburn's little theorem; I'm still shocked by this result.