Classification of All Associative Mono-n-ary Algebras with 2 Elements

We consider algebras with a single n-ary operation and a certain type of associativity.We prove that up to isomorphism there are exactly 5 of these associative mono-n-ary algebras with 2 elements for even n ≥ 2 and 6 for odd n ≥ 3. These algebras are described explicitly. It is shown that a similar result is impossible for algebras with at least 4 elements. An application concerning the assignment of a control bit to a string is given.