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

Abstract EN

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.