International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 678987, 16 pages
doi:10.1155/2009/678987

Classification of all associative mono-$n$-ary algebras with 2 elements

Abstract

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.