International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 678987, 16 pages
Research Article

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

Department of Mathematics and Computer Science, Fern Universität in Hagen, Lützowstraße 125, 58084 Hagen, Germany

Received 9 July 2009; Accepted 22 December 2009

Academic Editor: Robert Redfield

Copyright © 2009 Stephan Dominique Andres. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 n2 and 6 for odd n3. 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.