A set with a binary operation is a fundamental concept in algebra and one of the most fundamental properties of a binary operation is associativity. In this paper the authors discuss binary operations on a three-element set and show, by an inclusion-exclusion argument, that exactly 113 operations out of the 19,683 existing operations on the set are associative. Moreover these 113 associative operations are accounted for by means of their operation tables.
Diego, Friðrik and Jónsdóttir, Kristín Halla
"Associative Operations on a Three-Element Set,"
The Mathematics Enthusiast: Vol. 5
, Article 9.
Available at: http://scholarworks.umt.edu/tme/vol5/iss2/9