|
%I #6 Mar 30 2012 18:36:03
%S 0,0,10,24,2064,39961,1194828
%N The number of isomorphism classes of Szasz (uniquely non-associative) groupoids of order n.
%C A Szasz groupoid (S,*) is one for which there is exactly one ordered triple (a,b,c) of members of S that does not associate: (a*b)*c != a*(b*c). For any other triple (x,y,z), we have (x*y)*z = x*(y*z). Thus, a Szasz groupoid is as close to being a semigroup as possible, without actually being associative. G. Szasz proved that such groupoids exist on any set with at least four members. Every Szasz groupoid is non-commutative.
%D G. Szasz, Die Unabhangigkeit der Assoziativitatsbedingungen, Acta. Sci. Math. Szeged 15 (1953), 20-28.
%e The "smallest" Szasz groupoid of order 3 with elements {a,b,c} defines c*b = b, and the product of every other pair of elements is defined to be a. Then, (c*c)*b = a*b = b but c*(c*b) = c*b = b, but every triple other than (c,c,b) associates.
%K nonn,hard,more
%O 1,3
%A _James McCarron_, Jan 14 2011
|
