A180043 The number of isomorphism classes of Szasz (uniquely non-associative) groupoids of order n.

0, 0, 10, 24, 2064, 39961, 1194828

Offset: 1

Author

James McCarron, Jan 14 2011

Keywords

• nonn
• hard
• more

Comments

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.

Examples

			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.
		

References

• G. Szasz, Die Unabhangigkeit der Assoziativitatsbedingungen, Acta. Sci. Math. Szeged 15 (1953), 20-28.