

A180043


The number of isomorphism classes of Szasz (uniquely nonassociative) groupoids of order n.


0




OFFSET

1,3


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 noncommutative.


REFERENCES

G. Szasz, Die Unabhangigkeit der Assoziativitatsbedingungen, Acta. Sci. Math. Szeged 15 (1953), 2028.


LINKS

Table of n, a(n) for n=1..7.


EXAMPLE

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.


CROSSREFS

Sequence in context: A264296 A265150 A057462 * A251317 A251201 A048195
Adjacent sequences: A180040 A180041 A180042 * A180044 A180045 A180046


KEYWORD

nonn,hard,more


AUTHOR

James McCarron, Jan 14 2011


STATUS

approved



