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 non-commutative.
REFERENCES
G. Szasz, Die Unabhangigkeit der Assoziativitatsbedingungen, Acta. Sci. Math. Szeged 15 (1953), 20-28.
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
KEYWORD
nonn,hard,more
AUTHOR
James McCarron, Jan 14 2011
STATUS
approved