ON A COMBINATORIAL PROPERTY OF FIBONACCI SEMIGROUP

We say that a semigroup S has property P(n), n greater-than-or-equal- to 2, if, given elements x(i), ..., x(n) of S, at least two of the n! products of these elements coincide. In a recent paper, Restivo consid ered the Fibonacci semigroup (i.e. the Rees quotient of {a, b}+ by the ideal of nonfactors of the well-known infinite Fibonacci word abaabab aabaab ...) and proved that it has property P8. Aim of this paper is to prove that it has property P4. As it does not have property P3*, t his is the best possible result.