On a combinatorial property of Fibonacci semigroup

✍ Scribed by Giuseppe Pirillo

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 250 KB
• Volume: 122
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(93)90301-9

✦ Synopsis

We say that a semigroup S has property P.* , n > 2, if, given elements xi. , x, of S, at least two of the n! products of these elements coincide. In a recent paper, Restivo considered the Fibonacci semigroup (i.e. the Rees quotient of (a, h}+ by the ideal of nonfactors of the well-known infinite Fibonacci word ahaahahaahaab

) and proved that it has property Pg. Aim of this paper is to prove that it has property P$. As it does not have property P T, this is the best possible result.

-%(1)X0(2) ... %7(")=&(l)x*(z) "'XT(")

We say that S has property P (resp. P*) if there exists an integer n greater than 1 such that S has property P, (resp. P.*).

Proposition 1.2 (Restivo and Reutenauer [9]

). A jnite[y generated semigroup is jinite if and only if it is periodic and has property P.