Classes of non-commutative semigroups with finite Fibonacci invariants

M. Monsef, H. Doostie

For a 2-generated semigroup S we define the invariant λ(S) as the minimum of the Fibonacci lengths over all generating pairs of the semigroup S, where the Fibonacci length with respect to a generating pair (x, y) is the fundamental period (if exist) of the truncated periodic sequence z0=x, z1=y, z_k=zk-2zk-1, (k≥2) of the elements of S. We name this invariant as the Fibonacci invariant of S. Our used notation is the same as of the celebrated work of D.L. Johnson in 2005 on infinite groups. In this paper we examine two classes of semigroups for existence of this invariant. The considered semigroups are the finite semigroup S = ⟨ a, b | apα=a, bqβ=b, ab=a⟩ of order pαqβ-1 and the infinite semigroup T = ⟨ a, b | apα=a, ab=a⟩ for all integers α, β ≥ 2 and all distinct primes p and q. We prove the existence of the Fibonacci invariants of these semigroups, for all parameters. As a numerical result we show that λ(T) ≤ λ(S) = pα+1, if p is even.

Tbilisi Mathematical Journal, Vol. 13(3) (2020), pp. 153-159