On half-factoriality of transfer Krull monoids.

Let H be a transfer Krull monoid over a subset G₀ of an abelian group G with finite exponent. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · ... · u k. The set L (a) of all possible factorization lengths k is called the set of lengths of a, and H is said to be half-factorial if | L (a) | = 1 for all a ∈ H. We show that, if a ∈ H is a non-unit and | L (a ⌊ (3 exp (G) − 3) / 2 ⌋ ) | = 1 , then the smallest divisor-closed submonoid of H containing a is half-factorial. In addition, we prove that, if G₀ is finite and | L (∏ g ∈ G 0 g 2 ord (g) ) | = 1 , then H is half-factorial. [ABSTRACT FROM AUTHOR]