On the incomparability of systems of sets of lengths

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Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. We consider the system L(H) of all sets of lengths of H and study when L(H) contains or is contained in a system L(H′) of a Krull monoid H′ with finite class group G′, prime divisors in all classes and Davenport constant D(G′)=D(G). Among others, we show that if G is either cyclic of order m≥7 or an elementary 2-group of rank m−1≥6, and G′ is any group which is non-isomorphic to G but with Davenport constant D(G′)=D(G), then the systems L(H) and L(H′) are incomparable.