On the algebraic and arithmetic structure of the monoid of product-one sequences II

Abstract

Let G be a finite group and G ′ its commutator subgroup. By a sequence over G, we mean a finite unordered sequence of terms from G, where repetition is allowed, and we say that it is a productone sequence if its terms can be ordered such that their product equals the identity element of G. The monoid B(G) of all product-one sequences over G is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if G is abelian (equivalently, B(G) is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if |G ′ | ≤ 2 if and only if B(G) is seminormal, and we study sets of lengths in B(G).

Which sets are sets of lengths in all numerical monoids?

“…where the intersection • (1) is taken over all finite abelian groups G with |G| ≥ 3, • (2) is taken over all non-half-factorial transfer Krull monoids H over finite abelian groups, and • (3) is taken over all finite groups with |G| ≥ 3. We recall that Equation (a) easily follows from Equation (1.1), Equation (b) is proved in [19,Section 3], and Equation (c) can be found in [27,Proposition 4.1]. Now we consider numerical monoids, where by a numerical monoid, we mean an additive submonoid of (N 0 , +) whose complement in N 0 is finite.…”

Erdős–Ginzburg–Ziv theorem and Noether number for C⋉C

“…We define a sequence over G to be an element of the free abelian monoid F (G), · , see Chapter 5 of [23], Section 3.1 of [10] or [16] for detailed explanation. Our notation of sequences follows the notation in the papers [21,25,30]. Note that, since G is a multiplicative group, we denote by ∅ the unit element of F (G) which is called the empty sequence.…”

“…• minimal product-one sequence if 1 G ∈ π(S) and S cannot be factored as a product of two non-trivial product-one subsequences. For recent study on the algebraic and arithmetic structure of product-one sequence for non-abelian groups, we refer to [30]. Using the above concepts, let…”

A characterization of seminormal C-monoids

“…Both statements generalize to rings with zero-divisors ([12, Theorem 3.5 and Section 4]). We refer to [17,1,6,20] for C-monoids, that do not stem from ring theory, and to [18] for a more general concept.…”

“…But this difference in complexity stems from the fact that the structure of the class semigroup of a C-monoid H can be much more intricate than the structure of the class group C( H) of its complete integral closure. We provide an explicit example of a half-factorial seminormal C-monoid (Example 4.3) and we also refer to the explicit examples of class semigroups given in [20,Section 4].…”