The universal zero-sum invariant and weighted zero-sum for infinite abelian groups

Abstract

Let G be an abelian group, and let F ( G ) be the free commutative monoid with basis G. For Ω ⊂ F ( G ) , define the universal zero-sum invariant d Ω ( G ) to be the smallest integer l such that every sequence T over G of length l has a subsequence in Ω . The invariant d Ω ( G ) unifies many classical zero-sum invariants. Let B ( G ) be the submonoid of F ( G ) consisting of all zero-sum sequences over G, and let A ( G ) be the set consisting of all minimal zero-sum sequences over G. The empty sequence, which is the identity of B ( G ) , is denoted by ε . The well-known Davenport constant D ( G ) of the group G can be also represented as d B ( G ) ∖ { ε } ( G ) or d A ( G ) ( G ) in terms of the universal zero-sum invariant. Notice that A ( G ) is the unique minimal generating set of the monoid B ( G ) from the point of view of Algebra. Hence, it would be interesting to determine whether A ( G ) is minimal to represent the Davenport constant or not for a general finite abelian group G. In this paper, we show that except for a few special classes of groups, there always exists a proper subset Ω of A ( G ) such that d Ω ( G ) = D ( G ) . Furthermore, in the setting of finite cyclic groups, we discuss the distributions of all minimal sets by determining their intersections. By connecting the universal zero-sum invariant with weights, we make a study of zero-sum problems in the setting of infinite abelian groups. The universal zero-sum invariant d Ω ; Ψ ( G ) with weights set Ψ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant D Ψ ( G ) (being an special form of the universal invariant with weights) is also investigated for infinite abelian groups. Among other results, we obtain the necessary and sufficient conditions such that D Ψ ( G ) < ∞ in terms of the weights set Ψ when | Ψ | is finite. In doing this, by using the Neumann Theorem on Cover Theory for groups we establish a connection between the existence of a finite cover of an abelian group G by cosets of some given subgroups of G, and the finiteness of weighted Davenport constant.