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

AbstractLet <italic>G</italic> be an abelian group, and let F(G) be the free commutative monoid with basis <italic>G</italic>. For Ω⊂F(G), define the universal zero-sum invariant dΩ(G) to be the smallest integer l such that every sequence <italic>T</italic> over <italic>G</italic> of length l has a subsequence in Ω. The invariant dΩ(G) unifies many classical zero-sum invariants. Let B(G) be the submonoid of F(G) consisting of all zero-sum sequences over <italic>G</italic>, and let A(G) be the set consisting of all minimal zero-sum sequences over <italic>G</italic>. The empty sequence, which is the identity of B(G), is denoted by ε. The well-known Davenport constant D(G) of the group <italic>G</italic> can be also represented as dB(G)∖{ε}(G) or dA(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 <italic>G</italic>. 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 dΩ(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 <italic>infinite</italic> abelian groups. The universal zero-sum invariant dΩ;Ψ(G) with weights set Ψ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant DΨ(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 DΨ(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 <italic>G</italic> by cosets of some given subgroups of <italic>G</italic>, and the finiteness of weighted Davenport constant. [ABSTRACT FROM AUTHOR]