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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. [ABSTRACT FROM AUTHOR]

Let S be a nonempty commutative semigroup written additively. An element e of S is said to be idempotent if e + e = e. The Erdős-Burgess constant I (S) of the semigroup S is defined as the smallest positive integer ℓ such that any S -valued sequence T of length ℓ contains a nonempty subsequence the sum of whose terms is an idempotent of S. We make a study of I (S) when S is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions such that I (S) is finite, and we obtain sharp bounds of I (S) in case I (S) is finite, and determine the precise value of I (S) in some cases which unifies some well known results on the precise value of Davenport constant in the setting of commutative semigroups. [ABSTRACT FROM AUTHOR]