Fuzzy generators and fuzzy direct sums of Abelian groups

We define the concept of a fuzzy generating set and describe the fuzzy subgroup which it generates. We introduce the notion of a minimal fuzzy generating set and we show that any fuzzy subgroup whose support is cyclic has a minimal fuzzy generating set. We show by examples that this result breaks down if the support of A is a direct sum of two or more cyclic groups. These examples also show that a fuzzy subgroup of a group G need not be a fuzzy direct sum of subgroups whose supports are cyclic even if G is a direct sum of cyclic groups. We show that every fuzzy subgroup is a fuzzy direct sum of p-primary fuzzy subgroups.