Radicals Commuting with Bands of Semigroups

It is known that ring and semigroup theories have evolved many similar methods. The exchange of ideas between these theories has enriched both of them. Among ring concepts transfered to semigroups, the notion of a radical deserves mentioning. The analogy with radicals of rings suggests that semigroup radicals may be applicable in studying the structure of semigroups. On the other hand, many structure results involve bands of semigroups. Therefore in order to apply radicals it seems important to know what are the radicals interacting with bands well in a sense. Let ρ be a radical, B a band. Defering complete definitions to §1, now we only say that one of the most convenient cases arises when ρ(S) = ∪b∈Bρ(Sb) for every S = ∪b∈BSb, band of semigroups. The aim of the present paper is to describe radicals with this property.