On E-unitary (left regular) bands of groups

We study those semigroups S which are E-unitary (left regular) bands of groups. The main result of the paper says that any such S is isomorphic to the semigroup (under the induced multiplication of the direct product $$E\times G$$ ) $$(E,G,\varphi )=\{(e,a)\in E\times G:a\in e\varphi \},$$ where E is the set of idempotents of S, G is the maximum group homomorphic image of S and $$\varphi $$ is a function from E into the lattice of all subgroups of G that meets some natural additional condition. Moreover, we give a description of certain fundamental congruences on $$(E,G,\varphi )$$ . Finally, we characterize the structure of orthodox bands of groups.