In this paper, we introduce the notion of weak-group, hemi-group, weak-B-algebra and hemi-B-algebra\n as examples of constructions on general types of algebras. We show that in these cases the groupoid types obtained are (logically) equivalent. As a further consequence we are also able to show that they are logically equivalent to groups and B-algebras. Finally, some observations of a more general nature are also considered.