  The main purpose of this paper, is to introduce and study the classes
$(ab_{e})$ and $(aw_{e})$ which are strongly related to what has been recently
studied in \cite{aznay-zariouh}. Furthermore, we give the connection between
these classes and those that have been studied in \cite{berkani-zariouh0}. We
also give an affirmative answer to a question asked in \cite{aznay-zariouh}.
