  We introduce and study the notion of weak semi-greedy systems -which is
inspired in the concepts of semi-greedy and Branch semi-greedy systems and weak
thresholding sets-, and prove that in the context Markushevich bases in
infinite dimensional Banach spaces, the notions of \textit{ semi-greedy, branch
semi-greedy, weak semi-greedy, and almost greedy} Markushevich bases are all
equivalent. This completes and extends some results from \cite{Berna2019},
\cite{Dilworth2003b}, and \cite{Dilworth2012}. We also exhibit an example of a
semi-greedy system that is neither almost greedy nor a Markushevich basis,
showing that the Markushevich condition cannot be dropped from the equivalence
result. In some cases, we obtain improved upper bounds for the corresponding
constants of the systems.
