  A systematic study has been carried out to obtain the fundamental units of
triad interaction in Hall magnetohydrodynamic turbulence. Instead of finding
the elementary building blocks of non-unique mode-to-mode transfer rates, we
have investigated the fundamental units for uniquely defined combined transfers
and convincingly showed that the mode-to-mode transfers can act as a practical
base element for the same. In addition to the conventional field-specific
mode-to-mode transfers, here we have introduced the idea of mode-specific
transfers which is found to be important for the turbulent cascade and the
turbulent relaxed states. Whereas the Hall transfer is found to associate
mode-to-mode transfers for mode-specific interactions (with a three-member
basis), it presents a mixture of typical mode-to-mode (also with a three-member
basis) and non mode-to-mode (with a five-member basis) transfers for the
field-specific interactions. The non mode-to-mode transfers are shown to
satisfy the triad conservation differently from the mode-to-mode transfers.
However, they also possess an inherent non-uniqueness and hence cannot be
determined unambiguously unlike the combined transfer rates.
