  Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on
normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$
and we say that $f$ is completely regular when all natural extensions are
equal. In this manuscript, we have some results on the close-to-regular maps
and investigate the close-to-regularity of tri-linear maps. We investigate the
relation between Arens regularity of bounded bilinear maps and
close-to-regularity bounded tri-linear maps. We give a simple criterion for the
completely regularity of tri-linear maps. We provide a necessary and sufficient
condition such that the fourth adjoint $D^{****}$ of a tri-derivation is again
a tri-derivation.
