  While the set of Martin-L\"of random left-c.e. reals is equal to the maximum
degree of Solovay reducibility, Miyabe, Nies and
Stephan(DOI:10.4115/jla.2018.10.3) have shown that the left-c.e. Schnorr random
reals are not closed upwards under Solovay reducibility. Recall that for two
left-c.e. reals~$\alpha$ and~$\beta$, the former is Solovay reducible to the
latter in case there is a partially computable function $\varphi$ and
constant~$c$ such that for all rational numbers $q < \beta$ we have \[\alpha -
\varphi(q) < c(\beta - q).\] By requiring the translation function $\varphi$ to
be total, we introduce a total version of Solovay reducibility that implies
Schnorr reducibility. Accordingly, by Downey and Griffiths
(DOI:10.2178/jsl/1082418542), the set of Schnorr random left-c.e. reals is
closed upwards relative to total Solovay reducibility.
  Furthermore, we observe that the notion of speedability introduced by Merkle
and Titov (DOI:10.1007/978-3-030-50026-9_22) can be equivalently characterized
via partial computable translation functions in a way that resembles Solovay
reducibility. By requiring the translation function to be total, we obtain the
concept of total speedability. Like for speedability, this notion does not
depend on the choice of the speeding constant.
