In this article we solve two questions of Odifreddi on the r.e.
tt
{\text {tt}}
-degrees. First we construct an r.e.
tt
{\text {tt}}
-degree with anticupping property. In fact, we construct r.e.
tt
{\text {tt}}
-degrees
a
,
b
{\mathbf {a}},{\mathbf {b}}
with
0
>
a
>
b
{\mathbf {0}} > {\mathbf {a}} > {\mathbf {b}}
and such that for all (not necessarily r.e.)
tt
{\text {tt}}
-degrees
c
{\mathbf {c}}
if
a
∪
c
≥
b
{\mathbf {a}} \cup {\mathbf {c}} \geq {\mathbf {b}}
then
a
≤
c
{\mathbf {a}} \leq {\mathbf {c}}
. This result also has ramifications in, for example, the r.e.
wtt
{\text {wtt}}
-degrees. Finally we solve another question of Odifreddi by constructing an r.e.
tt
{\text {tt}}
-degree with no greatest r.e.
m
m
-degree.