The three quantifier theory of
(
R
,
≤
T
)
(\mathcal {R},\leq _{T})
, the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of
R
\mathcal {R}
that lies between the two and three quantifier theories with
≤
T
\leq _{T}
but includes function symbols. Theorem. The two quantifier theory of
(
R
,
≤
,
∨
,
∧
)
(\mathcal {R},\leq ,\vee ,\wedge )
, the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on
R
\mathcal {R}
) is undecidable. The same result holds for various lattices of ideals of
R
\mathcal {R}
which are natural extensions of
R
\mathcal {R}
preserving join and infimum when it exits.