Let L(R) denote the frame of reals [1], that is, the frame generated by all ordered pairs (p,q) of rationals, subject to the relations
(R1) (p,q) ^ (r,s) = (p v r, q ^ s),
(R2) (p,q) v (r,s) = (p,s) whenever p <= r < q <= s,
(R3) (p,q) =V{(r,s) | p < r < s < q},
(R4) V{(p,q) | p,q in Q} = 1.
For any frame L the real continuous functions on L are the frame homomorphisms L(R) --> L. They form a lattice-ordered ring (l-ring) C(L) [1]. An l-ring is Dedekind order complete, it may be recalled, in case each non-void set of elements that is bounded from above has a supremum. In general, due to axiom (R2) above, C(L) fails to be order complete. For that reason we introduced L(IR) in [2], the frame of partial reals, by removing this relation from the list.
The aim of this talk is to describe the frame of partial reals and the lattice of continuous partial real functions which plays a crucial role in the construction of the Dedekind completion of C(L).

References
[1] B. Banaschewski, The real numbers in pointfree topology, Textos de Matemática,
vol. 12, Departamento de Matemática da Universidade de Coimbra, 1997.
[2] I. Mozo Carollo, J. Gutiérrez García and J. Picado, On the Dedekind completion
of function rings, Forum Mathematicum, to appear.