  We consider the logarithmic minimal models LM(1,p) as `rational' logarithmic
conformal field theories with extended W symmetry. To make contact with the
extended picture starting from the lattice, we identify 4p-2 boundary
conditions as specific limits of integrable boundary conditions of the
underlying Yang-Baxter integrable lattice models. Specifically, we identify 2p
integrable boundary conditions to match the 2p known irreducible
W-representations. These 2p extended representations naturally decompose into
infinite sums of the irreducible Virasoro representations (r,s). A further 2p-2
reducible yet indecomposable W-representations of rank 2 are generated by
fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro
representations. The fusion rules in the extended picture are deduced from the
known fusion rules for the Virasoro representations of LM(1,p) and are found to
be in agreement with previous works. The closure of the fusion algebra on a
finite number of representations in the extended picture is remarkable
confirmation of the consistency of the lattice approach.
