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We consider the logarithmic minimal models LM(1,p) as `rational' logarithmicconformal field theories with extended W symmetry. To make contact with theextended picture starting from the lattice, we identify 4p-2 boundaryconditions as specific limits of integrable boundary conditions of theunderlying Yang-Baxter integrable lattice models. Specifically, we identify 2pintegrable boundary conditions to match the 2p known irreducibleW-representations. These 2p extended representations naturally decompose intoinfinite sums of the irreducible Virasoro representations (r,s). A further 2p-2reducible yet indecomposable W-representations of rank 2 are generated byfusion and these decompose as infinite sums of indecomposable rank-2 Virasororepresentations. The fusion rules in the extended picture are deduced from theknown fusion rules for the Virasoro representations of LM(1,p) and are found tobe in agreement with previous works. The closure of the fusion algebra on afinite number of representations in the extended picture is remarkableconfirmation of the consistency of the lattice approach.

Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)