  We construct new Yang-Baxter integrable boundary conditions in the lattice
approach to the logarithmic minimal model WLM(1,p) giving rise to reducible yet
indecomposable representations of rank 1 in the continuum scaling limit. We
interpret these W-extended Kac representations as finitely-generated W-extended
Feigin-Fuchs modules over the triplet W-algebra W(p). The W-extended fusion
rules of these representations are inferred from the recently conjectured
Virasoro fusion rules of the Kac representations in the underlying logarithmic
minimal model LM(1,p). We also introduce the modules contragredient to the
W-extended Kac modules and work out the correspondingly-extended fusion
algebra. Our results are in accordance with the Kazhdan-Lusztig dual of tensor
products of modules over the restricted quantum universal enveloping algebra
$\bar{U}_q(sl_2)$ at $q=e^{\pi i/p}$. Finally, polynomial fusion rings
isomorphic with the various fusion algebras are determined, and the
corresponding Grothendieck ring of characters is identified.
