  We introduce the notion of (graded) anchored Lagrangian submanifolds and use
it to study the filtration of Floer' s chain complex. We then obtain an
anchored version of Lagrangian Floer homology and its (higher) product
structures. They are somewhat different from the more standard non-anchored
version. The anchored version discussed in this paper is more naturally related
to the variational picture of Lagrangian Floer theory and so to the likes of
spectral invariants. We also discuss rationality of Lagrangian submanifold and
reduction of the coefficient ring of Lagrangian Floer cohomology of thereof.
