  We construct an algebraic version of Lagrangian Floer homology for immersed
curves inside the pillowcase. We first associate to the pillowcase an algebra
A. Then to an immersed curve L inside the pillowcase we associate an A infinity
module M(L) over A. Then we prove that Lagrangian Floer homology HF(L,L') is
isomorphic to a suitable algebraic pairing of modules M(L) and M(L'). This
extends the pillowcase homology construction - given a 2-stranded tangle inside
a 3-ball, if one obtains an immersed unobstructed Lagrangian inside the
pillowcase, one can further associate an A infinity module to that Lagrangian.
