  We study in this article a possible further structure of homotopic nature on
multiplicative spectral sequences. More precisely, since Kadeishvili's theorem
asserts that, given a dg (or A-infinity-)algebra, its cohomology has also a
structure of A-infinity-algebra such that both become quasi-isomorphic, and in
a multiplicative spectral sequence one considers the cohomology of dg algebras
when moving from a term to the next one, a natural problem that arises is to
study how this two possible structures intertwine. We give such a homotopic
structure proposal, called A-infinity-enhancement of multiplicative spectral
sequences, which could be of interest in our opinion. As far we know, this
construction was studied only recently by S. Lapin, even though he did not
state any definition. Seeing that the procedure considered by Lapin is rather
complicated to handle, we propose an equivalent but in our opinion easier
approach. In particular, from our definition we show that the canonical
multiplicative spectral sequence obtained from a filtered dg (or
A-infinity-)algebra, which could be viewed as the main example, has such an
A-infinity-enhancement.
