  When the Canonical Ramsey's Theorem by Erd\H{o}s and Rado is applied to
regressive functions one obtains the Regressive Ramsey's Theorem by Kanamori
and McAloon. Taylor proved a "canonical" version of Hindman's Theorem,
analogous to the Canonical Ramsey's Theorem. We introduce the restriction of
Taylor's Canonical Hindman's Theorem to a subclass of the regressive functions,
the $\lambda$-regressive functions, relative to an adequate version of
min-homogeneity and prove some results about the Reverse Mathematics of this
Regressive Hindman's Theorem and of natural restrictions of it.
  In particular we prove that the first non-trivial restriction of the
principle is equivalent to Arithmetical Comprehension. We furthermore prove
that this same principle strongly computably reduces the
well-ordering-preservation principle for base-$\omega$ exponentiation.
