H-spaces are examined by
studying left translations, actions and a homotopy version of left translations to be
called homolations. If (F,m) is an H-space, the map s : F → FF given by s(x) = Lx,
i.e. s(x) is left translation by x, is a homomorphism if and only if m is associative. In
general, s is an An-map if and only if (F,m) is an An+1 space. The action
r : FF× F → F is given by r(φ,x) = φ(x). The map s respects the action only
of left translations. In general, s respects the action of homolations up to
higherorder homotopies. Each homolation generates a family of maps to be
called a homolation family. Denoting the set of all homolation families by
H∞(F),s : F → FF factors through F → H∞(F) and this latter map is a homotopy
equivalence.
