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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. © 1971 Pacific Journal of Mathematics.

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