For every H-space
, the
set of homotopy classes
possesses a natural algebraic structure of a loop near-ring. Albeit one cannot say
much about general loop near-rings, it turns out that those that arise from
H-spaces are sufficiently close to rings to have a viable Krull–Schmidt type
decomposition theory, which is then reflected into decomposition results of
H-spaces. In the paper, we develop the algebraic theory of local loop near-rings
and derive an algebraic characterization of indecomposable and strongly
indecomposable H-spaces. As a consequence, we obtain unique decomposition
theorems for products of H-spaces. In particular, we are able to treat certain infinite
products of H-spaces, thanks to a recent breakthrough in the Krull–Schmidt
theory for infinite products. Finally, we show that indecomposable finite
–local
H-spaces are automatically strongly indecomposable, which leads to an easy
alternative proof of classical unique decomposition theorems of Wilkerson and
Gray.