On one-parameter semigroups generated by commuting continuous injections

The problem of the embeddability of two commuting continuous injections f,g:I=(0,b]→I in Abelian semigroups is discussed. We consider the case when there is no iteration semigroup in which f and g can be embedded. Explaining this phenomenon we modify the definition of an iteration semigroup introducing a new notion - a T-iteration semigroup of f and g, that is a family {ft:I→I,t∈T} of continuous injections for which fu∘fv=fu+v, u,v∈T, such that f=f1 and g=fs for an s∈T and s∉Q, where T⊊R+ is a dense semigroup which can be extended to a group. We determine a maximal semigroup of indices Sem(f,g)⊊R+ such that for every T-iteration semigroup T⊂Sem(f,g). We give also a construction of maximal T-iteration semigroups of f and g that is such semigroups for which T=Sem(f,g). We examine also some other Abelian semigroups of continuous functions containing f and g.