Let S be a topological
Hausdorff semigroup and s ∈ S be a strongly root compact element. Then there are
an algebraic morphism f : Q+∪{0}→ S with f(0) = e, f(1) = s, and a
one-parameter semigroup ϕ :H→ S which satisfy the following properties: If
K = ∩{f(Q) : 0 < 𝜖 < 1}, then K is a compact connected abelian subgroup of
ℋ(e), ϕ(0) = e, ϕ(H) is in the centralizer Z = {x ∈ eSe : xk = kx for all k ∈ K}
of K in eSe, and ϕ(t) ∈ f(t)K for each t ∈ Q+. Furthermore, if 𝒰 is any
neighborhood of s in S, then ϕ may be chosen so that ϕ(1) ∈𝒰: and, in
fact, if K is arcwise connected, then ϕ may be chosen so that ϕ(1) = s. The
above statements also hold for strongly p-th root compact elements almost
everywhere.
![]0,𝜖[](a270x.png)
