Let \({\varvec{S}}\) be a semigroup, and \({\varvec{D}}\) a finite directed graph with a nonempty edge set. The directed power graph of \({\varvec{S}}\) is the directed graph which has all elements of \({\varvec{S}}\) as vertices and has edges (x, y) for all x, \(y\in {\varvec{S}}\) such that \(x\ne y\) and y is a power of x. We say that \({\varvec{S}}\) is power \({\varvec{D}}\) -saturated if for every infinite subset \({\varvec{T}}\) of \({\varvec{S}},\) the directed power graph of \({\varvec{S}}\) contains a subgraph isomorphic to \({\varvec{D}}\) with all vertices in \({\varvec{T}}.\) Let \(\Delta \) be a group with identity \(\varepsilon ,\) and let \(\{{\varvec{S}}_\delta \}_{\delta \in \Delta }\) be a family of subsets of a semigroup \({\varvec{S}}\) with zero 0 such that: i) \({\varvec{S}}=\bigcup _{\delta \in \Delta }{\varvec{S}}_\delta ;\) ii) \({\varvec{S}}_\delta \cap {\varvec{S}}_\gamma =\{{\varvec{0}}\}\) for all distinct \(\delta ,\) \(\gamma \in \Delta ;\) iii) \({\varvec{S}}_\delta {\varvec{S}}_\gamma ={\varvec{S}}_{\delta \gamma }\) for all \(\delta ,\) \(\gamma \in \Delta .\) Then we say that \({\varvec{S}}\) is a strong homogeneous semigroup, and graded by \(\Delta ,\) or that \({\varvec{S}}\) is a strongly \(\Delta \) -graded semigroup. A semigroup \({\varvec{S}}\) without zero is said to be strongly \(\Delta \) -graded if \({\varvec{S}}^0={\varvec{S}}\cup \{{\varvec{0}}\}\) is. We characterize power \({\varvec{D}}\) -saturation of an infinite strongly \(\Delta \) -graded semigroup \({\varvec{S}}\) in terms of power \({\varvec{D}}\) -saturation of \({\varvec{S}}_\varepsilon .\) In particular, for an infinite group G, we prove that G is power \({\varvec{D}}\) -saturated if and only if the center \({\varvec{C}}(G)\) of G is not simple, both N and G/N are power D-saturated for every subgroup \({\varvec{N}}\) of \({\varvec{G}},\) contained in \({\varvec{C}}(G),\) and the index of \({\varvec{C}}(G)\) is not divisible by p for each quasicyclic p-subgroup of \({\varvec{G}}\) .
