Algebrability and Faà di Bruno conditions on M suffice for stability under product and composition in Roumieu and Beurling Carleman classes with uniform asymptotics, and are necessary in Roumieu when suitable characteristic functions exist in the class.
Thilliez, On quasianalytic local rings, Expo
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Stability under product and composition for uniform Carleman asymptotic expansions
Algebrability and Faà di Bruno conditions on M suffice for stability under product and composition in Roumieu and Beurling Carleman classes with uniform asymptotics, and are necessary in Roumieu when suitable characteristic functions exist in the class.