Algebraic semigroups, C0-semigroups and generalized convexity

An Altomare projection gives rise to a C0-semigroup and to generalized convexity. Besides other properties, it has one which can be described in geometric terms as well as in terms of algebraic semigroups. We study operators having only this last property and the types of convexity induced by them. In this framework we consider also a conjecture formulated by F. Altomare and I. Rasa in 1999 and present related results.