Flat unars

Flat modules over rings, acts over semigroups are modules or acts such that functor A⊗− preserves monomorphisms. A unar, i.e. a set with only an unary operation can be considered as an act over a free cycle semigroup. It is shown that a unar is flat if and only if it is a coproduct of unars, each of which is a line, ray or cycle.