Flat Unars

A flat act over a simigroup (as well as a flat module over a ring) is such act A that functor $$A \otimes - $$ preserves monomorhpisms. Flat modules over rings, acts over semigroups are modules or acts such that functor $$A \otimes - $$ preserves monomorphisms. A unar, that is, a set with only an unary operation can be considered to be an act over a free cycle semigroup. It is shown that a unar is flat iff it is a coproduct of unars, each of which is a line, ray, or cycle.