We study five extensions of the polymorphically typed lambda-calculus (systemF) by type constructs intended to model fixed-points of monotoneoperators. Building on work by Geuvers concerning the relation between termrewrite systems for least pre-fixed-points and greatest post-fixed-points ofpositive type schemes (i.e., non-nested positive inductive and coinductivetypes) and so-called retract types, we show that there are reduction-preservingembeddings even between systems of monotone (co)inductive types andnon-inter leav ing positive fixed-point types (which are essentially thoseretract types). The reduction relation considered is β- and η-reduction for systemFF plus either (full) primitive recursion on the inductive types or (full)primitive corecursion on the coinductive types or an extremely simple rule forthe fixed-point types. Monotonicity is not confined to the syntacticrestriction on type formation of havingonly positive occurrences of the type variable α in ρ forthe inductive type µαρ or the coinductive type ναρ. Instead of thatonly a“monotonicity witness” which is a term of type ∀α∀β.(α → β) → ρ → ρ[α:=β] is required. Thisterm may already use (co)recursion such that our monotone (co)inductive typesmay even be “interleaved” and not only nested.