Radicals of an ordered semigroup in terms of type of ordered semigroups

A type F of ordered semigroups is a class of ordered semigroups such that (i) if S belongs to F and S  is isomorphic to S, then S  belongs to F , and (ii) any one-element (ordered) semigroup belongs to F . Given a type F of ordered semigroups and an ordered semigroup S, the radS  F is the intersection of all pseudoorders of S having type F (a pseudoorder σ on S has type F if the quotient semigroup of S by the congruence 1 :       has also type F - we consider the quotient semigroup as an ordered semigroup under the induced order relation by σ). The derived type  F of F is the class of all ordered semigroups S such that radS  F is the order relation of S. An F maximal homomorphic image of an ordered semigroup of an ordered semigroup S is an ordered semigroup S  in F for which there exists a homomorphism η of S onto S  with the factorization property: if υ is a homomorphism of S onto an ordered semigroup of type F , then there exists a homomorphism θ of S  onto T such that      . We give sufficient and necessary condition under which an ordered semigroup admits an F maximal homomorphic image. We show that every ordered semigroup has an  F maximal homomorphic image and finally for a type F of ordered semigroups we prove thatevery ordered semigroup has an F maximal homomorphic image if and only if