the Algebras for monads in the category of subobjects

Resumen

For a given object Y in a category C, we construct the category
of T-Algebras (Eilenberg-Moore category) and Kleisli category corresponding
to the monad defined on partial order category SubC[Y ]. We
obtain sufficient condition for the right adjoint to be monadic for the string
of adjunction f(−) ⊣ f−1 ⊣ f#. Finally, given any adjunction the sufficient
condition for the comparison functor between the original category
and the category of T-Algebras derived from monad to have a left adjoint
is obtained.