  Strictly positive logics recently attracted attention both in the description
logic and in the provability logic communities for their combination of
efficiency and sufficient expressivity. The language of Reflection Calculus RC
consists of implications between formulas built up from propositional variables
and constant `true' using only conjunction and diamond modalities which are
interpreted in Peano arithmetic as restricted uniform reflection principles.
  We extend the language of RC by another series of modalities representing the
operators associating with a given arithmetical theory T its fragment
axiomatized by all theorems of T of arithmetical complexity $\Pi^0_n$, for all
n>0. We note that such operators, in a precise sense, cannot be represented in
the full language of modal logic.
  We formulate a formal system RC$^\nabla$ extending RC that is sound and, as
we conjecture, complete under this interpretation. We show that in this system
one is able to express iterations of reflection principles up to any ordinal
$<\epsilon_0$. On the other hand, we provide normal forms for its variable-free
fragment. Thereby, the variable-free fragment is shown to be decidable and
complete w.r.t. its natural arithmetical semantics.
  Whereas the normal forms for the variable-free formulas of RC correspond in a
unique way to ordinals below $\epsilon_0$, the normal forms of RC$^\nabla$ are
more general. It turns out that they are related in a canonical way to the
collections of proof-theoretic ordinals of (bounded) arithmetical theories for
each complexity level $\Pi^0_{n+1}$.
  Finally, we present an algebraic universal model for the variable-free
fragment of RC$^\nabla$ based on Ignatiev's Kripke frame. Our main theorem
states the isomorphism of several natural representations of this algebra.
