Square and non-reflection in the context of

We define □ P κ λ , a square principle in the context of P κ λ , and prove its consistency relative to ZFC by a directed-closed forcing and hence that it is consistent to have □ P κ λ hold when κ is supercompact, whereas □ κ is known to fail under this condition. The new principle is then extended to produce a principle with a non-reflection property. Another variation on □ P κ λ is also considered, this one based on a family of club subsets of P κ x ( x ) . Finally, a new square principle for cardinals, denoted □ κ f , is introduced. This principle is proved consistent with κ being supercompact. It is shown to yield a non-reflection result similar to that given by □ κ .