Square and non-reflection in the context of Pκλ

We define □Pκλ, a square principle in the context of Pκλ, and prove its consistency relative to ZFC by a directed-closed forcing and hence that it is consistent to have □Pκλ 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 □Pκλ is also considered, this one based on a family of club subsets of Pκx(x). Finally, a new square principle for cardinals, denoted , 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.