Effectively closed mass problems and intuitionism

Let Ps denote the lattice of all strong degrees (or Medvedev degrees) of nonempty (or effectively closed) subsets of the Cantor space 2ω, and let Pw denote the lattice of all weak degrees (or Muchnik degrees) of nonempty subsets of 2ω. We show that Ps is not a Brouwer algebra and Pw is not a Heyting algebra. These give solutions to problems presented by Simpson in 2008 and by Terwijn in 2006.