Exercise

13 February

• Define: Category, Functor
• Define: Preorder, partial order, least element, lattice, CPO, monotone, continuous
• Give the first few elements of the fixed-point approximation for each of the following recursive definitions:

  f k = if k < 1 then 1 else f (k - f (k - 2))
  
  ones = 1 : ones
  

  How can you use GHCi to check this?
• Prove: Each preorder can be seen as a category
• What are the order-related properties of a functor between preorder categories?
• Generalise the following order-related concepts into category-theoric concepts: least element, join, meet, lub