Halting Problem and Turing Degree and Reduction? This is a Local Olympiad question on computation and computer science on 2013. How can explain it and says some hint for understanding such an example question.


  for $ A \subseteq \mathbb{N}$ we have $a=deg_T(A)=\{B | B \equiv_T A \} $ and $D=\{deg_T(A)|  A \subseteq \mathbb{N} \}$. For $(D, \leq)$
  that has $A \leq_T B$ iff $ a \leq b$. which of the following is
  false:
  
  1) $(D, \leq)$ is a distributive lattice
  
  2) $(D, \leq)$ ‌ is bounded (has minimum and maximum)
  
  3) $(D, \leq)$ is a half disjunctive lattice. (may be I‌ worded this
  statement poorly, sorry)
  
  4) he maximum elements of $(D, \leq)$ is a degree of Halting Problem .


I think $deg_T$ means Turing Degree and $\leq_T‌$ means Turing Reduciblity.

Edit 1:
To complete LogicLove's answer, the solution for (1) can be found in a math.se question.

As there are pairs of degrees which have no greatest lower bound, the conditions for a distributive lattice can't be met.