Halting Problem and Turing Degree and Reduction?

Question

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:

Answer 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.

Answer 2

(2) and (4) are false as explained by Yuval's Comment.

The lattice of Turing degrees has a minimum (the recursive degree) but no maximum, since the jump of any degree (i.e., the corresponding halting problem) is larger than the degree.

(3) is true, because the Turing Degrees form an upper semi-lattice.

I have no idea for (1).