title: Turing Degree and Local Contest Questions?
body: <p>I ran into a olympiad question on Practical Computer Knowledge.</p>
  <blockquote>
  <p>For <em>A</em> ⊆ ℕ we have <em>a</em> = deg<sub><em>T</em></sub>(<em>A</em>) = {<em>B</em> | <em>B</em> ≡<sub><em>T</em></sub> A} and <em>D</em> = {deg<sub><em>T</em></sub>(<em>A</em>) | <em>A</em> ⊆ ℕ}. For (<em>D</em>, ≤) that has A ≤<sub><em>T</em></sub> <em>B</em> iff <em>a</em> ≤ <em>b</em>. Which of the following is false:</p>
  <ol>
  <li><p>(D,≤) is a distributive lattice</p>
  </li>
  <li><p>(D,≤)‌ is bounded (has minimum and maximum)</p>
  </li>
  <li><p>(D,≤) is a half disjunctive lattice. (may be I‌ worded this statement poorly, sorry)</p>
  </li>
  <li><p>he maximum elements of (D,≤) is a degree of Halting Problem .</p>
  </li>
  </ol>
  </blockquote>
  <p>I think deg<sub><em>T</em></sub> means <a href="https://en.wikipedia.org/wiki/Turing_degree" rel="nofollow noreferrer">Turing Degree</a> and ≤<sub><em>T‌</em></sub> means <a href="https://en.wikipedia.org/wiki/Turing_reduction" rel="nofollow noreferrer">Turing Reduciblity</a>.</p>
  <p>Who can give me a hint for these options, we want to check the answer
  sheet that wrote (1) is correct?</p>