Consider the following Hasse diagrams. Which all of the above represent a…
2008
Consider the following Hasse diagrams.
Which all of the above represent a lattice?
- A.
(i) and (iv) only
- B.
(ii) and (iii) only
- C.
(iii) only
- D.
(i), (ii) and (iv) only
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Correct answer: A
Answer: diagrams (i) and (iv) only.
Diagram (i): lattice. This is the diamond-shaped poset. For every pair of elements a and b the greatest lower bound (meet) and least upper bound (join) exist. For example, the two middle elements have meet equal to the bottom element and join equal to the top element.
Diagram (ii): not a lattice. There exist two elements (the two lower/incomparable elements) whose common upper bounds include more than one minimal element, so there is no unique least upper bound (no join) for that pair. Thus the lattice property fails.
Diagram (iii): not a lattice. Some pairs of elements do not have a unique least upper bound (or greatest lower bound), because there are multiple incomparable minimal upper bounds for those pairs. Therefore it fails to be a lattice.
Diagram (iv): lattice. This Hasse diagram is a totally ordered set (a chain). In any chain every two elements are comparable, so each pair has a unique meet (the lesser) and a unique join (the greater).
Therefore only diagrams (i) and (iv) satisfy the lattice condition.
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