Discussion
Consider the following statements about a lattice L :
(i) Every ideal of L is a prime ideal.
(ii) L is a chain.
Then which of the following is correct ?
Answer: C
The answer is (I) if and only if (II).
If L is a chain and \(a \wedge b \in I\) for an ideal I, then \(a \wedge b\) is the smaller of \(a\) and \(b\), so one of them lies in I: every ideal is prime.
Conversely, suppose every ideal is prime but \(a\) and \(b\) are incomparable. The principal ideal \((a \wedge b]\) contains \(a \wedge b\), so being prime it contains \(a\) or \(b\). If it contains \(a\), then \(a \le a \wedge b \le b\), contradicting incomparability; similarly for \(b\). So L is a chain. Each condition implies the other.
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