Exercise · ZW-LATTICE-OR-ORDER-THEORY-009 · ★★★ Advanced
Let \(L\) be the poset with elements \(\hat{0}\), \(a\), \(b\), \(c\), \(\hat{1}\) and \(\hat{0} < a, b < c < \hat{1}\) (no relation between \(a\) and \(b\) beyond those stated). How many pairs of elements of \(L\) have no supremum ?
Choose one option.
A 2 B 3 C 0 D 1 Show answer ← Back to the question
Explanation Every pair has a supremum : for \(\{a,b\}\) the upper bounds are \(c\) and \(\hat{1}\), and \(c\) is the least of them.
A lattice is a partially ordered set (L, ≤) in which every pair of elements x and y has a supremum (x ∨ y) and an infimum (x ∧ y).
\[x \vee y = \sup\{x,y\}, \quad x \wedge y = \inf\{x,y\}\]
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