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Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole

Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of upper bounds for $A$. A meet of $A$, if such exists, is a greatest element in the set of lower bounds for $A$.

Exercise: Make sure you understand the definition of meet and join in a preorder $X$. Think of some simple finite preordered sets in which meets and joins do not exist.

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    @Théophile My apologies. I quoted the first part of the exercise verbatim from the book. I have now added the definitions as well for clarification.2012-07-23

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Let $X = {a, b}$. Define a preorder on $X$ as $a \le a$ and $b \le b$. Now suppose that $\vee X$ is a join of $X$. Then $\vee X$ is an upperbound of $X$. So $a \le \vee X$ and $b \le \vee X$. So $\vee X = a$ and $\vee X = b$, which is a contradiction. Therefor, $\vee X$ does not exist. A similar proof will show that $X$ does not have a meet, either.