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I understand that join semilattices are upward directed sets, but why not the converse?

A simple counterexample would be very helpful.

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2 Answers 2

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The following set is an (upward) directed set, but not a join-semilattice: $\{\{a\}, \{d\}, \{a, b, d\}, \{a, c, d\}, \{a, b, c, d\}\}.$

It is a directed set, because every pair has at least one upper bound. It is not a join-semilattice, because not every pair has a least upper bound. The pair $(\{a\}, \{d\})$ has three upper bounds: $\{a, b, d\}, \{a, c, d\} \; and \; \{a, b, c, d\}.$

None of the three is a least upper bound.

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Suppose $P$ is a nonempty partially ordered set with no least element, like $(0,\infty)$ for example, and adjoin two incomparable minimums $m_1$ and $m_2$. They have no least upper bound since $P$ has no least element but they have upper bounds since $P$ is nonempty.

(Compare with the "line with two origins," which is $\mathbb{R}$ but with two kinds of $0$.)

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