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Are there lattices in which not every chain has an upper bound?

I think not but I think I'm wrong wrong because I read the sentence "Let $L$ be a lattice in which every chain has an upper bound...."

Thanks for your help.

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1 Answer 1

up vote 6 down vote accepted

A lattice can even have chains that are unbounded in both directions. A lattice is simply a partial order $\langle L,\le \rangle$ in which every two elements $x$ and $y$ have both a least upper bound, called their join and written $x \lor y$, and a greatest lower bound, called their meet and written $x \land y$. In particular, $\langle \mathbb{Z},\le\rangle$ is a lattice that is itself a chain with neither an upper bound nor a lower bound: for $m,n \in \mathbb{Z}$, $m\lor n = \max\{m,n\}$, and $m\land n = \min\{m,n\}$.

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