Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 $m\land n = \min\{m,n\}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.