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I'm looking for a set $M$ which is partially ordered by $\subseteq$. $M$ should have a lower bound but no infimum. Is that possible?

A lower bound is an element $x \in N$ with $M \subset N$ such that for all elements $y \in M$, $x \subseteq y$ holds.

An infimum is a maximal lower bound with respect to $\subseteq$.

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    @Brian and joachim: Got it...2012-11-04

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It doesn’t make sense to say that $M$ has a lower bound but no infimum if $M$ is the partially ordered set, so I’m assuming that you want a set $N$ partially ordered by $\subseteq$ in which there is a subset $M$ with a lower bound but no infimum. (This is now clear from the revised version of your question.)

That is possible. Let $N=\{(x,\to):x\in\Bbb Q\}$, the set of all open rays unbounded on the right; $N$ is even linearly ordered by $\subseteq$. Let $M=\{(x,\to)\in N:x>\sqrt2\}$.