In order to find the supremum or infimum of a Hasse diagram we follow the outgoing lines from the elements up for supremum or down for infimum until the lines meet each other.

My question is, do we have to go only straight or are we allowed to turn left or right until we reach the meeting point. In this diagram for example I know the Sup ({c,d}) = 1 but what about the Inf? Is the Inf ( {c,d} ) = 0 or doesn't it exists?

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Is there any difference in the meaning if two elements are directly connected or not? What is for example the difference between Sup ({b,e}) = e and Sup ({b,c}) = e or Sup({d,1}) = 1 and Sup({d,c}) = 1?


It's not quite as simple as that. In my opinion, it is better to use (or at least think of) the terms least upper bound instead of supremum and greatest lower bound instead of infimum.

So for example, $1$ is an upper bound of $b$ and $c$ (go from $b$ to $d$ to $1$ and from $c$ to $e$ to $1$), but it is not the least upper bound, because $e$ is also an upper bound of $b$ and $c$, and it is lower than $1$. So yes, you are allowed to go left or right, but you can not just pick an arbitrary upper bound, you have to find the least one.

$\inf\{c,d\} = 0$, because $0$ is the only lower bound of $c$ and $d$. When you say "directly connected", it simply means that one of the elements is a lower bound of both of them, and the other element is an upper bound of both of them. For example, when taking the supremum of $b$ and $e$, notice that $e$ is an upper bound of $b$, and it is also an upper bound of itself. Clearly, for any element $x$ there can be no element which is both an upper bound of $x$ and less than $x$, so $e$ is also the least upper bound. So for "directly connected" elements, one of the two must be the supremum and one of the two must be the infimum. This is also how one can recover the ordering using supremums and infimums: $a \leq b$ iff $\inf\{a,b\} = a$ iff $\sup\{a,b\} = b$.

  • $\begingroup$ Thanks for the answer. It would be nice if you could also explain the second part of the question - the meaning if two elements are directly connected or not but have the same Sup. or Inf. $\endgroup$
    – Infinity
    Apr 2 '16 at 11:09
  • $\begingroup$ @Infinity that is what my last part of the answer is about. $\endgroup$
    – mrp
    Apr 2 '16 at 13:26

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