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Consider this Hasse Diagram of a poset: Hasse Diagram

This poset is not a complete Lattice.

From definition of complete lattice, we need to have a least upper bound and a greatest lower bound for each pair of elements (is this right)?

From what I understood, this is not a complete Lattice because we have for example no lower bound for (a,b).

But my question is whether in this poset we have a least upper bound for (a,b).

The candidates are $c,d,e$ but there is no least one because $$ c \leq e \quad d\leq e \quad c \nleq d \quad d \nleq c $$

Or can we just choose c or d as our least upper bound?

EDIT:

So, in the figure above, we don't have a lattice, since there is no least upper bound for (a,b), and there is no greater lower bound for (c,d).

So now I decided to change the diagram in this way:

Modified Diagram

In this case can we say that it is a complete lattice? Can we correctly say that we can choose c or d to be our least upper bound for (a,b). And choose a or b to be our greatest lower bound for (c,d)?

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  • $\begingroup$ least means that its smaller then all other upper bounds. is $c<d$ or the other way around? $\endgroup$
    – user2520938
    Aug 9, 2017 at 23:07
  • $\begingroup$ Right the first time; $a$ and $b$ have no least upper bound. The given poset is not an upper semilattice, nor is it a lower semilattice, since $c$ and $d$ have no greatest lower bound. $\endgroup$
    – bof
    Aug 9, 2017 at 23:14
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    $\begingroup$ (After your edit) What do the red lines represent? Take the lower one. Does it say that $a \leq b$ and $b \leq a$? If so, then $a = b$, since the order in a poset is anti-symmetric, so it doesn't make sense.You might want to check the definition of preorder. $\endgroup$
    – amrsa
    Aug 10, 2017 at 15:47
  • $\begingroup$ oh my... this cannot even happen you're completely right.. I think it was a too long day yesterday, should take the habit to post questions in the morning and not midnight :-/ $\endgroup$
    – Ergo
    Aug 10, 2017 at 15:53

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Among the upper bounds, there is no least element as you correctly pointed out. A least element of a set is an element that is less than any other element in the set (so no, we can't just choose $c$ or $d$, they aren't the least)

Your definition of complete lattice is slightly wrong tough. You gave de definition for a lattice: every pair of elements has a least upper bound and greatest lower bound). For a complete lattice you demand that every subset has a least upper bound and greatest lower bound. This is a stronger condition.

So in the beginning of your post you actually showed that your poset isn't even a lattice. (let alone a complete lattice)

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