Take the 2-minute tour ×  Let $A=\{1,2,3,4,5,6\}$ and let's
consider the usual order relation given by $\leq$. My textbook
includes an image representing this structure:  enter image
description here  Given the definition of lattice, I don't
understand how this isn't it. When I am looking for the supremum
and the infimum for every pair it is obvious that $\forall
(a,b)\in A$  $$\begin{aligned} \;a\leq6 \text{ and } b\leq6,
1\leq a \text{ and } 1\leq b\end{aligned}$$  so what is that I
don't take into account?  share|improve this question I think
2,3 don't have a join, and 4,5 don't have a meet? –  sperners
lemma Nov 13 '12 at 18:13 @spernerslemma I'm trying to figure
out this thing myself... and it's kind of confusing. I don't
know why they aren't joined! –  haunted85 Nov 13 '12 at 18:15
Sperners observation is correct. Both $4$ and $5$ are upper
bounds for $2$ and $3$. There is no least upper bound. Similarly
there is no greatest lower bound for $4$ and $5$. –  Jay Nov 13
'12 at 18:23 add comment  2 Answers