# Lattice GLB uniqueness .

In the above lattice is the GLB of b and c unique? I think the GLB of b and c should be only d. In my book it's given that the GLB of b and c is not unique as GLB of b and c can even be a.

I think that GLB of b and c would not have been unique if d and a would have been at same level which is not the case here.

Can someone tell me whether I am right?

By definition, the GLB (or $\inf$, for infimum), is the greatest lower bound. Looking at your Hasse diagram, the set $\{b,c\}$ has two lower bounds, namely $a$ and $d$. However, these two (i.e. $a$ and $d$) are not comparable, so there is no maximum among the lower bounds. Hence, the GLB of $b$ and $c$ does not exist. In particular, this tells you that your ordered set is not a lattice.
• @Zephyr As William says, the order relation is indicated by the lines; the height at which the elements are drawn means nothing if there's no line. If there were a line joining $a$ and $d$ and $a$ was lower than $d$, then we would have $a\le d$, but this is not the case. – Reveillark Jul 23 '17 at 16:50