Simplest Example of a Poset that is not a Lattice A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a partially ordered set can fail to be a lattice. In $\mathbb{R}$, for example, once two elements are selected the completeness of the real numbers guarantees the existence of both the infimum and supremum. Now, if we restrict our attention to a nondegenerate interval $(a,b)$ it is clear that no two points in $(a,b)$ have either a suprememum or infimum in $(a,b)$. 
Is this the right way to think of a poset that is not a lattice? Is there perhaps a more fundamental example that would yield further clarity?

Here a simple example, consider the $\{0,1,2\}$ with the order 
$$\{(0,0),(1,1),(2,2),(0,1),(0,2)\}$$
this is indeed a poset (the verification is simple) but it is not a lattice because it doesn't have any supremum of $1$ and $2$, i.e  $1 \lor 2$.
