Upper and lower bounds

An upper bound of a subset is an element above every member, and a lower bound is below every member. They need not lie in the subset or be unique. Infimum and supremum are the greatest lower and least upper bounds when those exist.

An upper bound of a subset $A$ of a partially ordered set is an element $u$ satisfying $a\le u$ for every $a\in A$.

The least upper bound in a partially ordered set of a subset $A$, when it exists. It bounds every element of $A$ and is at most every other upper bound. For two elements it is written $x\vee y$; an empty join is a least element.

For $D=\{(x,y):x^2+y^2<1\}$, the product order gives least upper bound $(1,1)$, because each coordinate has supremum one. The lexicographic order instead has no least upper bound. Every point $(1,v)$ is an upper bound, but $(1,v-1)$ is a strictly smaller upper bound; a point with first coordinate below one cannot bound the disc. The same geometric set therefore has different order-completeness behavior under the two partial orders. Neither conclusion is determined by the usual drawing alone.