This is a very dumb example, but suppose we augment the real line by adding two additional elements, $\infty$ and $\widetilde{\infty}$. We order this set by declaring $x < \infty$ and $x < \widetilde{\infty}$ for all real numbers $x$. (But we declare no order between these two new elements, so, in particular, the ordering is no longer total).
Note that both $\infty$ and $\widetilde{\infty}$ are upper bounds of $\mathbb{R}$ (the real reals) in this augmented order.
- Since there is no upper bound of $\mathbb{R}$ strictly below either, they are actually minimal upper bounds.
- However neither $\infty$ nor $\widetilde{\infty}$ is a least upper bound, because neither $\infty < \widetilde{\infty}$ nor $\widetilde{\infty} < \infty$ holds. (A least upper bound must be strictly below any other upper bound.)
This is the difference between the adjectives minimal and least.
- To be minimal with respect to some property an object must have that property, and no object strictly below it can share that same property.
- To be least with respect to some property an object must have that property, and it must be strictly below any other object having that property.
In terms of partially ordered sets, every least object will be a minimal object, but the converse may not hold. The two concepts do coincide in linear (total) orders.