Subset of a partially ordered set: Least upper bound and greatest lower bound.

Question

Let $(A, \leq)$ be a poset and $B \subseteq A$. I need to show

i) B may have at most one least upper bound in A

ii) B may have at most one greatest lower bound in A.

Answer 1

This is really a matter of unpacking and applying the definitions of what it means

• for $A$ to be a poset (poset: "partially ordered set")

  • (See also the section of that entry subtitled Extrema)
• for $B$ to be a subset of $A$,
• for a set to have a least upper bound (lub) and
• for a set to have greatest lower bound (glb).

Recall the definition of a least upper bound as:

Let S be a poset:

1. $\alpha$ is an upper bound for S if $x \leq \alpha \space \forall x \in S$, (upper and lower bounds are NOT unique) and
2. $\beta$ is the least upper bound for S if $\beta$ is an upper bound, and $\beta \leq \alpha$ whenever $\alpha$ is an upper bound for S.

A symmetrical definition defines the greatest lower bound.