Definition:Bounded Below/Ordered Set

Definition

Let $\left({S, \preceq}\right)$ be a poset.

A subset $T \subseteq S$ is bounded below (in $S$) if:

$\exists m \in S: \forall a \in T: m \preceq a$

That is, there is an element of $S$ (at least one) that precedes all the elements in $T$.

If there is no such element, then $T$ is unbounded below (in $S$).

Also see

• Lower Bound
• Bounded Above
• Upper Bound
• Bounded

Sources

• James M. Hyslop: Infinite Series (1942): $\S 3$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): $\S 1.1$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 2.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 10$