Question

# Give an example of a poset which has exactly one maximal element but does not have a greatest element.

Discrete math
Give an example of a poset which has exactly one maximal element but does not have a greatest element.

2021-03-12

Here we consider a poset which has exactly one maximal element but does not have a greatest element.
Recall Minimal and Maximal Element of a Poset: Let $$\displaystyle{\left({S},\le\right)}$$ be a poset. An element $$\displaystyle{a}∈{S}$$ said to be a minimal element if x $$(\displaystyle{x}∈{S})$$. An clement $$\displaystyle{b}∈{S}$$ is said to be a maximal clement if $$b < x$$ for no $$\displaystyle{x}∈{S}$$.
Clearly, the least element in $$\leq$$ poset is a minimal clement and the greatest clement in a poset is a maximal element but the converse is not true.
Let $$\displaystyle{\left({N},\le\right)}$$ is a poset where $$m \leq n$$ means “m is a divisor of n” for m,$$\displaystyle{n}∈{N}$$.
This poset $$\displaystyle{\left({N},\le\right)}$$ contains no greatest element and no maximal element.
The least $$\displaystyle{\left({N},\le\right)}$$ contains no greatest element and no maximal element.