Question

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

Discrete math
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asked 2021-03-11
Give an example of a poset which has exactly one maximal element but does not have a greatest element.

Answers (1)

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.

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