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## Explanation: Notes on Special Elements of Posets

Please note that a maximum or minimum of $S$ do not have to exist. For instance, the chain of integers $(Z,\ge)$ neither has a maximum, nor a minimum. However, if a maximum or a minimum of $S$ exist, then they are unique, which follows immediately from their definition. For instance, if $g$ and $g’$ were two maxima, then $g\prec g’$ and $g\succ g’$ would follow by definition. Therefore, $g’=g,$ and thus $g$ is unique.

Moreover, lower and upper bounds of $S$ do not have to be the elements of $S.$ For instance, $1$ is an upper bound of the set of all negative real numbers, but $1$ is not a negative real number. The same holds for the infimum and the supremum. To take the same example, $0$ is a supremum of all negative real numbers but it is still not a negative real number.

Below, a Hasse diagram of a simple poset $V:\{a,b,c,d,e,f,g\}$ is given, in which the elements of the subset $S=\{a,b,c,d,e\}$ are drawn as dark nodes:

In this poset, the following observations can be made:

• $g$ is a maximum of $V$ and, at the same time, its supremum.
• $a$ is a minimum of $V$ and of $S.$ It is also a minimal element and an infimum of both sets.
• $S$ has no maximal element, but $d$ and $e$ are maximal in $S.$
• $\sup(S)=f,$ $\sup(V)=g,$
• $S^u=\{g,f\},$ $V^u=\{g\},$ $S^l=V^l=\{a\}.$

| | | | Contributors: bookofproofs | References: [577], [979]

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