Greatest element

Relationship to upper/lower bounds

If ${\displaystyle g\in P}$ then ${\displaystyle g}$ is a greatest element of ${\displaystyle S}$ if and only if ${\displaystyle g}$ is an upper bound of ${\displaystyle S}$ in ${\displaystyle (P,\leq )}$and${\displaystyle g\in S.}$ In particular, any greatest element of ${\displaystyle S}$ is also an upper bound of ${\displaystyle S}$ (in ${\displaystyle P}$) but an upper bound of ${\displaystyle S}$ in ${\displaystyle P}$ is a greatest element of ${\displaystyle S}$ if and only if it belongs to ${\displaystyle S.}$ In the particular case where ${\displaystyle P=S,}$ the definition of "${\displaystyle u}$ is an upper bound of ${\displaystyle S}$in ${\displaystyle S}$" becomes: ${\displaystyle u}$ is an element such that ${\displaystyle u\in S}$ and ${\displaystyle s\leq u}$ for all ${\displaystyle s\in S,}$ which is completely identical to the definition of a greatest element given before. Thus ${\displaystyle g}$ is a greatest element of ${\displaystyle S}$ if and only if ${\displaystyle g}$ is an upper bound of ${\displaystyle S}$in ${\displaystyle S}$.