Supremum

In mathematics, the infimum (abbreviated inf; pl.: infima) of a subset${\displaystyle S}$ of a partially ordered set${\displaystyle P}$ is the greatest element in ${\displaystyle P}$ that is less than or equal to each element of ${\displaystyle S,}$ if such an element exists. If the infimum of ${\displaystyle S}$ exists, it is unique, and if b is a lower bound of ${\displaystyle S}$, then b is less than or equal to the infimum of ${\displaystyle S}$. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; pl.: suprema) of a subset ${\displaystyle S}$ of a partially ordered set ${\displaystyle P}$ is the least element in ${\displaystyle P}$ that is greater than or equal to each element of ${\displaystyle S,}$ if such an element exists. If the supremum of ${\displaystyle S}$ exists, it is unique, and if b is an upper bound of ${\displaystyle S}$, then the supremum of ${\displaystyle S}$ is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB).