Definition:Supremum

Ordered Set
Let $$\left({S; \preceq}\right)$$ be a poset.

Let $$T \subseteq S$$.

An element $$c \in S$$ is the supremum of $$T$$ in $$S$$ if:


 * 1) $$c$$ is an upper bound of $$T$$ in $$S$$;
 * 2) $$c \preceq d$$ for all upper bounds $$d$$ of $$T$$ in $$S$$.

Plural: Suprema.

The supremum of $$T$$ is denoted $$\sup \left({T}\right)$$.

The supremum of $$x_1, x_2, \ldots, x_n$$ is denoted $$\sup \left\{{x_1, x_2, \ldots, x_n}\right\}$$.

If there exists a supremum of $$T$$ (in $$S$$), we say that $$T$$ admits a supremum (in $$S$$).

The supremum of $$T$$ is often called the least upper bound of $$T$$ and denoted $$\mathrm{lub} \left({T}\right)$$.

Mapping
Let $$f$$ be a mapping defined on a poset $$\left({S; \preceq}\right)$$.

Let $$f$$ be bounded above on $$S$$.

It follows from the Continuum Property that the range of $$f$$ has a supremum on $$S$$.

Thus $$\sup_{x \in S} f \left({x}\right) = \sup f \left({S}\right)$$.