# Definition:Upper Semilattice on Classical Set

## Definition

Let $\left({S, \le}\right)$ be an ordered set with the property that:

- $\forall x, y \in S: \sup \left\{{x, y}\right\} \in S$

where $\sup$ denotes supremum.

Then $\left({S, \vee}\right)$ is called an **upper semilattice**, where $\vee: S \times S \to S$ is defined by:

- $x \vee y := \sup \left\{{x, y}\right\}$

An **upper semilattice** hence is a particular kind of algebraic structure.

## Also see

- Upper Semilattice on Classical Set is Semilattice, the justification for the nomenclature.