Definition:Upper Semilattice on Classical Set

Definition
Let $\left({S, \le}\right)$ be a poset 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.