Definition:Upper Semilattice on Classical Set

Definition

Let $\struct {S, \le}$ be an ordered set with the property that:

$\forall x, y \in S: \sup \set {x, y} \in S$

where $\sup$ denotes supremum.

Then $\struct {S, \vee}$ is called an upper semilattice, where $\vee: S \times S \to S$ is defined by:

$x \vee y := \sup \set {x, y}$

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