Coprime elements are finitary analogues of compact elements.
Let (D,⊑)(D, \sqsubseteq)(D,⊑) be a partial order.
An element c∈Dc \in Dc∈D is coprime just if
c⊑x⊔y⟹c⊑x or d⊑yc \sqsubseteq x \sqcup y \quad\Longrightarrow\quad c \sqsubseteq x \text{ or } d \sqsubseteq y c⊑x⊔y⟹c⊑x or d⊑y
whenever x⊔yx \sqcup yx⊔y exists.
The definition is usually phrased in terms of complete lattices.