Coprime element

Coprime elements are finitary analogues of compact elements.

Definition

Let (D, \sqsubseteq) be a partial order.

An element c \in D is coprime just if

c \sqsubseteq x \sqcup y
  \quad\Longrightarrow\quad
c \sqsubseteq x \text{ or } d \sqsubseteq y

whenever x \sqcup y exists.

The definition is usually phrased in terms of complete lattices.

See also

• Prime algebraic lattice