EDB — 2B2

[2B2]Suppose that on the set \(X\) there is a relation \(R\) that is reflexive and transitive and satisfies

(as seen in [(3.96)]↺↻)

This pair \((X,R)\) is a "Directed Set" according to the usual definition (see [ 17 ] or other references in [ 43 ] ).

Show that there exists another relation \(≤\) such that

• \(≤\) is a partial order and it satisfies [(3.96)]↺↻;

• \(R\) extends \(≤\) that is;

  \[ ∀ x,y∈ X ~ x≤ y⇒~ x R y\quad ; \]

• moreover \((X,≤)\) is cofinal in \((X,R)\).