Dugundji extenders and retracts on generalized ordered spaces

For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an ${\displaystyle L_{ch}}$-extender (resp. ${\displaystyle L_h}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X\A; (iii) there is a continuous ${\displaystyle L_{ch}}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and the pointwise convergence topology, for each space Y; (iv) A × Y is C*-embedded in X × Y for each space Y; (v) there is a continuous linear extender ${\displaystyle \phi :C{\cdot }_k\left(A\right)\to C_p\left(X\right)}$; (vi) there is an ${\displaystyle L_{ch}}$-extender φ:C(A) → C(X); and (vii) there is an ${\displaystyle L_h}$-extender φ:C(A) → C(X). We prove that these conditions are related as follows: (i)⇒(ii)⇔(iii)⇔(iv)⇔(v)⇒(vi)⇒(vii). If A is paracompact and the cellularity of A is nonmeasurable, then (ii)-(vii) are equivalent. If there is no connected subset of X which meets distinct convex components of A, then (ii) implies (i). We show that van Douwen's example of a separable GO-space satisfies none of the above conditions, which answers questions of Heath-Lutzer [9], van Douwen [1] and Hattori [8].