Switchability and collapsibility of Gap Algebras

Let A be an idempotent algebra on a 3-element domain D that omits a G-set for a factor. Suppose A is not αβ-projective (for some α, β ⊂ D so that α ∪ β = D and α ∩ β 6= ∅) and is not collapsible. It follows that A is switchable. We prove that, for every finite subset ∆ of Inv(A), Pol(∆) is collapsible. We also exhibit an algebra that is collapsible from a non-singleton source but is not collapsible from any singleton source.