Switchability and collapsibility of Gap Algebras

TL;DR

This paper investigates the properties of certain 3-element domain algebras, establishing conditions under which they are switchable or collapsible, and explores their polymorphism structures.

Contribution

It proves that non-collapsible, non-projective algebras on 3-element domains are switchable and characterizes their polymorphism collapsibility.

Findings

A non-projective, non-collapsible algebra on a 3-element domain is switchable.

Pol(Delta) is collapsible for every finite subset Delta of Inv(A).

An algebra can be collapsible from a non-singleton source but not from any singleton source.

Abstract

Let A be an idempotent algebra on a 3-element domain D that omits a G-set for a factor. Suppose A is not \alpha\beta-projective (for some alpha, beta subsets of D) and is not collapsible. It follows that A is switchable. We prove that, for every finite subset Delta of Inv(A), Pol(Delta) is collapsible. We also exhibit an algebra that is collapsible from a non-singleton source but is not collapsible from any singleton source.