Dualizing Clones as Models of Lawvere Theories

Abstract

While universal algebraists are well aware of the equivalence between abstract clones and Lawvere theories as well as that of concrete clones and models of Lawvere theories in the category of sets, they almost never use the category-theoretic framework. It seems as if they simply do not see a reason why it might be beneficial to use category theory in order to study the problems that they are interested in. In this paper, we argue that the possibility of applying duality theory might be such a reason, and we support this claim by outlining how treating and dualizing clones as models of Lawvere theories can be beneficial for the classical problem of studying the lattice of clones on a given set. In particular, we give several examples of concrete results that are obtained with this method.