Abstract clones and operads

Abstract

We establish a connection between abstract clones and operads, which implies that both clones and operads we particular instances of a more general notion. The latter is called W-operad (due to a close resemblance with operads) and can be regarded as a functor on a certain subcategory W, of the category of finite ordinals, with some rather natural properties. When W is a category whose morphisms are the various bijections, the variety of W-operads is rationally equivalent to the variety of operads in the traditional sense. Our main result claims that if W coincides with the category of all finite ordinals then the variety of W-operads is rationally equivalent to the variety of abstract clones.