Finitary approximations and metric structure of the space of clones

Let O_k denote the set of all multivariable functions over E_k into E_k where E_k is a k element set (k⩾2). A clone over E_k is a subset of O_k which is closed under composition. The set L_k of all clones over E_k is called the clone space. The structure of L₂ is completely known since E.L. Post (1941), but for k⩾3, the structure of L_k seems extremely complicated and is still mostly unknown. In order to get a better perspective on L_k, we firstly propose to define finitary approximation of L_k, which is some simplified structure of L_k. Then, motivated by this concept, a metric function is introduced into the clone space L_k. As a metric space, L_k is shown to have such properties as completeness, totally boundedness and compactness. Moreover, it is shown that if a clone C is an isolated point in L_k , it is finitely generated