The clone space as a metric space
Let \(O_k\), \(k\in\mathbb{N}\), be the set of all functions \(E^n_k\to E_k\) for some \(n\in\mathbb{N}\), where \(E_k=\{0,1,\dots,k-1\}\). Any subset \(C\) of \(O_k\) which contains all projections, i.e. functions defined by \(\text{pr}^n_i(x_1,x_2,\dots,x_n)=x_i\), \(1\leq i\leq n\), \(n\in\mathbb{N}\), and closed under superpositions is called a clone over \(E_k\). The set of all clones over \(E_k\) forms a lattice \(L_k\). Let \(C^{(n)}\) denote the set of all \(n\)-ary functions from \(C\). The author shows that the function \(d_k:L_k\times L_k\to\mathbb{R}\), defined by \(d_k(C_1,C_2)=1/\min\{n:C_1^{(n)}=C_2^{(n)}\}\), is a metric in \(L_k\). Moreover \(d_k\) is complete, totally bounded, and if \(C\) is an isolated point in \(L_k\), then \(C\) is finitely generated.
scientific article; zbMATH DE number 1218153