Relative completeness with respect to two unary functions

Abstract

One of the most important results is the fact that the number of clones is a continuum for k≥3, while the corresponding set for k=2 is countable. This shows a sharp difference when we go from the binary to the ternary case. This paper discusses the relative completeness with respect to the clone generated by two unary functions and show the sharp difference when we go from four-valued logic to k-valued logic for k>4, as well. The number of maximal clones over a finite set is finite and increases when k increases. However, there are two relative maximal clones if k=3,4 and there is one relative maximal clone if k>4. © 2001 Elsevier Science B.V.