On the number of operations in a clone

Abstract

A clone C on a set A is a set of operations on A containing the projection operations and closed under composition. A combinatorial invariant of a clone is its p n {p_n} -sequence ⟨ p 0 ( C ) , p 1 ( C ) , … ⟩ \langle {p_0}(C),{p_1}(C), \ldots \rangle , where p n ( C ) {p_n}(C) is the number of essentially n-ary operations in C. We investigate the links between this invariant and structural properties of clones. It has been conjectured that the p n {p_n} -sequence of a clone on a finite set is either eventually strictly increasing or is bounded above by a finite constant. We verify this conjecture for a large family of clones. A special role in our work is played by totally symmetric operations and totally symmetric clones. We show that every totally symmetric clone on a finite set has a bounded p n {p_n} -sequence and that it is decidable if a clone is totally symmetric.