On the composition of idempotent functions

Abstract

W. Sierpifiski has shown [1] that given any set A and any function f:An~A, f can be obtained by an appropriate composition of binary functions. In this paper we consider the corresponding problem for idempotent functions; f.'A"~A is idempotent iff (a, a ..... a)=a for all aeA. The motivation for this investigation is a sequence of papers by G. Gratzer and J. Plonka [2]-[5] characterizing the number p, of n-ary polynomials in an algebra. Satisfactory answers are known for all but idempotent algebras (an algebra is idempotent if each of its operations is idempotent). For a finite idempotent algebra it is easy to calculate the maximum asymptotic rate of growth of thepn'S and it is clear this rate is achieved when all idempotent functions are polynomials. Can this rate be achieved by an idempotent algebra with only a finite number of operations? Obviously the answer is yes if every idempotent function can be obtained by an appropriate composition of, say, binary idempotent functions. Let IAI be the cardinality of A. We shall now prove the following theorem: If IAI>2 then every idempotent function can be obtained by composition of binary idempotent functions; if IAI =2 then every idempotent function can be obtained by composition of ternary idempotent functions but not by binary idempotent functions. LEMMA 1: If IAI ~>No then for all n>~O every n-ary idempotent function can be obtained by composition of ternary idempotent functions.