Iteration properties of switching functions
by
Klaus Denecke
University of Potsdam, Germany

For a fixed finite set A, let n: = |A| >= 2 denote the cardinality of A. For f: A --> A let Imf: = { f(a) | a in A } be the image of f and let \lambda(f) be the least non-negative integer m such that Imf^m = Imf^m+1. The number \lambda(f) is called the pre-period of f. Clearly, 0 <= \lambda(f) <= n-1. If \lambda(f) = n-1, then f is called an LT-function. It is well-known that f is a LT-function if and only if there exists an element d in A such that

A = {d, f(d), f2(d), ..., fn-1(d)}, fn(d) = fn-1(d).

In [Den-R;03] we determined all kinds of partial order relations on A which are invariant under LT-functions. In a similar way we can ask for all equivalence relations on A which are invariant under LT-functions.

This can be done also for functions f:A --> A with \lambda(f) = n-2 or other values for \lambda(f). The first step is always to give a good description of such functions. Then we wil determine all equivalence relations and all partial order relations which are preserved by these functions.

[Den-R;03] K. Denecke, Ch. Ratanaprasert, Hyperidentities in order-primal algebras, preprint 2004.

Date received: May 5, 2004