Associative,Idempotent, Symmetric,and Order-preserving Operations on Chains

We characterize the associative, idempotent, symmetric, and order-preserving binary operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n-element chain is the n^th Catalan number.

     

Associative and preassociative functions

We investigate the associativity property for functions of multiple arities and introduce and discuss the more general property of preassociativity, a generalization of associativity which does not involve any composition of functions.     

A Characterization of Barycentrically Preassociative Functions

We provide a characterization of the variadic functions which are barycentrically preassociative as compositions of length-preserving associative string functions with one-to-one unary maps. We also discuss some consequences of this characterization.     

On the generalized associativity equation

The so-called generalized associativity functional equation

$$\begin{aligned} G(J(x,y),z) = H(x,K(y,z)) \end{aligned}$$

has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real, continuous, and strictly monotonic in each variable. In this note we investigate the following related problem: given the functions J and K, find every function F that can be written in the form

$$\begin{aligned} F(x,y,z) = G(J(x,y),z) = H(x,K(y,z)) \end{aligned}$$

for some functions G and H. We show how this problem can be solved when any of the inner functions J and K has the same range as one of its sections.