[Paper Review] The clone generated by the median functions

Michael Pinsker|ArXiv.org|Jan 10, 2004

Optimization and Mathematical Programming3 citations

TL;DR

This paper proves that the clone generated by any odd-arity median function on a linearly ordered set contains all median functions of odd arity, establishing that median functions generate each other. The key result is that the clone generated by a single median function is minimal and contains all odd-arity medians, regardless of the size of the underlying set.

ABSTRACT

Let X be a linearly ordered set of arbitrary size (finite or infinite). Natural functions on such a set one can define using the linear order include maximum, minimum and median functions. While it is clear what the clone generated by the maximum or the minimum looks like, this is not obvious for median functions. We show that every clone on X contains either no median function or all median functions, that is, the median functions generate each other.

Motivation & Objective

• To determine the structure of the clone generated by median functions on a linearly ordered set.
• To investigate whether median functions generate each other within a clone, especially in comparison to max/min functions.
• To characterize which functions in the clone generated by a median are minimal or non-minimal.
• To clarify the role of odd-arity median functions in clone generation and their interdefinability.

Proposed method

• Define $ m^n_k $ as the $ k $-th smallest element in an $ n $-tuple, with $ \operatorname{med}_n = m^n_{(n+1)/2} $ for odd $ n $.
• Use variable identification to construct $ \operatorname{med}_k $ from $ \operatorname{med}_n $ when $ n $ is almost divisible by $ k $, based on remainder conditions.
• Introduce a recursive construction using $ b $-ary majority functions to generate larger majority functions from smaller ones.
• Apply a recurrence relation $ r_{j+1} > r_j (\frac{3}{2} - \frac{1}{2}r_j^2 - \frac{1}{n_j-1}) $ to show $ r_j \to 1 $, enabling median construction.
• Prove that $ \operatorname{med}_3 \in \langle \operatorname{med}_n \rangle $ for any odd $ n \geq 3 $, then extend to all odd $ k \geq 3 $.
• Use the fact that $ \min_2 $ and $ \max_2 $ are definable from $ m^n_k $ when $ k \leq \lfloor n/2 \rfloor $ or $ k \geq \lceil n/2 \rceil $, to show non-minimality of non-extremal $ m^n_k $.

Experimental results

Research questions

• RQ1 Does the clone generated by a single median function contain all other median functions of odd arity?
• RQ2 Can median functions of different odd arities be interdefined using composition and variable identification?
• RQ3 Which functions $ m^n_k $ are minimal, and under what conditions is the clone they generate minimal?
• RQ4 What is the role of the majority function in generating larger median functions?
• RQ5 How do lower and upper medians (for even $ n $) compare to true odd-arity medians in terms of definability?

Key findings

• The clone generated by any odd-arity median function $ \operatorname{med}_n $ contains all median functions $ \operatorname{med}_k $ for odd $ k \geq 3 $, regardless of the size of the underlying set.
• The clone generated by $ \operatorname{med}_3 $ is minimal, and since $ \operatorname{med}_3 \in \langle \operatorname{med}_n \rangle $, the clone generated by $ \operatorname{med}_n $ is also minimal.
• For any odd $ n \geq 3 $, $ \operatorname{med}_3 $ can be constructed from $ \operatorname{med}_n $ by identifying variables: $ \operatorname{med}_3(x_1,x_2,x_3) = \operatorname{med}_n(x_1,\dots,x_1,x_2,\dots,x_2,x_3,\dots,x_3) $ with $ \lfloor (n-1)/2 \rfloor $ or $ \lfloor (n-1)/2 \rfloor + 1 $ repetitions.
• Functions $ m^n_k $ for $ 2 \leq k \leq \lfloor n/2 \rfloor $ or $ \lceil n/2 \rceil < k < n $ are not minimal, as they generate $ \min_2 $ or $ \max_2 $, respectively.
• The only minimal functions among $ m^n_k $ are the maximum ($ k=n $), minimum ($ k=1 $), and odd-arity median functions ($ k=(n+1)/2 $).
• Lower and upper medians for even $ n $, such as $ \operatorname{med}^{\text{low}}_n = m^n_{n/2} $, are not definable from odd-arity medians, and $ \operatorname{med}^{\text{low}}_4 $ cannot generate any odd-arity median.