Strong Convergence in Posets

TL;DR

This paper studies a solitaire game on finite posets involving element swaps and proves that the final ordering is uniquely determined by the initial ordering, using invariants to establish convergence.

Contribution

It introduces a new analysis of a leapfrog-like game on posets and proves strong convergence to a unique final ordering.

Findings

Final ordering is uniquely determined by initial ordering.

The proof employs a system of invariants to establish convergence.

The game always terminates with a unique order regardless of swap choices.

Abstract

We consider the following solitaire game whose rules are reminiscent of the children's game of leapfrog. The player is handed an arbitrary ordering $\pi=(x_1,x_2,...,x_n)$ of the elements of a finite poset $(P,\prec)$. At each round an element may "skip over" the element in front of it, i.e. swap positions with it. For example, if $x_i \prec x_{i+1}$, then it is allowed to move from $\pi$ to the ordering $(x_1,x_2,...,x_{i-1},x_{i+1},x_i,x_{i+2},...,x_n)$. The player is to carry out such steps as long as such swaps are possible. When there are several consecutive pairs of elements that satisfy this condition, the player can choose which pair to swap next. Does the order of swaps matter for the final ordering or is it uniquely determined by the initial ordering? The reader may guess correctly that the latter proposition is correct. What may be more surprising, perhaps, is that this question is not trivial. The proof works by constructing an appropriate system of invariants.