Against Hirose's Argument for Saving the Greater Number

Faced with the choice between saving one person and saving two others, what should we do? It seems intuitively plausible that we ought to save the two, and many forms of consequentialists offer a straightforward rationale for the intuition by appealing to interpersonal aggregation. But still many other philosophers attempt to provide a justification for the duty to save the greater number without combining utilities or claims of separate individuals. I argue against one such attempt proposed by Iwao Hirose. Despite being consequentialist, his argument is aggregation-free since it relies on a non-aggregative value judgement method, instead of interpersonal aggregation, to establish that (other things being equal) a state of affairs is better when more people survive therein. I do not take issue with its consequentialist element; rather, I claim that there is no good reason to adopt the method in question, and thus no good reason to be moved by his argument overall. What we are in search of is not merely a logically possible method that can produce the conclusion that we already want, but one that we have good reason to adopt. Hirose's argument elegantly demonstrates how it could possibly be true that it is right to save the greater number; but it fails to show that we have reason to believe so - even when we do not combine the interests of different individuals

Some power of an element in a Garside group is conjugate to a periodically geodesic element

We show that for each element $g$ of a Garside group, there exists a positive integer $m$ such that $g^m$ is conjugate to a periodically geodesic element $h$, an element with |h^n|_\D=|n|\cdot|h|_\D for all integers $n$, where |g|_\D denotes the shortest word length of $g$ with respect to the set \D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside group, its stable super summit set.Comment: Subj-class of this paper should be Geometric Topology; Version published by BLM

Periodic elements in Garside groups

Let $G$ be a Garside group with Garside element $\Delta$, and let $\Delta^m$ be the minimal positive central power of $\Delta$. An element $g\in G$ is said to be 'periodic' if some power of it is a power of $\Delta$. In this paper, we study periodic elements in Garside groups and their conjugacy classes. We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\Delta^a$; every finite subgroup of the quotient group $G/$ is cyclic. By a classical theorem of Brouwer, Ker\'ekj\'art\'o and Eilenberg, an $n$-braid is periodic if and only if it is conjugate to a power of one of two specific roots of $\Delta^2$. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of $\Delta^m$. We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type $A_n$, $B_n$, $D_n$, $I_2(e)$ and the braid group of the complex reflection group of type $(e,e,n)$, endowed with the dual Garside structure, we may further assume the precentrality.Comment: The contents of the 8-page paper "Notes on periodic elements of Garside groups" (arXiv:0808.0308) have been subsumed into this version. 27 page