Finite Domination Type for Monoid Presentations

Abstract

In [5], Squier, Otto and Kobayashi explored a homotopical property for monoids called finite derivation type (FDT) and proved that FDT is a necessary condition that a finitely presented monoid must satisfy if it is to have a finite canonical presentation. In the latter development in [2], Kobayashi proved that the property <img src=image/13428295_01.gif> is equivalent with what is called in [2] finite domination type. It was indicated in the end of [2] that there are <img src=image/13428295_01.gif> monoids which are not even finitely generated, and as a consequence are not of FDT. It was this indication that inspired us to look for the possibility of defining a property of monoids which encapsulates both, FDT and finite domination type. This is realized in the current paper by extending the notion of finite domination from monoids to rewriting systems, and to achieve this, we are based on the approach of Isbell in [1], who defined the notion of the dominion of a subcategory <img src=image/13428295_02.gif> of a category <img src=image/13428295_03.gif> and characterized that dominion in terms of zigzags in <img src=image/13428295_03.gif> over <img src=image/13428295_02.gif>. The reason we followed this approach is that to every rewriting system <img src=image/13428295_04.gif> which gives a monoid <img src=image/13428295_06.gif>, there is always a category <img src=image/13428295_05.gif> associated to it which contains three types of information at the same time: (i) all the possible ways in which the elements of <img src=image/13428295_06.gif> are written in terms of words with letters from <img src=image/13428295_07.gif>, (ii) all the possible ways one can transform a word with letters from <img src=image/13428295_07.gif> into another one representing the same element of <img src=image/13428295_06.gif> by using rewriting rules from <img src=image/13428295_08.gif>. Each of such way gives is in fact a path in the reduction graph of <img src=image/13428295_04.gif>. The last information (iii) encoded in <img src=image/13428295_05.gif> is that <img src=image/13428295_05.gif> contains all the possible ways that two parallel paths of the reduction graph are linked to each other by a series of compositions of whiskerings of other parallel paths. This category <img src=image/13428295_05.gif> turns out to have the advantage that it can "measure" the extent to which a set <img src=image/13428295_09.gif> of parallel paths is sufficient to express any pair of parallel paths by composing whiskers from <img src=image/13428295_09.gif>. The gadget used to measure this, is the Isbell dominion of the whisker category <img src=image/13428295_10.gif> generated by <img src=image/13428295_09.gif> over <img src=image/13428295_05.gif>. We then define the monoid <img src=image/13428295_06.gif> given by <img src=image/13428295_04.gif> to be of finite domination type (FDOT) if both <img src=image/13428295_07.gif> and <img src=image/13428295_08.gif> are finite and there is a finite set <img src=image/13428295_09.gif> of morphisms such that <img src=image/13428295_11.gif> is exactly <img src=image/13428295_05.gif>. The first main result of our paper is that likewise FDT, FDOT is an invariant of the monoid presentation, and the second one is that that FDT implies FDOT, while remains open whether the converse is true or not. The importance of FDOT stands in the fact that not only it generalizes FDT, but the way it is defined has a lot in common with <img src=image/13428295_01.gif>, giving thus hope that FDOT is the right tool to put FDT and <img src=image/13428295_01.gif> into the same framework.