Mathematics and Statistics Vol. 10(5), pp. 1105 - 1110
DOI: 10.13189/ms.2022.100520
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Finite Domination Type for Monoid Presentations

Elton Pasku ¹, Anjeza Krakulli ^2,*
1 Department of Mathematics, Faculty of Natural Sciences, University of Tirana, 1001, Tirana Albania
² Department of Mathematics, Faculty of Technology and Information, Aleksand¨er Moisiu University, 2002, Durr¨es, Albania

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 is equivalent with what is called in [2] finite domination type. It was indicated in the end of [2] that there are 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 of a category and characterized that dominion in terms of zigzags in over . The reason we followed this approach is that to every rewriting system which gives a monoid , there is always a category associated to it which contains three types of information at the same time: (i) all the possible ways in which the elements of are written in terms of words with letters from , (ii) all the possible ways one can transform a word with letters from into another one representing the same element of by using rewriting rules from . Each of such way gives is in fact a path in the reduction graph of . The last information (iii) encoded in is that 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 turns out to have the advantage that it can "measure" the extent to which a set of parallel paths is sufficient to express any pair of parallel paths by composing whiskers from . The gadget used to measure this, is the Isbell dominion of the whisker category generated by over . We then define the monoid given by to be of finite domination type (FDOT) if both and are finite and there is a finite set of morphisms such that is exactly . 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 , giving thus hope that FDOT is the right tool to put FDT and into the same framework.

KEYWORDS
Monoid, Presentation, Category, Homology, Dominion

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Elton Pasku , Anjeza Krakulli , "Finite Domination Type for Monoid Presentations," Mathematics and Statistics, Vol. 10, No. 5, pp. 1105 - 1110, 2022. DOI: 10.13189/ms.2022.100520.

(b). APA Format:
Elton Pasku , Anjeza Krakulli (2022). Finite Domination Type for Monoid Presentations. Mathematics and Statistics, 10(5), 1105 - 1110. DOI: 10.13189/ms.2022.100520.