A Categorical Approach of FDT for Ω-Monoids

Mathematics and Statistics, 2022

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 bi-FP1 is equivalent with what is called in [2] finite domination type. It was indicated in the end of [2] that there are bi-FP1 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 C of a category D and characterized that dominion in terms of zigzags in D over C. The reason we followed this approach is that to every rewriting system (x, r) which gives a monoid M, there is always a category D(x, r) associated to it which contains three types of information at the same time: (i) all the possible ways in which the elements of M are written in terms of words with letters from x, (ii) all the possible ways one can transform a word with letters from x into another one representing the same element of M by using rewriting rules from r. Each of such way gives is in fact a path in the reduction graph of (x, r). The last information (iii) encoded in D(x, r) is that D(x, r) 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 D(x, r) turns out to have the advantage that it can ”measure” the extent to which a set U of parallel paths is sufficient to express any pair of parallel paths by composing whiskers from U. The gadget used to measure this, is the Isbell dominion of the whisker category W(U) generated by U over D(x, r). We then define the monoid M given by (x, r) to be of finite domination type (FDOT) if both x and r are finite and there is a finite set U of morphisms such that DomD(x,r)(W(U)) is exactly D(x, r). 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 bi-FP1, giving thus hope that FDOT is the right tool to put FDT and bi-FP1 into the same framework.

2006

When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given.

Journal of the Australian Mathematical Society, 1989

AbsractIn this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

Fixed Point Theory and Applications, 2013

In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal.

Electronic Notes in Theoretical Computer Science, 1998

A concrete monoid over a category C is a subset of the endomorphisms of an object of C, containing the identity and closed under composition. To contrast, an abstract monoid is just a one object category.

The Electronic Journal of Combinatorics, 2010

The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or

Journal of Algebra, 1987

Journal of Algebra, 1998

Conditions are found under which a general product of two finitely presented monoids is itself finitely presented. Presentations in terms of the presentations of the factors are given, subject to these conditions.

Journal of Computer and System Sciences, 1967

An overcategory with base category C is merely any functor into C. In this paper we extend the work of Dominique Bourn and Jacques Penon [4] on overcategories. In particular we show that Freyd's adjoint theorem, a theorem of Barr and Wells in [6] are still valid in the overcategorical context. We also show that a free monoid construction remains valid in the context of overcategories. The motivation for this study is the development of higher categories as found in [4] and in [10].