Homotopy reduction systems for monoid presentations
type: Item
instance of: scholarly article
title: Homotopy reduction systems for monoid presentations
zbMATH Open document ID: 0932.20053
DOI: 10.1016/S0022-4049(97)00095-9
author: Yuji Kobayashi
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-11T00:00:00Z
review text: Let \(\Sigma\) be a finite set and \(E\) a set of equations on the free monoid \(\Sigma^*\). Let \(M=\Sigma^*/_{=_E}\) where \(=_E\) is the congruence on \(\Sigma^*\) generated by \(E\). A graph \(\Gamma=\Gamma(\Sigma,E)\) is defined as follows: the set of vertices is \(\Sigma^*\) and for \(e=(u,v)\in E\), \(x,y\in\Sigma^*\), \((x;u,v;y)\) is an edge from \(xuy\) to \(xvy\). The set of all paths in \(\Gamma\) is denoted by \(P(\Gamma)\); two paths \(p\) and \(q\) with the same source and the same target are called parallel and written as \(p\parallel q\). Any equivalence relation on \(P(\Gamma)\) contained in \(\parallel\) and satisfying certain conditions is called a homotopy relation. For a subset \(B\) of \(\parallel\) -- called homotopy reduction system -- and for any two paths \(p,q\) write \(p\sim_Bq\), if \(q\) can be obtained from \(p\) by replacing a subpath \(r\) by \(s\) for some \((r,s)\in B\). The smallest homotopy relation containing \(B\) is denoted by \(\sim_B\). The reflexive and transitive closure and reflexive, symmetric and transitive closure of \(\sim_B\) is denoted by \(\sim^*_B\) and \(\lesssim^*_B\), respectively. \(B\) is Noetherian, if there is no infinite sequence of reductions \(p_1\lesssim_Bp_2\lesssim_B\cdots\lesssim_Bp_i\lesssim_B\cdots\); \(B\) is confluent, if for any paths \(p,q\) and \(r\) such that \(p\lesssim_B^*q\) and \(p\lesssim_B^*r\), there is a path \(s\) such that \(q\lesssim_B^*s\), \(r\lesssim_B^*s\). A system \(B\) is called complete, if it is Noetherian and confluent and \(\lesssim^*_B=\sim_B\). The article developes homotopy theory for monoids. It is proved, for example, that if a finite monoid presentation \((\Sigma,E)\) has a complete homotopy reduction system that is essentially finite, then the monoid \(M(\Sigma,E)\) satisfies the homological finiteness property \(FP_4\).
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55P99
zbMATH DE Number: 1308208
zbMATH Keywords: monoid presentations
zbMATH Keywords: homotopy theory for monoids
zbMATH Keywords: graphs
zbMATH Keywords: homotopy reduction systems
zbMATH Keywords: homological finiteness conditions
reviewed by: Peeter Normak
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00095-9
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description: scientific article; zbMATH DE number 1308208