Homotopy reduction systems for monoid presentations
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 \(FP_4\).
scientific article; zbMATH DE number 1308208
scientific article