Let \(\mathcal P=[\mathbf x,\mathbf r]\) be a monoid presentation, where \(\mathbf x\) is a set of generating symbols and \(\mathbf r\) is a set of ordered pairs of words on \(\mathbf x\). Let \(\Gamma\) be the graph whose vertices are words on \(\mathbf x\) and whose edges are ordered pairs \((UR_1V,UR_2V)\) of words, where \(U,V\) are arbitrary words and \((R_1,R_2)\in\mathbf r\) or \((R_2,R_1)\in\mathbf r\). So the edge set has a natural orientation. This situation is represented geometrically by means of a so called atomic (monoid) picture. A path in \(\Gamma\) is represented geometrically by means of a so called monoid picture. Considering special paths in this graph, a 2-complex is defined and then corresponding homology groups. The monoid isoperimetric (or Dehn) function for \(\mathcal P\) is defined and it is proved that for two finite presentations of a given monoid these functions are equivalent (§2).
Three so called “research” programs are highlighted. The material presented in §3 (“Modules associated with \(\mathcal P\)”) is given in more detail in the author’s paper [in Int. J. Algebra Comput. 5, No. 6, 631-649 (1995; Zbl 0838.20075)]. The last §4 (the relationship between the monoid and group defined by a given monoid presentation) is discursive.
For the entire collection see [Zbl 0858.00023].