Presentations of general products of monoids
Let \(S\) and \(T\) be two semigroups, \(\lambda\colon T\times S\to T\) be a right action of \(S\) on \(T\) and \(\rho\colon T\times S\to S\) be a left action of \(T\) on \(S\). If \(\lambda\) and \(\rho\) satisfy some conditions the set \(S\times T\) endowed with the binary operation \((a,x)\circ(b,y)=(a\rho(x,b),\lambda(x,b)y)\) is a semigroup, called a general product of \(S\) and \(T\), and denoted \(S\times T\). If \(S\) and \(T\) are monoids and \(\lambda,\rho\) satisfy some supplementary conditions then \(S\times T\) is also a monoid. The main results of the paper give conditions under which a general product of two finitely presented semigroups is finitely presented. Two examples that illustrate the theoretical results are included. Presentations are obtained for: (1) the Bruck-Reilly extension of a finitely presented monoid; (2) the symmetric groups.
scientific article; zbMATH DE number 1218682