Finite presentability of Bruck-Reilly extensions of semigroups

In this study, we survey some recent result concerning finite presentability ofseveral semigroup constructions. Namely we study problems regarding finitepresentability of an arbitrary subgroup of a monoid defined by a finite presentation;and of Bruck-Reilly extensions BR(M,q ) of a monoid (group) M with respect to anendomorphism q of M . It is proved that any subgroup of a finitely presentedmonoid with finite index is also finitely presented as a monoid. Also it is proved thatif M is a finitely generated / presented monoid than BR(M,q ) is finitely generated /presented. Finally, it is proved that when M is a group, BR(M,q ) is finitelygenerated if and only if G can be generated by a set of the form 00iiAq³ U, whereAÍ G is finite; and BR(M,q ) is finitely presented if and only if G can be definedby a presentation with relations of the form 00iiR Rq³= U for some finite set ofrelations 0 R .