Finite Presentability of Bruck–Reilly Extensions of Groups

Abstract

The purpose of this paper is to consider finite generation and finite presentability of a Bruck–Reilly extension S=BR(G,θ) of a group G with respect to an endomorphism θ. It is proved that S is finitely generated if and only if G can be generated by a set of the form ⋃∞i=0Aθi, where A⊆G is finite. The main result states that S is finitely presented if and only if G can be defined by a presentation of the form 〈A∣R〉 where A is finite and R is of the form ⋃∞i=0R̄θi for some finite set of relations R̄. Finally, it is proved that S is finitely presented as an inverse monoid if and only if it is finitely presented as an ordinary monoid.