Coset enumeration in a finitely presented semigroup

Andrzej Jura
1978 Canadian mathematical bulletin  
1. Introduction. The enumeration method for finite groups, the so-called Todd-Coxeter process, has been described in [2] , [3] . Leech [4] and Trotter [5] carried out the process of coset enumeration for groups on a computer. However Mendelsohn [1] was the first to present a formal proof of the fact that this process ends after a finite number of steps and that it actually enumerates cosets in a group. Dietze and Schaps [7] used Todd-Coxeter's method to find all subgroups of a given finite
more » ... a given finite index in a finitely presented group. B. H. Neumann [8] modified Todd-Coxeter's method to enumerate cosets in a semigroup, giving however no proofs of the effectiveness of this method nor that it actually enumerates cosets in a semigroup. The present paper presents a proof of the fact that given a finite and finitely presented semigroup, the coset enumeration process in that semigroup ends after a finite number of steps and that it actually does enumerate the semigroup cosets. In this proof we make use of the lemma on stabilization of r initial rows in all tables after a finite number of steps (Mendelsohn, [la]) as well as some fragments of Schreier's theory-appropriately modified for semigroups-which has been described e.g. in [6] and used by Mendelsohn in his proof. The final result in this paper is the description of an effective way of determining the semigroup P s of transformations of a set Z into itself, where Z = {1,2,..., fc}, isomorphic to a given finite and finitely presented semigroup S of order k -1. The paper is self-consistent, however Mendelsohn's terminology is widely used.
doi:10.4153/cmb-1978-007-x fatcat:f32we4qqwjhifjh3j4v3ol2xru