On the Efficiency and Deficiency of Rees Matrix Semigroups

In this paper we combine results from [1] and [3] in order to determine the relationship between the deficiency of a group G and the deficiency of a Rees matrix semigroup S over G; in particular, we investigate how the efficiency of S depends on the efficiency of G. We start by defining semigroup and group presentations. Let A be an alphabet. We denote by A the free semigroup on A consisting of all nonempty words over A, and by F (A) the free group of all freely reduced words over A ∪ A−1 (including the empty word), where A−1 is an alphabet whose elements represent the inverses of elements of A. A semigroup presentation is an ordered pair 〈 A | R 〉, where R ⊆ A × A. If both A and R are finite then we have a finite presentation. A semigroup S is said to be defined by the semigroup presentation 〈 A | R 〉 if S ∼= A/ρ , where ρ is the congruence on A generated by R. Replacing A by F (A) in the above definitions yields the notions of a group presentation and of a group defined by a presentation. For basic facts about semigroup and group presentations see any standard introductory texts on semigroups and groups, such as [6], [8] and [10]. We define the deficiency of a finite presentation P = 〈A|R〉 to be |R|−|A|. The semigroup deficiency of a finitely presented semigroup S is the minimum deficiency of any semigroup presentation P defining S: defS(S) = min{ def(P) | P is a finite semigroup presentation that defines S }. We define the group deficiency defG(G) of a finitely presented group analogously, using finite group presentations: