SEMIGROUP PRESENTATIONS FOR CONGRUENCES ON GROUPS

Abstract

Abstract. We consider a congruence ρ on a group G as a subsemigroupof the direct product G × G. It is well known that a relation ρ on G isa congruence if and only if there exists a normal subgroup N of G suchthat ρ =(s,t) : st −1 ∈ N . In this paper we prove that if G is a finitelypresented group, and if N is a normal subgroup of G with finite index,then the congruence ρ =(s,t) : st −1 ∈ N on G is finitely presented. 1. IntroductionFinite presentability of semigroup constructions has been widely studied inrecent years (see, for example [1, 2, 6, 8, 9]). One construction is an extensionof a semigroup by a congruence. Let S and T be semigroups and let ρ bea congruence on S. If S/ρ is isomorphic to T, then S is called an extensionof T by ρ. There is a similar construction in group theory. An extension ofa group H by a group N is a group G having N as a normal subgroup andG/N ∼= H. It is known that if H and N are both finitely presented groups, thenthe extension of them is finitely presented (see [7, Corollary 10.2]). Recently,it is proved in [3] that, for given a semigroup S and a congruence ρ on S, if ρis finitely presented as a subsemigroup of the direct product S×S, then S andS/ρ are finitely presented. In [3] finite presentability of ρ on a finitely presentedinfinite semigroup is an open problem. More recently, for inverse semigroupsS and T, and for a surjective homomorphism π : S → T with kernel K whichis a congruence on S, it is showed in [4] that how to the obtain a presentationfor K from a given a presentation for S and vice versa. It is also investigatedin [4] the relationship between finite presentability of inverse semigroups andtheir kernels.Let G be a group and let N be a normal subgroup of G with finite index.Then it is known that if G is finitely presented, then N is also finitely presented