FINITE PRESENTABILITY OF HNN EXTENSIONS OF INVERSE SEMIGROUPS

HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely generated or finitely presented base is again finitely generated or finitely presented. Our main results depend upon properties of the [Formula: see text]-preorder in the associated subsemigroups. Let S be a finitely generated inverse semigroup and let U, V be inverse subsemigroups of S, isomorphic via φ: U → V, that are order ideals in S. We prove that the HNN extension S*U,φ is finitely generated if and only if U is finitely [Formula: see text]-dominated. If S is finitely presented, we give a necessary and suffcient condition for S*U,φ to be finitely presented. Here, in contrast to the theory of HNN extensions of groups, it is not necessary that U be finitely generated.

On defining groups efficiently without using inverses

Let G be a group, and let 〈A[mid ]R〉 be a finite group presentation for G with [mid ]R[mid ][ges ][mid ]A[mid ]. Then there exists a finite semigroup presentation 〈B[mid ]Q〉 for G such that [mid ]Q[mid ]- [mid ]B[mid ] = [mid ]R[mid ]- [mid ]A[mid ]. Moreover, B is either the same generating set or else it contains one additional generator.

On the Algebra of Semigroup Diagrams

In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid [Formula: see text] of reduced semigroup diagrams over the presentation S = <X:R> may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid [Formula: see text] are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

REWRITING A SEMIGROUP PRESENTATION

Let [Formula: see text] be a finitely presented semigroup having a minimal left ideal L and a minimal right ideal R. The main result gives a presentation for the group R∩L. It is obtained by rewriting the relations of [Formula: see text], using the actions of [Formula: see text] on its minimal left and minimal right ideals. This allows the structure of the minimal two-sided ideal of [Formula: see text] to be described explicitly in terms of a Rees matrix semigroup. These results are applied to the Fibonacci semigroups, proving the conjecture that S(r, n, k) is infinite if g.c.d.(n, k)>1 and g.c.d.(n, r+k−1)>1. Two enumeration procedures, related to rewriting the presentation of [Formula: see text] into a presentation for R∩L, are described. The first enumerates the minimal left and minimal right ideals of [Formula: see text], and gives the actions of [Formula: see text] on these ideals. The second enumerates the idempotents of the minimal two-sided ideal of [Formula: see text].