John Meakin

E-unitary inverse monoids and the Cayley graph of a group presentation

Abstract Geometric methods have played a fundamental and crucial role in combinatorial group theory almost from the inception of that field. In this paper we initiate a study of the use of some of these methods in inverse semigroup theory. We modify a lemma of I. Simon and show how to construct E -unitary inverse monoids from the free idempotent and commutative category over the Cayley graph of the maximal group image. The construction provides an expansion from the category of X -generated groups to the category of X -generated E -unitary inverse monoids and specializes to a construction of certain relatively free E -unitary inverse monoids. We show more generally that this construction is the left adjoint of the maximal group image functor. Munns solution to the word problem for the free inverse monoids and several of the results of McAlister and McFadden on the free inverse semigroup with two commuting generators may be obtained fairly easily from the construction. We construct the free product of E -unitary inverse monoids, thus providing an alternate construction to that of Jones.

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

Inverse monoids, trees and context-free languages

This paper is concerned with a study of inverse monoids presented by a set X subject to relations of the form e i = f i , i ∈ I, where e i and f i are Dyck words, i.e. idempotents of the free inverse monoid on X. Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on X. In the finitely presented case the word problem is solved by using Rabins theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored

The fundamental four-spiral semigroup

Pastijn [17] showed that every semigroup may be embedded in a bisimple idempotent-generated semigroup. This resolves in the negative the conjecture of Eberhart, Williams, and Kinch [4] that a simple idempotent-generated semigroup is completely simple and suggests a study of the structure of bisimple idempotent-generated semigroups which are not completely simple. In this paper we introduce and analyse the structure of a semigroup which is a basic building block for bisimple non-completely simple idempotent-generated semigroups. This semigroup is fundamental (in the sense of Munn [13] and Nambooripad [15]) and has a biordered set of idempotents which can be described as a “spiral” biordered set: It is idempotent-generated, generated by four idempotents and can be described as a rectangular band of four semigroups, three of which are isomorphic to the bicyclic semigroup and one of which is a union of a bicyclic semigroup and an infinite cyclic semigroup.

PSPACE-complete problems for subgroups of free groups and inverse finite automata

We investigate the complexity of algorithmic problems on finitely generated subgroups of free groups. Margolis and Meakin showed how a finite monoid Synt(H) can be canonically and effectively associated with such a subgroup H. We show that H is pure (that is, closed under radical) if and only if Synt(H) is aperiodic. We also show that testing for this property of H is PSPACE-complete. In the process, we show that certain problems about finite automata which are PSPACE-complete in general remain PSPACE-complete when restricted to injective and inverse automata (with single accept state), whereas they are known to be in NC for permutation automata (with single accept state).

Subgroups of Free Groups: a Contribution to the Hanna Neumann Conjecture

We prove that the strengthened Hanna Neumann conjecture, on the rank of the inter-section of finitely generated subgroups of a free group, holds for a large class of groups characterized by geometric properties. One particular case of our result implies that the conjecture holds for all positively finitely generated subgroups of the free group F(A) (over the basis A), that is, for subgroups which admit a finite set of generators taken in the free monoid over A.

The word problem for inverse monoids presented by one idempotent relator

Abstract We study inverse monoids presented by a finite set of generators and one relation e = 1, where e is a word representing an idempotent in the free inverse monoid, and 1 is the empty word. We show that (1) the word problem is solvable by a polynomial-time algorithm; (2) every congruence class (in the free monoid) with respect to such a presentation is a deterministic context-free language. Such congruence classes can be viewed as generalizations of parenthesis languages; and (3) the word problem is solvable by a linear-time algorithm in the more special case where e is a “positively labeled” idempotent.

Amalgams of free inverse semigroups

We study inverse semigroup amalgams of the formS *U T whereS andT are free inverse semigroups andU is an arbitrary finitely generated inverse subsemigroup ofS andT. We make use of recent work of Bennett to show that the word problem is decidable for any such amalgam. This is in contrast to the general situation for semigroup amalgams, where recent work of Birget, Margolis and Meakin shows that the word problem for a semigroup amalgamS *U T is in general undecidable, even ifS andT have decidable word problem,U is a free semigroup, and the membership problem forU inS andT is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for theD-classes of such an amalgam to be finite and we show that the amalgam is combinatorial in such a case. For example every one-relator amalgam of this type has finiteD-classes and is combinatorial. We also obtain information concerning when such an amalgam isE-unitary: for example every one relator amalgam of the formInv whereA andB are disjoint andu (resp.v) is a cyclically reduced word overA ∪A−1 (resp.B ∪B−1) isE-unitary.

On one-relator inverse monoids and one-relator groups

Abstract It is known that the word problem for one-relator groups and for one-relator monoids of the form Mon 〈A || w=1〉 is decidable. However, the question of decidability of the word problem for general one-relation monoids of the form M= Mon 〈A || u=v〉 where u and v are arbitrary (positive) words in A remains open. The present paper is concerned with one-relator inverse monoids with a presentation of the form M= Inv 〈A || w=1〉 where w is some word in A ∪ A −1 . We show that a positive solution to the word problem for such monoids for all reduced words w would imply a positive solution to the word problem for all one-relation monoids. We prove a conjecture of Margolis, Meakin and Stephen by showing that every inverse monoid of the form M= Inv 〈A || w=1〉 , where w is cyclically reduced, must be E -unitary. As a consequence the word problem for such an inverse monoid is reduced to the membership problem for the submonoid of the corresponding one-relator group G= Gp 〈A || w=1〉 generated by the prefixes of the cyclically reduced word w . This enables us to solve the word problem for inverse monoids of this type in certain cases.

Free objects in certain varieties of inverse semigroups

It is shown how the graphical methods developed by Stephen for analyzing inverse semigroup presentations may be used to study varieties of inverse semigroups. In particular, these methods may be used to solve the word problem for the free objects in the variety of inverse semigroups generated by the five-element combinatorial Brandt semigroup and in the variety of inverse semigroups determined by laws of the form x n =x n+1 . Covering space methods are used to study the free objects in a variety of the form V∨G where is a variety of inverse semigroups and G is the variety of groups