F-birestriction monoids in enriched signature

On a Class of Lattice Ordered Inverse Semigroups

Every $F$-inverse monoid can be equipped with the unary operation which maps each element to the maximum element of its $\sigma$-class. In this enriched signature, the class of all $F$-inverse monoids forms a variety of algebraic structures. We describe universal objects in several classes of $F$-inverse monoids, in particular free $F$-inverse monoids. More precisely, for every $X$-generated group $G$ we describe the initial object in the category of all $X$-generated $F$-inverse monoids $F$ for which $F/\sigma=G$.

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids. For a monoid [Formula: see text] and a class of partial actions of [Formula: see text], determined by a set, [Formula: see text], of identities, we define [Formula: see text] to be the universal [Formula: see text]-generated two-sided restriction monoid with respect to partial actions of [Formula: see text] determined by [Formula: see text]. This is an [Formula: see text]-restriction monoid which (for a certain [Formula: see text]) generalizes the Birget–Rhodes prefix expansion [Formula: see text] of a group [Formula: see text]. Our main result provides a coordinatization of [Formula: see text] via a partial action product of the idempotent semilattice [Formula: see text] of a similarly defined inverse monoid, partially acted upon by [Formula: see text]. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that some properties of [Formula: see text] agree well with suitable properties of [Formula: see text], such as being cancellative or embeddable into a group. We observe that if [Formula: see text] is an inverse monoid, then [Formula: see text], the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion [Formula: see text]. This gives a presentation of [Formula: see text] and leads to a model for [Formula: see text] in terms of the known model for [Formula: see text].

This paper surveys some of the authors’ recent and ongoing work aimed at developing a theory of presentations of inverse monoids analogous to the theory of generators and relations for groups. We regard inverse monoids as a class of algebras of type 〈2,1,0〉 and study presentations of inverse monoids from this point of view. The paper is concerned with two basic decision problems for inverse monoid presentations: the word problem and the E-unitary problem. We develop the general construction of a birooted word graph associated with an inverse monoid presentation and show how it can be used as a basic tool in the study of the word problem. We indicate several cases in which the word problem can be solved using these techniques. We study the E-unitary problem for inverse monoids of the form M=Inv〈X|w=1〉 where w is in the free inverse monoid on X. We show how the Lyndon diagrams of combinatorial group theory may be used to analyze the problem and we study several examples and special cases in detail.

There is a well-known correspondence between varieties of algebras and fully invariant congruences on the appropriate term algebra. A special class of varieties are those which are balanced, meaning they can be described by equations in which the same variables appear on each side. In this paper, we prove that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids. In the case of unary algebras, we recover the theorem of Meakin and Sapir that establishes a bijection between congruences on the free monoid with n generators and wide, positively self-conjugate inverse submonoids of the polycyclic monoid on n generators. In the case of varieties generated by linear equations, meaning those equations where each variable occurs exactly once on each side, we can replace the clause monoid above by the linear clause monoid. In the case of algebras with a single operation of arity n, we prove that the linear clause monoid is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated essential right ideals of the free monoid on n letters, a monoid previously studied by Birget in the course of work on the Thompson group V and its analogues. We show that Dehornoy's geometry monoid of a balanced variety is a special kind of inverse submonoid of ours. Finally, we construct groups from the inverse monoids associated with a balanced variety and examine some conditions under which they still reflect the structure of the underlying variety. Both free groups and Thompson's groups Vn,1 arise in this way.

The word problem for inverse monoids presented by one idempotent relator

It is known that the problem of determining consistency of a finite system of equations in a free group or a free monoid is decidable, but the corresponding problem for systems of equations in a free inverse monoid of rank at least two is undecidable. Any solution to a system of equations in a free inverse monoid induces a solution to the corresponding system of equations in the associated free group in an obvious way, but solutions to systems of equations in free groups do not necessarily lift to solutions in free inverse monoids. In this paper, we show that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are able to use this to solve the consistency problem for certain classes of single-variable equations in free inverse monoids.

An inverse monoidM is an idempotent-pure image of the free inverse monoid on a setX if and only ifM has a presentation of the formM=Inv , whereei,fi are idempotents of the free inverse monoid: every inverse monoid is an idempotent-separating image of one of this type. IfR is anR-class of such an inverse monoid, thenR may be regarded as a Schreier subset of the free group onX. This paper is concerned with an examination of which Schreier subsets arise in this way. In particular, ifI is finite, thenR is a rational Schreier subset of the free group. Not every rational Schreier set arises in this way, but every positively labeled rational Schreier set does.

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After reviewing the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture — also verified by Ash — it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH1H2…Hn, where g ∈ G and each Hi is a finitely generated subgroup of G. This significantly extends classical results of M. Hall. Finally, we return to the roots of this problem and give connections with the complexity theory of finite semigroups. We show that the largest local complexity function in the sense of Rhodes and Tilson is computable.

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

The word problem for inverse monoids generated by a set Γ subject to relations of the form e = f, where e and f are both idempotents in the free inverse monoid generated by Γ, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, it is shown that the Cayley-graph of the free inverse monoid has an undecidable monadic second-order theory.

A monoidSisright coherentif every finitely generated subact of every finitely presented rightS-act is finitely presented. This is the non-additive notion corresponding to that for a ringRstating that every finitely generated submodule of every finitely presented rightR-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of amodel companionof the class of rightS-acts (rightR-modules) and hence that the class of existentially closed rightS-acts (rightR-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω)isright coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of rightS-acts and of leftS-acts.

This paper is concerned with a study of inverse monoids presented by a set X X subject to relations of the form e i = f i {e_i} = {f_i} , i ∈ I i \in I , where e i {e_i} and f i {f_i} are Dyck words, i.e. idempotents of the free inverse monoid on X 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 X . In the finitely presented case the word problem is solved by using Rabin’s 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 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.

Expansions, free inverse semigroups, and Schützenberger product