Polycyclic Monoids Research Articles

Gauge Inverse Monoids

The paper introduces a class of inverse (sub)monoids which contains Jones–Lawson’s gauge inverse (sub)monoid. The aim is to give examples and the basic properties of these monoids. Jones–Lawson’s gauge inverse monoid, as an inverse submonoid of the polycyclic monoid, is the prototype in our development line. The generalization leads also to Meakin–Sapir type results involving bijections between special congruences and special wide inverse submonoids.

On universal objects in the class of graph inverse semigroups

We show that polycyclic monoids are universal objects in the class of graph inverse semigroups. In particular, we prove that a graph inverse semigroup G(E) over a directed graph E embeds into the polycyclic monoid $${\mathscr {P}}_{\lambda }$$ where $$\lambda =|G(E)|$$. We show that each graph inverse semigroup G(E) admits the coarsest inverse semigroup topology $$\tau $$. Moreover, each injective homomorphism from $$(G(E),\tau )$$ to $$({\mathscr {P}}_{|G(E)|},\tau )$$ is a topological embedding.

Decidability and independence of conjugacy problems in finitely presented monoids

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by ∼p, ∼o, and ∼c) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, p-conjugacy is “almost” transitive, ∼c is strictly included in ∼p, and the p- and c-conjugacy problems are decidable with linear complexity on a two-tape Turing Machine. For other classes of monoids, the situation is more complicated. We show that there exists a monoid M defined by a finite complete presentation such that the c-conjugacy problem for M is undecidable, and that for finitely presented monoids, the c-conjugacy problem and the word problem are independent, as are the c-conjugacy and p-conjugacy problems. On other hand, we show that for finitely presented monoids, the o-conjugacy problem is reducible to the c-conjugacy problem.

On locally compact semitopological graph inverse semigroups

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for locally compact semitopological polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_{\lambda}$, respectively.

On strong affine representations of the polycyclic monoids

Jones and Lawson have discovered in (Period Math Hungar 64(1):53–87 2012) that certain representations of the so-called polycyclic monoids are closely related to some permutative representations of the Cuntz algebras $${\mathcal {O}}_{n}$$ studied by Bratteli and Jorgensen in (Mem Am Math Soc 139(663):x+89 1999). We investigate these representations of the polycyclic monoids, and we generalize some results from Jones and Lawson (2012). We give a (sharp) upper bound on the number of atoms in case one of the parameters tends to infinity and present an infinite family of representations having only one atom. Furthermore, by making use of a C++ program we present some observations regarding the number of atoms in the case $$n=3$$ .

Closed inverse subsemigroups of graph inverse semigroups

ABSTRACTAs part of his study of representations of the polycylic monoids, Lawson described all the closed inverse submonoids of a polycyclic monoid Pn and classified them up to conjugacy. We show that Lawson’s description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply Schein’s theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when it is finite.

On a complete topological inverse polycyclic monoid

We give sufficient conditions when a topological inverse $\lambda$-polycyclic monoid $P_{\lambda}$ is absolutely $H$-closed in the class of topological inverse semigroups. For every infinite cardinal $\lambda$ we construct the coarsest semigroup inverse topology $\tau_{mi}$ on $P_\lambda$ and give an example of a topological inverse monoid $S$ which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.

Topological graph inverse semigroups

To every directed graph E one can associate a graph inverse semigroupG(E), where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C⁎-algebras, and Toeplitz C⁎-algebras. We investigate topologies that turn G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E)∖{0} must be discrete for any directed graph E. On the other hand, G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E, G(E) admits a T1 semigroup topology in which G(E)∖{0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E) in larger topological semigroups.

Automata with Counters that Recognize Word Problems of Free Products

An M-automaton is a finite automaton with a blind counter that mimics a monoid M. The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M-automata play a central role in understanding the family 𝔏(M) of all languages accepted by M-automata. If G1 and G2 are finitely generated groups whose word problems are languages in 𝔏(M), in general, the word problem of the free product G1 * G2 is not necessarily in 𝔏(M). However, we show that if M is enlarged to the free product M*P2, where P2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M1 * M2 lies in 𝔏(M * P2) whenever M1 and M2 are finitely generated monoids with special word problems in 𝔏(M * P2). We also observe that there is a monoid without zero, denoted by CF2, that can be used in place of P2 for this purpose. The monoid CF2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P2. The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P2 in the above result and under what circumstances P1 (or the bicyclic monoid) is enough to do the job of P2.

Ordered groupoids and the holomorph of an inverse semigroup

We present a construction for the holomorph of an inverse semigroup, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that preserve the ternary heap operation on an inverse semigroup: for groups these two constructions coincide. We present detailed calculations for semilattices of groups and for the polycyclic monoids.

Möbius monoids and their connection to inverse monoids

Mobius monoids are Mobius categories in the sense of Leroux having a single object. The paper explores several properties of Mobius monoids and inverse monoids as well as the links between them. The right cancellative, left rigid Mobius monoids have a special place in our study. The paper has two goals: first, an examination of Mobius functions and Mobius inversion under suitable conditions, and second, a development of special inverse monoids and inverse submonoids which arise from Mobius monoids and Mobius categories. The prototype of this development is the polycyclic monoid, more precisely the free monoid as a Mobius monoid. This leads us to the generalization of some significant results on polycyclic monoids as Meakin–Sapir’s result involving self-conjugate inverse submonoids, or as Jones-Lawson’s results involving gauge inverse submonoids of the polycyclic monoids. Regarding Mobius functions we create links between several types of these functions.

Strong representations of the polycyclic inverse monoids: Cycles and atoms

This paper was inspired by a monograph by Bratteli and Jorgensen, and the work of Kawamura. We introduce two new semigroups: a wide inverse submonoid of the polycyclic inverse monoid, called the gauge inverse monoid, and a Zappa-Szep product of an arbitrary free monoid with the free monoid on one generator. Both these monoids play an important role in studying arbitrary, not necessarily transitive, strong actions of polycyclic inverse monoids. As a special case of such actions, we obtain some new results concerning the strong actions of P2 on ℤ determined by the choice of one positive odd number. We explain the role played by Lyndon words in characterising these repesentations and show that the structure of the representation can be explained by studying the binary representations of the numbers \(\frac{1} {p},\frac{2} {p}, \ldots \frac{{p - 1}} {p}\). We also raise some questions about strong representations of the polycyclic monoids on free abelian groups.

Primitive partial permutation representations of the polycyclic monoids and branching function systems

We characterise the maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse semigroups, and so determine all their primitive partial permutation representations. We relate our results to the work of Kawamura on certain kinds of representations of the Cuntz C*-algebras and to the branching function systems of Bratteli and Jorgensen.

Rational subsets of polycyclic monoids and valence automata

We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 (that is, the bicyclic monoid), they accept a class of languages strictly including the partially blind one-counter languages. Key to the proof is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem, and the closure of the class of rational subsets under intersection and complement.

The Polycyclic Monoids P n and the Thompson Groups V n,1

We construct what we call the strong orthogonal completion C n of the polycyclic monoid P n on n generators. The inverse monoid C n is congruence free and its group of units is the Thompson group V n,1. Copies of C n can be constructed from partitions of sets into n blocks each block having the same cardinality as the underlying set.

Orthogonal Completions of the Polycyclic Monoids

We introduce the notion of an orthogonal completion of an inverse monoid with zero. We show that the orthogonal completion of the polycyclic monoid on n generators is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated right ideals of the free monoid on n generators, and so we can make a direct connection with the Thompson groups Vn,1.

HNN extensions of inverse semigroups with zero

AbstractWe study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups –including the polycyclic monoids –admit HNN decompositions in a natural way, and that this leads to concise presentations for them.

A CORRESPONDENCE BETWEEN BALANCED VARIETIES AND INVERSE MONOIDS

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.

Coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids

We obtain several new results about the coding of a monoid by an appropriate submonoid of a polycyclic monoid. In particular, we characterize groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids.

The Structure of 0-E-unitary inverse semigroups I: the monoid case

This is the first of three papers in which we generalise the classical McAlister structure theory for E-unitary inverse semigroups to those 0-E-unitary inverse semigroups which admit a 0-restricted, idempotent pure prehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary and sufficient conditions for the existence of such prehomomorphisms in the case of 0-E-unitary inverse monoids. A class of inverse monoids which satisfy our conditions automatically are those which are unambiguous except at zero, such as the polycyclic monoids.