Left Cancellative Monoids Research Articles

Köthe's Problem, Kurosh-Levitzky Problem and Graded Rings

Let R be an associative ring graded by left cancellative monoid S, and e the neutral element of S. We study the following problem: if Re is nil, then is R nil/nilpotent? We proved that if Re is nil (of bounded index) and f-commutative, then R is nil (of bounded index). Later, we showed that Re being nilpotent implies R is nilpotent. Consequently, we exhibited a generalization of Dubnov-Ivanov-Nagata-Higman Theorem for the graded algebras case. Furthermore, we exhibited relations between graded rings and the problems of Köthe and Kurosh-Levitzky. We proved that f-commutative rings provide positive solutions to these problems, and we also present a generalization of Kurosh-Levitzky Problem for the graded rings whose neutral components are f-commutative.

Cayley graphs of basic algebraic structures

We present simple graph-theoretic characterizations for the Cayley graphs of monoids, right-cancellative monoids, left-cancellative monoids, and groups.

Graded near-rings

AbstractIn this paper, we consider graded near-rings over a monoid G as generalizations of graded rings over groups, and study some of their basic properties. We give some examples of graded near-rings having various interesting properties, and we define and study the Gop-graded ring associated to a G-graded abelian near-ring, where G is a left cancellative monoid and Gop is its opposite monoid. We also compute the graded ring associated to the graded near-ring of polynomials (over a commutative ring R) whose constant term is zero.

Free completely J (ℓ)-simple Semigroups

A semigroup is called completely J(l)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J(l)-simple semigroups form a quasivarity. Moreover, the construction of free completely J(l)-simple semigroups is given. It is found that a free completely J(l)-simple semigroup is just a free completely J*-simple semigroup and also a full subsemigroup of some completely simple semigroups.

A correspondence between a class of monoids and self-similar group actions II

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

On C*-algebras of irreversible algebraic dynamical systems

Extending the work of Cuntz and Vershik, we develop a general notion of independence for commuting group endomorphisms. Based on this concept, we initiate the study of irreversible algebraic dynamical systems, which can be thought of as irreversible analogues of the dynamical systems considered by Schmidt. To each irreversible algebraic dynamical system, we associate a universal C*-algebra and show that it is a UCT Kirchberg algebra under natural assumptions. Moreover, we discuss the structure of the core subalgebra, which is closely related to generalised Bunce-Deddens algebras in the sense of Orfanos. We also construct discrete product systems of Hilbert bimodules for irreversible algebraic dynamical systems which allow us to view the associated C*-algebras as Cuntz-Nica-Pimsner algebras. Besides, we prove a decomposition theorem for semigroup crossed products of unital C*-algebras by semidirect products of discrete, left cancellative monoids.

A strong geometric hyperbolicity property for directed graphs and monoids

We introduce and study a strong “thin triangle” condition for directed graphs, which generalises the usual notion of hyperbolicity for a metric space. We prove that finitely generated left cancellative monoids whose right Cayley graphs satisfy this condition must be finitely presented with polynomial Dehn functions, and hence word problems in NP. Under the additional assumption of right cancellativity (or in some cases the weaker condition of bounded indegree), they also admit algorithms for more fundamentally semigroup-theoretic decision problems such as Green's relations L, R, J, D and the corresponding pre-orders.In contrast, we exhibit a right cancellative (but not left cancellative) finitely generated monoid (in fact, an infinite class of them) whose Cayley graph is essentially a tree (hence hyperbolic in our sense and probably any reasonable sense), but which is not even recursively presentable. This seems to be strong evidence that no geometric notion of hyperbolicity will be strong enough to yield much information about finitely generated monoids in absolute generality.

QUASI-ISOMETRY AND FINITE PRESENTATIONS OF LEFT CANCELLATIVE MONOIDS

We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterization of finite presentability for left cancellative monoids. We also give examples to show that this characterization does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.

IF-AUTOMORPHISMS OF A FREE SOLVABLE Sp-PERMUTABLE ALGEBRA OF A FINITE RANK OVER A LEFT CANCELLATIVE MONOID ARE PSEUDO-TAME

Consider a variety V of acts over a left cancellative monoid S that have a ternary Maltsev operation p(〈p, S〉- algebras ). Using the Magnus–Artamonov representation, the construction of the free Abelian extension of an arbitrary V-algebra was obtained and explored by the author. In the present article we prove that each IF-automorphism of a free k-step solvable V-algebra of a finite rank (k > 1) is pseudo-tame, i.e. it is presented as a product of elementary automorphisms of a special module over a ring with several objects.

本原可分rpp半群

完全0-单半群是一类非常重要的正则半群，也是正则半群结构基础。本原可分rpp半群是完全0-单半群在rpp半群理论中推广。本文主要研究本原可分rpp半群，给出了这类半群的若干特征，并证明了：半群为本原可分rpp半群当且仅当它是满足一些条件的左消幺半群上的Rees矩阵半群的同态像。此外，还研究了若干特类。 Completely 0-simple semigroup is a very important class of regular semigroups, and it’s also the basis for the structure of regular semigroups. Primitively decomposable rpp semigroups are generalizations of completely 0-simple semigroups in the range of rpp semigroups. In this paper, the author studied primitively decomposable rpp semigroups, gave some characterizations for these semigroups and proved that a semigroup S was primitively decomposable rpp semigroup if and only if it was a homomorphic image of Rees matrix semigroup over a left cancellative monoid satisfying some conditions. In addition, some special primitively decomposable rpp semigroups were considered and discussed.

REES MATRIX THEOREM FOR $\mathcal{D}^{(\ell)}$-SIMPLE STRONGLY RPP SEMIGROUPS

A semigroup S is called rpp if all right principal ideals of S, regarded as S1-systems, are projective. An rpp semigroup S is said to be strongly rpp if for any a ∈ S, there exists a unique idempotent e such that [Formula: see text] and a = ea. In this paper, we show that a [Formula: see text]-simple strongly rpp semigroup can be expressed by a Rees matrix semigroup over a left cancellative monoid and conversely. Our result generalizes the classical theorem of Rees in 1940 and also amplifies the Rees theorem in semigroup given by Lallement and Petrich in 1969.

A correspondence between a class of monoids and self-similar group actions I

We describe a correspondence between a class of left cancellative monoids and self-similar group actions in the sense of Nekrashevych et al. This correspondence originated in Perrot’s 1972 thesis, and developed the ideas to be found in Rees’ 1948 paper.

Factorizable Rpp monoids

It is well known that the rpp semigroups are generalized regular semigroups. In order to further investigate the structure of rpp semigroups, we introduce the type-(I, L ∗ ) factorizable monoids. In this paper, we study such factorizable rpp monoids admitting a left univocal type-(I, L ∗ ) factorization. In particular, we prove that a semigroup S is a factorizable rpp monoid admitting a left univocal type-(I, L ∗ ) factorization if and only if S is isomorphic to a semidirect product of a band and a left cancellative monoid. Some results of Catino and Tolo on factorizable semigroups are extended and amplified.

The Structure of Right C-rpp Semigroups

We study the strongly rpp-semigroups whose set of idempotents forms a right regular band and the relation ${\cal L}^* \vee {\cal R}$ is a congruence. This kind of strongly rpp-semigroups was called by Y.Q. Guo the right C-rpp semigroups and he characterized them as semilattice of direct products of left cancellative monoid with right zero semigroup. The aim of this paper is to improve upon Guo’s result, giving a tighter description of the semilattice decomposition involving a system of connecting maps. By using our result, we can always construct a right C-rpp semigroup by glueing up the given ingredients by the connecting maps. An example of an infinite right C-rpp semigroup is constructed.

On left C-wrpp semigroups

It is known that a C-rpp semigroup can be described as a strongsemilattice of left cancellative monoids. In this paper, weintroduce the class of left C-wrpp semigroups which includes theclass of left C-rpp semigroups as a subclass. We shallparticularly show that the semi-spined product of a left regularband and a C-wrpp semigroup forms a curler which is a leftC-wrpp semigroup and vice versa. Results obtained by Fountainand Tang on C-wrpp semigroups are extended and strengthened.

On monoids related to braid groups

OUTLINE. In the first half of the thesis, we introduce the class of'chainable monoids' and describe general techniques for solving the word and division problems and for obtaining normal forms for monoids in this class. Their growth functions are shown to be rational, and calculable. We also show that a monoid with a certain central 'fundamental element' embeds in a group or related monoid. Motivating examples of monoids in this class are positive braid monoids and positive singular braid monoids. Much of the rest of the thesis is devoted to applying the techniques described to singular Artin monoids, and we obtain further results regarding conjugacy and parabolic submonoids. In the final chapter we apply the techniques of chainable monoids to braid groups of complex reflection groups. A chainable monoid is a monoid which may be defined by a finite presentation with four properties; three of which can be determined immediately from the length of the relations and their starting letters, and the fourth, the reduction property, which is more involved to verify. It has been shown that positive Artin monoids, including braid monoids, have these properties. Other examples of monoids in this class include free monoids, free commutative monoids, positive singular braid monoids (see later), a large class of one relation monoids, trace monoids and divisibility monoids. The class of chainable monoids is closed under taking direct and free products. Chainable monoids are always left cancellative. The method of chains enables the construction of a calculable partial map on pairs of words, which returns the left quotient of the first word by the second, whenever it exists, thus giving a solution to the division problem, and hence word problem, in a chainable monoid. Again using chains, we construct another partial map on pairs of words which is shown to return their least common left multiple, whenever it exists. We show that in a chainable monoid, greatest common left divisors always exist, and least common left multiples exist whenever common left multiples exist. [The author is grateful to Dietrich Kuske for recently pointing out that chainable monoids may be algebraically characterised as left cancellative monoids that

PERFECT rpp SEMIGROUPS

The aim of this paper is to study a class of rpp semigroups, namely the perfect rpp semigroups. We obtain some characterization theorems for such semigroups. In particular, the spined product structure of perfect rpp semigroups is established. As an application of spined product structure, we prove that a perfect rpp semigroup is a strong semilattice of left cancellative planks. By a left cancellative plank, we mean a product of a left cancellative monoid and a rectangular band. Thus, the work of J.B. Fountain on C-rpp semigroups is further developed. *The research of the first author is supported by the Natural Science Foundation and the Education Committee Foundation of Yunnan Province, China and also by the Foundation of Yunnan Province, China. #The research of the second author is partially supported by UGC grant (Hong Kong) #2260126. †The research of the third author is supported by a grant of NSF, China and a grant from the Basic Science Research Council of Yunnan Province, China.

On the Refined Decomposition Structure of Left C-rpp Semigroups

Let Mα be a left cancellative monoid and Iα a left zero band. Then we call the direct product (Mα × Iα) the left stripe. In this paper, we prove that a left C-rpp semigroup S is a refined strong semilattice of left stripes if and only if S is a spined product of a C-rpp component and a left regular band.

A characterization of left cancellative monoids by flatness properties

A characterization of left cancellative monoids by flatness properties