The monoid of orientation-preserving mappings on a chain

We introduce new kinds of regular monoids of transformations consisting of mappings which preserve or reverse the orientation of a finite cycle.     

The Monoid of All Injective Order Preserving Partial Transformations on a Finite Chain

In this paper we study several structural properties of the monoids \poi _n of all injective order preserving partial transformations on a chain with n elements. Our main aim is to give a presentation for these monoids. January 27, 1999     

Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application

In this paper we compute the abelian kernels of the monoids POI_n and POPI_n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI_n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI_n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.     

On the existence of special homogeneous monoids

The paper contains constructions of finite transformation monoids such that the number of transformations from one point into another is a constant and whose groups of units are not transitive.Dedicated to E. S. Ljapin and Stefan Schwarz on their 60 ^th birthdays.     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

Generation of infinite factorizable inverse monoids

We investigate the generation of factorizable inverse monoids, paying special attention to the factorizable parts of the symmetric and dual symmetric inverse monoids. Key ideas covered include rank, relative rank, Sierpiński rank, and the semigroup Bergman property. The results for finite monoids are well-known or follow quickly from well-known facts, so most of the paper concerns the infinite case.     

Monoides de comptage et langages rationnels

We study some syntactic properties of languages obtained from rational languages through counting operations. These counting operations use the congruences-threshold p, modulo n—on the set of integers. We show that in some cases they preserve the properties of syntactic monoids: the structure of idempotent and commutative monoids is not modified, aperiodic monoids with central idempotents keep their central idempotents and their regular D-classes still are groups although not trivial, aperiodic left (right) nilsimple semigroups still have regular R (L)-classes which are (non trivial) groups. If possible results are expressed in terms of varieties of semigroups or monoids.     

Möbius inversion formula for monoids with zero

The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, i.e., with an absorbing element, in particular the so-called Rees quotients of locally finite monoids. Some relations between the Möbius functions of a monoid and its Rees quotient are also provided.     

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.     

On the Arithmetic of Strongly Primary Monoids

In this paper we study the arithmetic of strongly primary monoids. Numerical monoids and the multiplicative monoids of one-dimensional local Mori domains are main examples of strongly primary monoids. Our investigations focus on local tameness, a basic finiteness property in the theory of non-unique factorizations. It is well-known that locally tame strongly primary monoids have finite catenary degree and finite set of distances.     

图的强收缩核与图的强自同态幺半群的正则性

本文研究图及其强自同态幺半群．首先刻画了图的强自同态幺半群的正则元，然后给出了此幺半群正则的充要条件．这推广了［１］和［２］中关于有限图的强自同态幺半群正则的结果．     

Constructions and presentations for monoids

Presentations are found for the wreath product of two monoids, the Schützenberger product of two monoids, the Bruck-Reilly extension of a monoid, strong semilattices of monoids and Rees matrix semigroups of monoids.     

右（左）逆半群的半直积及圈积

给出了两个幺半群的半直积及圈积为右（左）逆半群的充分必要条件，从而推广了[2]中两个幺半群的半幺直积和圈积为逆半群的充分必要条件．     

Localization with respect to endomorphisms

We introduce the class of localizable monoids. It contains inverse monoids. Then we define localizations of monoids with respect to localizable submonoids of their monoid of endomorphisms. These constructions can be applied to a category of left modules or to a category of A-rings. As a result, we are able to invert endomorphisms within the original category, unlike inversive localizations of Cohn’s type which need a base change.     

Strongly Flat and Condition (P) Covers of Acts Over Monoids

In this article we characterize monoids over which every right S-act has a strongly flat (condition (P)) cover. Similar to the perfect monoids, such monoids are characterized by condition (A) and having strongly flat (condition (P)) cover for each cyclic right S-act. We also give a new characterization for perfect monoids as monoids over which every strongly flat right S-act has a projective cover.     

A Survey of Lipschitz Monoids

Lipschitz monoids (or semi-groups) have been investigated since 1975, and have received more attention when more general Vahlen matrices were taken into consideration. Yet few people know the extent and the effectiveness of the already available theory about Lipschitz monoids. The present survey intends to let more people become acquainted with them.