On monoids over which all generators satisfy a flatness property

This paper addresses conditions under which all generators in the category of right S-acts (where S is a monoid) satisfy a flatness property. There are characterizations for monoids over which all generators satisfy a flatness property α where α can stand for freeness, projectivity, strong flatness, Condition (P), principal weak flatness and torsion freeness. To our knowledge, the problem has not been studied for other flatness properties such as weak flatness, Condition (E) and regularity. The present paper addresses this gap.     

On Flatness Properties of Torsion Free Right Rees Factor Acts

In this paper, we investigate the monoids over which all torsion free right Rees factor acts satisfy some properties that follow from projectivity(such as (weak) flatness, strong flatness, condition (P), etc.). These results answer the questions in [1].     

On a Generalization of Weak Pullback Flatness

In 1998, Laan introduced weakly pullback flat acts and showed that weak pullback flatness is equivalent to the conjunction of Conditions (P) and (E′). In 2013, Golchin introduced Condition (P′), which is a generalization of Condition (P). In this article, we first define Condition (PF″), which lies strictly between weak pullback flatness and Condition (P′), and prove that Condition (PF″) coincides with the conjunction of Conditions (P′) and (E′). Furthermore, we give a classfication of monoids by Condition (PF″) of (cyclic, Rees factor) acts, and find a necessary and sufficient condition under which Condition (PF″) coincides with Condition (P′) (resp., weak pullback flatness, strong flatness) for Rees factor acts. Finally, we investigate Conditions (P′), (E′) and (PF″) using some new types of epimorphisms.     

条件(wp)对幺半群的刻画

本文引入了条件（wp)，研究了其性质，建立了它与弱平坦性之间的联系，并利用其给出了某些幺半群的S－系范畴特征。     

When Torsion Free Acts are Principally Weakly Flat

A necessary and sufficient condition for principal weak flatness of all right torsion free acts over a monoid is found. It is shown that the corresponding class of monoids is a proper subclass of left PP monoids.     

n-正则幺半群的平坦性刻画

本文研究了主弱平坦性质的推广问题.利用张量积相等的等式组,以及同调分类方法,获得了对广义正则的幺半群的刻画结果,推广了关于正则幺半群刻画的主要的结果.     

Homoflatness on Ideal Extensions

We consider monoids $S=G\;\dot{\cup}\; I$ where $G$ is a group and $I$ is an ideal of $S$ and show that if an $S$-act is principally weakly homoflat or weakly homoflat as an $I^1$-act, then it has these properties as an $S$-act. We also show that an $S$-act which is (weakly) pullback flat, equalizer flat, (principally) weakly kernel flat, translation kernel flat or satisfies Condition $(E)$ as an $I^1$-act may not generally have these properties as an $S$-act. The flatness notions considered in this paper were introduced in {\it V. Laan, Pullbacks and flatness properties of acts I, Comm. Alg. ${\bf 29}(2)$ (2001), 829--850}.     

On a generalization of principal weak flatness property

In this article a generalization of principal weak flatness of acts is defined. Using this new property the characterization of some new classes of monoids are investigated. Furthermore, many known results are generalized.     

On Retractable $S$-Acts

In this paper we introduce a class of right S-acts called retractable S-acts which are right S-acts with homomorphisms into their all subacts. We also give some classifications of monoids by comparing such acts with flatness properties.     

On Retractable S-Acts

In this paper we introduce a class of right S-acts called retractable S-acts which are right S-acts with homomorphisms into their all subacts. We also give some classifications of monoids by comparing such acts with flatness properties.     

Classification of monoids by injectivities I. C-injectivity

A new kind of injectivity, namely, C-injectivity is investigated. Classifications of a monoid by properties of C-injective acts and right ideals are presented. It is shown that the class of monoids over which all acts are C-injective generalizes the class of self-injective monoids. On the way we complete an early result by Johnson and McMorris (Proc. Am. Math. Soc. 36(2), 385–388, 1972). Research of X. Zhang supported by the China Scholarship Council No. 2006101056. Research of Y. Chen supported by the NNSF of China (No. 10771077) and the NSF of Guangdong Province (No. 06025062).     

Principally Weakly and Weakly Coherent Monoids

We shall call a monoid S principally weakly (weakly) left coherent if direct products of nonempty families of principally weakly (weakly) flat right S-acts are principally weakly (weakly) flat. Such monoids have not been studied in general. However, Bulman-Fleming and McDowell proved that a commutative monoid S is (weakly) coherent if and only if the act S ^I is weakly flat for each nonempty set I. In this article we introduce the notion of finite (principal) weak flatness for characterizing (principally) weakly left coherent monoids. Also we investigate monoids over which direct products of acts transfer an arbitrary flatness property to their components.     

Examples concerning absolute flatness and amalgamation in pomonoids

Flatness properties of acts over monoids have been studied for almost four decades and a substantial literature is now available on the subject. Analogous research dealing with partially ordered monoids acting on posets was begun in the 1980s in two papers by S.M. Fakhruddin, and, after a dormancy period of some 20 years, has recently been rekindled with the appearance of several research articles. In comparing flatness properties of S-acts and S-posets, it has been noted that the imposition of order results in severe restrictions as far as absolute flatness is concerned. For example, whereas every inverse monoid is absolutely flat (meaning all of its left and right acts are flat), even the three-element chain in its natural order, considered as a pomonoid, fails to have this property. It has long been understood that absolutely flat monoids, in particular, inverse monoids, are amalgamation bases in the class of all monoids. The purpose of the present article is to further investigate absolute flatness of pomonoids and to begin to study its connection with amalgamation in that context. T.E. Hall’s results, that amalgamation bases in the class of all monoids have the so-called representation extension property (REP), which in turn implies the right congruence extension property, are first adapted to the ordered context. A detailed study of the compatible orders (of which there are exactly 13) on the three-element chain semilattice U then reveals a wide range of possibilities: exactly four of these orders render U absolutely flat as a pomonoid, two more give it the right order-congruence extension property in every extension (RCEP) (but fail to make it an amalgamation base because of the failure of the ordered analogue of (REP)), and for the remaining seven, even (RCEP) fails.     

幺半群的内射类及投射类刻画

类比于一般环上模的内射类,定义了幺半群上的S-系的内射类和投射类,并利用它们刻画了几类特殊的幺半群．证明了完全内射幺半群和完全拟内射幺半群是等价的．并且证明了对于标致幺半群S,它是完全投射的当且仅当它是完全拟投射的当且仅当它上面的投射S-系构成了一个投射类．     

半群半直积的主投影性质(英文)

本文研究了半群半直积的主投影性质.利用适当半群,获得了右主投影半群半直积的充分和必要条件,推广了半直积的一些结果.     

Some 3/2-transitive permutation groups in which the stabilizer of two points is always Abelian

A monoidS is susceptible to having properties bearing upon all right acts overS such as: torsion freeness, flatness, projectiveness, freeness. The purpose of this note is to find necessary and sufficient conditions on a monoidS in order that, for example, all flat rightS-acts are free. We do this for all meaningful variants of such conditions and are able, in conjunction with the results of Skornjakov [8], Kilp [5] and Fountain [3], to describe the corresponding monoids, except in the case all torsion free acts are flat, where we have only some necessary condition. We mention in passing that homological classification of monoids has been discussed by several authors [3, 4, 5, 8].In the following,S will always stand for a monoid. A rightS-act is a setA on whichS acts unitarily from the right in the usual way, that is to saya(rs) = (ar)s, a1 =a (a A,r,s S) where 1 denotes the identity ofS.     

An approach to injective acts over monoids based on indecomposability

The main purpose of this paper is to investigate classes of acts that are injective relative to all embeddings with indecomposable domains or codomains. We give some homological classifications of monoids in light of such kinds of injectivity. Our approach to the indecomposability property provides a new characterization of right absolutely injective monoids such that all indecomposable acts are injective.     

Flatness and Coproducts

In this paper by a new proof we will show that most of flatness properties of acts over monoids can be transferred to their coproduct and vice versa.