The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.     

On self-mapping degrees of <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>-geometry manifolds

In this paper we determined all of the possible self-mapping degrees of the manifolds with S3-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self-mapping degrees of all closed orientable 3-manifold in Thurston＇s picture.     

Flatness properties of diagonal acts over monoids

If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S ^I (for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness. Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats, etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems. S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494. Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007).     

Vector Variational Inequalities and the (S)<Subscript>+</Subscript> Condition

Let $${\cal Z}$$ and X be Hausdorff real topological vector spaces and let $${\cal L}_b(X,{\cal Z})$$ be the space of continuous linear mappings from X into $${\cal Z}$$ equipped with the topology of bounded convergence. In this paper, we define the (S)₊ condition for operators from a nonempty subset of X into $${\cal L}_b(X,{\cal Z})$$ and derive some existence results for vector variational inequalities with operators of the class (S)₊. Some applications to vector complementarity problems are given.     

S-types of Global Towers of Spaces and Exterior Spaces

The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact spaces and manifolds. The notion of exterior weak ℕ-S-equivalences is given by exterior maps which induce isomorphisms on the k-th ℕ-exterior homotopy groups for k ∈ S, where S is a set of non negative integers. The category of exterior spaces with a base ray localized by exterior weak ℕ-S-equivalences is called the category of exterior ℕ-S-types. The existence of closed model structures in the category of exterior spaces permits to establish equivalences between homotopy categories obtained by dividing by exterior homotopy relations, and categories of fractions (localized categories) given by the inversion of classes of week equivalences. The family of neighbourhoods ‘at infinity’ of an exterior space can be interpreted as a global prospace and under the condition of first countable at infinity we can consider a global tower instead of a prospace. The objective of this paper is to use localized categories to find the connection between S-types of exterior spaces and S-types of global towers of spaces. The main result of this paper establishes an equivalence between the category of S-types of rayed first countable exterior spaces and the category of S-types of global towers of pointed spaces. As a consequence of this result, categories of global towers of algebraic models localized up to weak equivalences can be used to give some algebraic models of S-types. The authors acknowledge the financial support given by the projects FOMENTA 2007/03 and MTM2007-65431.     

Projective and indecomposableS-acts

LetS be a semigroup andE the set of all idempotents inS. LetS-Act be the category of allS-acts. LetC be a full subcategory ofS-Act which contains_s S and is closed under coproducts and summands. It is proved that, inC, anS-actP is projective and unitary if and only ifP≅ ∐ _{jϕ I} Se _i ,e _i ϕE. In particular,P is a projective, indecomposable and unitary object if and only ifP ≅Se for somee ∈E. These generalize some results obtained by Knauer and Talwar. Research partially supported by a UGC (HK) (Grant No. 2160092).     

On flatness properties of cyclic S-posets

In this paper, we discuss flatness properties of cyclic S-posets. As applications, some partially ordered monoids are characterized and some results on S-acts can be also obtained. Research supported by the National Natural Science Foundation of China (No.10626012).     

Topological André-Quillen homology for cellular commutative S-algebras

Topological André-Quillen homology for commutative S-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative S-algebras and apply it to obtain results on minimal atomic p-local S-algebras which generalise those of Baker and May for p-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic commutative S-algebras. A. Baker was partially supported by a YFF Norwegian Research Council grant while at the University of Oslo in 2007–8, a Carnegie Trust for the Universities of Scotland grant, and Intas grants 03-51-3251 and 06-1000017-8609. H. Gilmour was supported by an EPSRC studentship. P. Reinhard was supported by an ORS grant. We would like to thank M. Basterra, P. Kropholler, M. Mandell, P. May, B. Richter, J. Rognes and S. Sagave for numerous helpful comments. We are also very grateful to the referee for encouraging us to rethink significantly issues of notation and structure, thus improving the structure of the paper.     

The (<Emphasis Type="Italic">S</Emphasis>)<Subscript>+</Subscript> condition on generalized variational inequalities

In this paper, we derive some existence results for generalized variational inequalities associated with mappings satisfying the (S)₊ condition. The relation between the (S)₊ and (S)₊¹ conditions is discussed. As an application, we also consider multivalued complementarity problems associated with mappings satisfying the (S)₊ condition, and prove a theorem to characterize the solvability of such problems in terms of exceptional families of elements.     

Fundamental groups of manifolds with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-category 2

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁,W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹ → M ⁿ . We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.      

Classification of monoids by injectivities II. CC-injectivity

We characterize monoids over which all S-acts are CC-injective and find conditions under which CC-injectivity of all acts implies that all acts are C-injective. 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).     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

On <Emphasis Type="Italic">S</Emphasis>-quasinormal and <Emphasis Type="Italic">c</Emphasis>-normal subgroups of a finite group

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001). Corresponding author. Supported in part by the Natural Science Foundation of China (10571181), NSF of Guangdong Province (06023728) and ARF(GDEI).     

Analytic properties of the S-matrix for interactions with Yukawa-potential tails

We give an explicit analytic expression for the S-matrix in the case of an arbitrary central interaction inside a sphere of finite radius with a Yukawa-potential tail at large distances. The method uses the completeness of the wave functions outside the finite sphere and also the unitarity and the symmetry conditions for the S-matrix. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 113–128, April, 2006.     

(Homo) Flatness of posets on poideal extensions

We consider pomonoids , where G is a pogroup and I is a poideal of S and show that if an S-poset is principally weakly flat, (weakly) flat, po-flat, (principally) weakly po-flat, (po-) torsion free or satisfies Conditions (P), (P _E ), (P _w ), (PWP), (PWP) _w , (WP) or (WP) _w as an I ¹-poset, then it has these properties as an S-poset. We also show that an S-poset which is free, projective or strongly flat as an I ¹-poset may not generally have these properties as an S-poset.     

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R., _S P _R,R Q _S ,〈〉 , ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζ _S = {(s ₁, s ₂) ∈S×S|ss ₁ = ss ₂, ∀s∈S}, S' = S/ζ _S and US-FAct = { _S M∈S− Act |SM = M and SHom _S (S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom _S (S, ∐ _i∈I S) →∐ _i∈I S, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism. The research is partially supported by a UGC(HK) grant #2160092. Project is supported by the National Natural Science Foundation of China     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)     

New examples of Cantor sets in <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> that are not <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-minimal

Although every Cantor subset of the circle (S¹) is the minimal set of some homeomorphism of S¹, not every such set is minimal for a C¹ diffeomorphism of S¹. In this work, we construct new examples of Cantor sets in S¹ that are not minimal for any C¹-diffeomorphim of S¹.     

Cohomologie <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> en degré 1 des espaces homogènes

The L ^p -cohomology in degree 1 of Riemannian homogeneous spaces is computed. It turns out that reduced cohomology does not vanish exactly for spaces quasiisometric to negatively curved homogeneous spaces.      

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)