The Similarity Between Finitary and Artinian-Finitary Groups

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is and the Artinian-finitary automorphism group of M over R is We investigate further the surprisingly close relationship between these two types of automorphism groups. Their group theoretic properties seem practically identical.     

On Generalized Finitary Groups

We introduce a wide range of generalized finitary automorphism groups of an arbitrary module M over an arbitrary ring R. The largest such subgroup of Aut_RM that we seriously consider here is the subgroup of all R-automorphisms g of M such that M(g − 1) has Krull dimension. We also consider the subgroup of all R-automorphisms g of M such that M(g − 1) is Artinian as an R-module. The results are vaguely analogous to the genuine finitary case but are somewhat weaker.     

Artinian-Finitary Groups over Commutative Rings and Non-Commutative Rings

Let M be a module over the ring R. Extensive use is made ofKrull codimension to study further the Artinian-finitary automorphismgroup of M over R. Substantial progress is made where either M isresidually Noetherian or R is commutative. There are some group-theoreticconsequences of the two main structure theorems.     

Locally Finite Finitary Skew Linear Groups

Let V be a vector space over the division ring D of infinitedimension. We study locally finite, primitive groups G of finitarylinear automorphisms of V. We show that the derived group G'of G is infinite, simple, and lies in every non-trivial normalsubgroup of G, and that G' G Aut G'. Moreover if char D =0, then G is either the finitary symmetric group or the alternatinggroup on some infinite set. If D is commutative, that is, ifD is a field, then all these results are known and are the consequenceof the collective work of a number of people: in particular(in alphabetical order) V. V. Belyaev, J. I. Hall, F. Leinen,U. Meierfrankenfeld, R. E. Phillips, O. Puglisi, A. Radfordand quite probably others. 2000 Mathematics Subject Classification:20H25, 20H20, 20F50.     

Finitary Automorphism Groups Over Non-Commutative Rings

The notion of a group of finitary automorphisms of an arbitrarymodule over an arbitrary ring is introduced, and it is shownhow properties of such groups can be derived from the case wherethe ring is a division ring (that is, from the properties offinitary skew linear groups). The results are particularly strongif either the group is locally finite or the module is Noetherian.     

The Complete Reducibility of Locally Completely Reducible Finitary Linear Groups

Let V be a vector space over some division ring D, and G a finitarysubgroup of GL(V). If G is locally completely reducible, thenthe D-G modules V, [V, G] and V/C_V(G) need not be completelyreducible, even if dim_DV is finite. Moreover, if F is a field,then V and V/C_V(G) need not be completely reducible. We provehere that if D is a finite-dimensional division algebra andG is locally completely reducible, then [V, G] is always a completelyreducible D-G module. 1991 Mathematics Subject Classification20H25.     

Periodic Groups Covered by Transitive Subgroups of Finitary Permutations or by Irreducible Subgroups of Finitary Transformations

Let be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive -groups are countably recognizable, while totally imprimitive -groups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive -subgroups. It turns out that totally imprimitive -groups in the class are countably recognizable.

     

Isomorphisms of Certain Locally Nilpotent Finitary Groups and Associated Rings

For any chain Γ the ring NT(Γ,K) of all finitary Γ-matrices ‖a _ij ‖ _i,jεΓ over an associative ring K with zeros on and above the main diagonal is locally nilpotent and hence radical. If R′=NT(Γ′,K′),R=NT(Γ,K) and either |Γ|<∞ or K is a ring with no zero-divisors, then isomorphisms between rings R and R′, their adjoint groups and associated Lie rings are described.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

Almost Isomorphic Torsion-Free Abelian Groups and Similarity of Homogeneously Decomposable Groups

In the paper the concept of the similarity of homogeneously decomposable groups is introduced. One investigates the cases in which almost isomorphic groups are similar or isomorphic. Mathematics Subject Classification (2000) 20K99.     

Fuzzy幂群与Fuzzy商群的相互关系

In this paper,we extend the concept of fuzzy quotient groups.The structures of fuzzy power groups and fuzzy quotient groups are discussed.The relationship between fuzzy power groups and fuzzy quotient groups are considered.     

Finitary monoids

Abstract. No abstract.     

Finitary bases and formal generating functions

Let k be a nonzero, commutative ring with 1, and let R be a k-algebra with a countably-infinite ordered free k-basis B = [p_n: n 0]. We characterize and analyze those bases from which one can construct a k-algebra of ‘formal B-series’ of the form f=∑c_np_n, with c_n ε k, showing inter alia that many classical polynomial bases fail to have this property.     

Finitary Incidence Algebras

It is well known that incidence algebras can be defined only for locally finite partially ordered sets (Doubilet et al., 1972 Doubilet , P. , Rota , G.-C. , Stanley , R. ( 1972 ). On the foundations of combinatorial theory (VI): The idea of generating function . In: Proc. of the Sixth Berkely Symp. on Math. Stat. and Probab . v. II , Univ. of Calif. Press , pp. 267 – 318 . [Google Scholar]; Stanley 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]). At the same time, for example, the poset of cells of a noncompact cell partition of a topological space is not locally finite. On the other hand, some operations, such as the order sum and the order product (Stanley, 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]), do not save the locally finiteness. So it is natural to try to generalize the concept of incidence algebra.

In this article, we consider the functions in two variables on an arbitrary poset (finitary series), for which the convolution operation is defined. We obtain the generalization of incidence algebra—finitary incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.     

企业集团与科研机构的融合机制

随着科技经济一体化的发展，作为其中重要方面的企业集团与科研机构的融合便受到了关注．本文首先论述了双方融合的必要性和必然性，同时对市场经济中出现和未出现的可行性融合模式分类进行了论述．最后，本文从科技经济一体化发展的趋势，得到结论：企业集团与科研机构必将从“融合”为主流走向“结合”为主流．