Products and h-homogeneity

Building on work of Terada, we prove that h-homogeneity is productive in the class of zero-dimensional spaces. Then, by generalizing a result of Motorov, we show that for every non-empty zero-dimensional space X there exists a non-empty zero-dimensional space Y such that X×Y is h-homogeneous. Also, we simultaneously generalize results of Motorov and Terada by showing that if X is a space such that the isolated points are dense then Xκ is h-homogeneous for every infinite cardinal κ. Finally, we show that a question of Terada (whether Xω is h-homogeneous for every zero-dimensional first-countable X) is equivalent to a question of Motorov (whether such an infinite power is always divisible by 2) and give some partial answers.     

Zero-Dimensional Imbeddability

In this article, we try to understand which generically complete intersection monomial ideals with fixed radical are Cohen–Macaulay. We are able to give a complete characterization for a special class of simplicial complexes, namely the Cohen–Macaulay complexes without cycles in codimension 1. Moreover, we give sufficient conditions when the square-free monomial ideal has minimal multiplicity.     

First order logic without equality on relativized semantics

Let $\alpha \ge 2$ be any ordinal. We consider the class ${\mathsf{Drs}}_{\alpha }$ of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of ${\mathsf{Drs}}_{\alpha }$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class ${\mathsf{Drs}}_{\alpha }$ corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.     

Lukasiewicz语义集上的紧Hausdorff拓扑

以Ω记从全体命题之集F（S）到单位区间的全体Lukasiewicz赋值之集.本文通过一种自然的方法在Ω上引入了Fuzzy拓扑δ,证明了其为第二可数的零维良紧空间,并证明了δ在Ω上生成的截拓扑空间是第二可数的紧Hausdorff空间,从而是可度量化的空间.     

General Jacobi Identity Revisited Again

Synthetic differential geometry occupies a unique position in topos-theoretic physics. Nevertheless it has appeared somewhat too conceptual to physicists in general, partly because it has appeared to lack computational aspects. Its computational facets are really concerned with computation of the quasi-colimit of a finite diagram of infinitesimal spaces, or equivalently, with computation of the limit of a finite diagram of Weil algebras. Indeed we have been forced to do a highly involved computation of the above kind by hand in our previous papers (Nishimura, H. in Int. J. Theor. Phys. 36:1099–1131, 1997 and Nishimura, H. in Int. J. Theor. Phys. 38:2163–2174, 1999). The principal objective in this paper is to show that Grö bner bases techniques provide us with means that relegate such computations to computers.     

Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.     

Clean Semiprime f-Rings with Bounded Inversion

Abstract

An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently,Anderson and Camillo (Anderson,D. D.,Camillo,V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover,every clean ring is a pm-ring,that is every prime ideal is contained in a unique maximal ideal. In the same article,the authors give an example of a commutative ring which is a pm-ring yet not clean,e.g.,C(?). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1.     

On Prime Spectrums of Commutative Rings

We show that π-regular rings and clean rings can be completely characterized by topological properties of their prime spectrums respectively. In addition, we give some applications of those result. Among others, we improve the main result of Samei (2004 Samei , K. ( 2004 ). Clean elements in commutative reduced rings . Comm. Algebra 32 ( 9 ): 3479 – 3486 . [CSA] [CROSSREF] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and give a new criterion for a clean ring that a commutative ring is clean if and only if idempotents lifts modulo every radical ideal.