Now showing 1 - 10 of 29
  • Publication
    Open Access
    δ-primary ideals of commutative rings
    In this paper we investigate -primary ideals which unify prime ideals and primary ideals. A number of main results about prime ideals and primary ideals are extended into this general framework. Prime ideals and primary ideals are two of the most important structures in commutative algebra. Although different from each other in many aspects, they share quite a number of similar properties as well( see [1] ). However, these two structures have been treated rather differently, and all of their properties were proved separately. It is therefore natural to examine whether it is possible to have a unified approach to studying these two structures. In this short paper we introduce the notion of -primary ideals where is a mapping that assigns to each ideal I an ideal (I) of the same ring. Such -primary ideals unify the prime and primary ideals under one frame. This approach clearly reveals how similar the two structures are and how they are related to each other. In the first section, we introduce ideal expansion and define primary ideals with respect to such an expansion. Besides the familiar expansions 0, 1 and B, we also have a new expansion M defined by means of maximal ideals. In the second section, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized primary ideals with respect to such expansions. In this paper, all the rings used are commutative rings with an multiplication identity and all the ring homomorphisms preserve the identity. We shall use Id(R) to denote the set of all ideals of the ring R.
      185  194
  • Publication
    Open Access
    Real-life mathematics tasks: A Singapore experience
    (2012)
    Wong, Khoon Yoong
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    ;
    Cheang, Wai Kwong
    ;
    ;
    Lee, P. Y. (Peng Yee)
    ;
    Yen, Yeen Peng
    ;
    Fan, Lianghuo
    ;
    ;
    Quek, Khiok Seng
    ;
    So, Hyo-Jeong
    ;
    Ng, Yvonne Qiu Ting
    ;
    Cheong, Jim Siew Kuan
    ;
    Oh, Kwang Shin
      845  9489
  • Publication
    Open Access
    Lattices of Scott-closed sets
    A dcpo P is continuous if and only if the lattice C(P) of all Scott-closed subsets of P is completely distributive. However, in the case where P is a non-continuous dcpo, little is known about the order structure of C(P). In this paper, we study the order-theoretic properties of C(P) for general dcpo's P. The main results are: (i) every C(P) is C-continuous; (ii) a complete lattice L is isomorphic to C(P) for a complete semilattice P if and only if L is weak-stably C-algebraic; (iii) for any two complete semilattices P and Q, P and Q are isomorphic if and only if C(P) and C(Q) are isomorphic. In addition, we extend the function P 7! C(P) to a left adjoint functor from the category DCPO of dcpo's to the category CPAlg of C-prealgebraic lattices.
      348  138
  • Publication
    Open Access
    Topologies generated by families of sets and strong poset models
    (2020) ;
    Xi, Xiaoyong
    ;
    Chen, Yixiang
    A poset model of a topological space X is a poset P such that X is homeomorphic to the maximal point space of P (the set Max(P) of all maximal points of P equipped with the relative Scott topology of P). The poset models of topological spaces based on other topologies, such as Lawson topology and lower topology, have also been investigated by other people. These models establish various types of new links between posets and topological spaces. In this paper we introduce the strong Scott topology on a poset and use it to de ne the strong poset model: a strong poset model of a space X is a poset P such that Max(P) (equipped with the relative strong Scott topology) is homeomorphic to X. The main aim is to establish a characterization of T1 spaces with T-generated topologies (such as the Hausdor k-spaces) in terms of maximal point spaces of posets. A poset P is called ME-separated if for any elements x; y of P, x y i " y \ Max(P) "x \ Max(P). We consider the topological spaces that have an ME-separated strong poset model. The main result is that a T1 space has an ME-separated strong poset model i its topology is T-generated. The class of spaces whose topologies are T-generated include all Scott spaces and all Hausdor k-spaces.
      313  170
  • Publication
    Open Access
    Asking converse questions and looking for extensions to Gauss's method for summing arithmetic progressions
    (Association of Mathematics Educators, 2002) ;
    Posing good problems is important for learning, teaching and research in mathematics. In this paper, the converse problem posing strategy is applied to Gauss's method that has been used to obtain the summation formula of an Arithmetic Progression. The work here serves as a simple but typical example to demonstrate the use of this strategy. The results obtained may also help the reader see to what extent Gauss's method can be applied, thus enriching one's understanding of this famous method.
      152  204
  • Publication
    Open Access
    On topological spaces that have a bounded complete DCPO model
    (2018) ;
    Xi, Xiaoyong
    A dcpo model of a topological space X is a dcpo (directed complete poset) P such that X is homeomorphic to the maximal point space of P with the subspace topology of the Scott space of P. It has been proved previously by X. Xi and D. Zhao that every T1 space has a dcpo model. It is, however, still unknown whether every T1 space has a bounded complete dcpo model (a poset is bounded complete if each of its upper bounded subsets has a supremum). In this paper we rst show that the set of natural numbers equipped with the co- nite topology does not have a bounded complete dcpo model, then prove that a large class of topological spaces (including all Hausdorff k-spaces) have a bounded complete dcpo model. We shall mainly focus on the model formed by all the nonempty closed compact subsets of the given space.
      108  109
  • Publication
    Open Access
    Problem-posing in teaching university algebra
    (1999-12) ;
    Lee, P. Y. (Peng Yee)
    Posing or raising appropriate problems is necessary and important for active and deep learning in mathematics. However, students rarely make effort to find thinking problems by themselves. Such an attitude and behavior often lead to passive learning and cause various difficulties and problems in mathematics teaching. The main objective of this paper is to explore the ways to develop students’ ability to find and pose good mathematical problems and thus to promote more active learning in mathematics.
      125  115
  • Publication
    Open Access
    Assessing mathematical competencies using disciplinary tasks
    (2012)
    Cheang, Wai Kwong
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    ;
    The Singapore Mathematics Assessment and Pedagogy Project (SMAPP) is a research project conducted by the National Institute of Education and funded by the Ministry of Education. It aims to make assessment practices an integral part of teaching and learning, and broaden student learning outcomes by using authentic disciplinary tasks. As part of the project, some guidelines are provided for designing disciplinary tasks which have the distinctive features of their emphasis on contextual aspects. One of the criteria of a good disciplinary task is its ability to assess multiple mathematical competencies of students. In this paper, we will present some examples to illustrate how these competencies can be assessed. Another aim is to find out to what extent these tasks serve the purpose of assessing these competencies, by analyzing the students’ performance in a sample SMAPP task.
      220  310
  • Publication
    Open Access
    The topological structure of the set of fuzzy numbers with the supremum metric
    (2022)
    Liu, Dongming
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    Yang, Zhongqiang
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    We study the family of all fuzzy sets of the n-dimensional Euclidean space, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in a non-degenerate convex subset Y, and prove that the following statements are equivalent: (i) The family of fuzzy sets with the topology induced by the supremum metric is homeomorphic to a non-separable Hilbert space whose weight is the cardinality of the set of all real numbers; (ii) the non-degenerate convex subset Y is topologically complete, or equivalently, it is a countable intersection of open sets in the n-dimensional Euclidean space.
    WOS© Citations 3Scopus© Citations 3  55  5
  • Publication
    Open Access
    Some open problems on well-filtered spaces and sober spaces
    (2021)
    Xu, Xiaoquan
    ;
    In the past few years, the research on sober spaces and well-filtered spaces has got some breakthrough progress. In this paper, we shall present a brief summarizing survey on some of such development. Furthermore, we shall pose and illustrate some open problems on well-filtered spaces and sober spaces.
    WOS© Citations 7Scopus© Citations 7  46  43