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    Theory of frames
    The study of frames can be traced back to as early as Wallman's work in 1938, in which he initiated the study of topological properties from a lattice-theoretical point of view. C. Ehresmann and J. Benabou firstly regarded complete Heyting algebras as generalised topological spaces in their own right. Such lattices were called 'local lattices'. It was Dowker and Strauss who first used the term 'frame' in their systematic study of such structure. After that many people have made a significant contribution to the study of frames (or locales: the opposite categorical version of frames), such as Isbell, Banaschewski, Joyal, Johnstone, Simmons, etc.

    On the other hand, inspired by frames, several types of generalised frames have been introduced and studied in relatively recent times. Three notable examples of generalised frames are σ-frames, κ-frames and preframes. The κ-frames, which were first systematically studied by Madden recently, generalise both frames and σ-frames. While preframes, which have been carefully studied by Johnstone and Vickers, belong to a different type of generalisation.

    The emergence of Ζ-continuous posets which unifies various "continuous" structures and discussed most of their basic properties. In 1992, D. Zhao launched a similar programme in attempt to make a uniform approach to various frame-like structure by introducing Ζ-frames. The approach turns out to be very convenient and effective for further categorical treatment.

    Category theory is an economical tool that provides a common framework for many branches of mathematics, especially in topology and algebra. In the process of my study of generalised frames, categorical concepts are employed extensively.

    My three-years course of study has been constantly motivated by many important papers and publications. The first one, Nuclearity by K.A. Rowe ([19]), is an important paper. The concept of nuclearity aims to characterise finite-dimensionality in symmetric monoidal closed categories. Rowe made a systematic study of nuclearity via many different examples.

    The second one is on Nuclearity in the category of complete semilattices by D.A. Higgs and KA. Rowe ([11]). This paper demonstrated that the nuclear objects of the category of complete lattices are precisely the completely distributive lattices (CDL for short). In lattice theory, completely distributive lattices have always attracted special attention. Thus, the CDLs, became one of the most important classes of lattices and have been extensively studied by many authors. This fact, together with many other examples in [19], lead to the following question: Are nuclear objects projective? The first part of my project indicates a positive answer with some minimal assumptions.

    The book A compendium of continuous lattices ([10]), written by six expert lattice-theorists (G. Gierz et al.), is an excellent guidebook for me in learning the ropes of continuous lattice theory. Difficult book it is indeed, but it gives a concise and in-depth treatment of continuity in lattice theory. It gives me a very sound foundation that prepares me to understand D.Zhao's approach to generalised frame theory via Ζ-theory.

    The doctoral dissertation Generalisation of Continuous Lattices and Frames by D.Zhao ([21]) gives a detailed and clear introduction of Ζ-frames. It opened up a completely new and exciting area of research for me because the concepts and mathematical concepts that arise from Ζ-frames are very rich.

    One natural question is whether the concept of nuclearity may be defined for the category of Ζ-frames. The very first step is, of course, to understand how tensor products may be set up in order that we have an autonomous categorical structure.

    So the third paper Tensor products and bimorphisms by B.Banaschewski and E. Nelson ([1]) provides very handy information about conditions which will guarantee the existence of tensor multiplication in a concrete category.

    Despite the promises that the Ζ-theory seemed to offer, there is one main obstacle that hinders a natural autonomous structure on ZFrm : It is not even clear how the internal hom may be established, let alone the tensor product. However, in the categories of complete join-semilattice, frames and preframes, various constructions have been made to show that they are autonomous categories (see [11], [16] and [17]). This branches off to two alternatives. One of them is to simplify the problem and focus on a less intricate category, namely the category of the Ζ-complete posets and the morphisms that preserve Ζ-sups. Although the internal hom exists, we still cannot enjoy the luxury of having a tensor product.

    Another approach, which may be more difficult, is to generalise P. Johnstone's work ([16]). It seems that we can take advantage of the monadic nature of the category of Ζ-frames. Much work, involving Universal Algebra and Proof Theory, remains to be done in this direction.

    The sixth is the paper On projective z-frames by D.Zhao ([23]) which characterises the E-projective objects in the category zFrm in adjunction to the category of semilattices. This leads to the study of E-projective objects in the category of frames in adjunction to the category of Ζ-frames. While working furiously at this problem, I ventured into the topic of generalised Scott-topology.
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  • Publication
    Open Access
    Pre-service teachers’ use of symmetry of quadratic graphs in problem solving
    (2012) ;
    Ho, Foo Him
    ;
    Dindyal, Jaguthsing
    This paper reports on the performance of pre-service mathematics teachers with regards to the use of symmetry in problem solving. The current study reveals that pre-service teachers do not make use of symmetry as their main problem-solving tool, even in situations where symmetry is the obvious notion to consider. In addition, a quantitative comparison of the effectiveness of problem solving among approaches (those relying on symmetry versus those relying on other conventional methods) is reported herein. This formal comparison validates the opinion that active usage of symmetry in problem solving significantly enhances the chance of solving non-routine problems.
      156  186
  • Publication
    Open Access
    When exactly is Scott sober?
    A topological space is sober if every nonempty irreducible closed set is the closure of a unique singleton set. Sobriety is precisely the topological property that allows one to recover completely a topological space from its frame of opens. Because every Hausdor space is sober, sobriety is an overt, and hence unnamed, notion. Even in non-Hausdor settings, sober spaces abound. A well-known instance of a sober space appears in domain theory: the Scott topology of a continuous dcpo is sober. The converse is false as witnessed by two counterexamples constructed in the early 1980's: the first by P.T. Johnstone and the second (a complete lattice) by J. Isbell. Since then, there has been limited progress in the quest for an order-theoretic characterization of those dcpo's for which their Scott topology is sober. This paper provides one answer to this open problem.
      229  212
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    Open Access
    Putnam grasshopper problem: Collaborative problem solving, generalisations, and computational thinking
    (Association of Mathematics Educators, 2023) ; ; ;
    Tong, Cherng Luen
    This paper considers several generalisations of an interesting problem posed in the 2021 William Lowell Putnam Mathematical Competition. We describe how the authors embarked on a collaborative problem-solving journey which resulted in solving two of these generalisations and how computational thinking guided our approaches.
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    Open Access
      210  143
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    K-12 school mathematics curriculum: Insights on development, renewal and future orientation
    In Singapore, nationwide educational policies and movements have taken place frequently and within a short space of time from each other. In turn, such educational initiatives get translated into changes in curricula of every school subject – mathematics inclusive. In this chapter, we make an explicit connection between Singapore students’ PISA performance and the aforementioned curricular shifts by highlighting the major changes that have taken place in K-12 Singapore school mathematics curriculum, analysing them in terms of the shifts in curriculum ideologies. Then we map each of the dimensions of the PISA assessment framework with the components of the Singapore Mathematics Curriculum Framework to further substantiate the claim that “the [Singapore] education system and school mathematics curriculum contribute in part towards the success of Singapore’s students in … PISA” (Kaur et al., Mathematics education in Singapore. Springer, Singapore, 2019, p. 134). Additionally, we give some answers to the “Ten Questions for Mathematics Teachers … and how PISA can help answer them” (OECD, PISA, OECD Publishing, Paris, 2016) that are relevant to the Singapore context. Based on the twenty-first-century competencies identified, respectively, by the OECD and Ministry of Education (Singapore), we explore possible new directions the national mathematics curriculum may head towards and hope to peek into the future education landscape for Singapore mathematics.
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  • Publication
    Open Access
    Teaching undergraduate mathematics: A problem solving course for first year
    In this paper we describe a problem solving course for first year undergraduate mathematics students who would be future school teachers.
      50  82
  • Publication
    Open Access
    An introduction to topology and its applications: A new approach
    (2012-10)
    The present paper aims to introduce the topic of topology to a reader with little background knowledge of university mathematics. Unlike the traditional approach which begins by abstracting of real analysis to metric spaces and then routing to topological spaces, our present exposition of opology exploits some basic experience with computations, scientific measurements and observations, such as those encountered in the use of a scientific calculator, a computer program or a measuring device. It is hoped that, in the present age of computers and information technology, topology, which is known to be an abstract mathematical subject, can be made palatable and relevant to a wider audience. Conjugate to our main discourse, we introduce some necessary knowledge of functional programming which is needed by the reader to understand how applications of topology can be realized in a concrete manner.
      137  226
  • 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.
      347  133
  • Publication
    Open Access
    Strong completions of spaces
    (2016)
    Andradi, Hadrian
    ;
    A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible, (2) directed subsets (which of fundamental status in domain theory) of a poset are exactly its Alexandroff irreducible sets, (3) directed subsets (with respect to the specialization order) of a T0 space are always irreducible, and (4) the topological closure of every irreducible set is again irreducible. In recent years, the usefulness of irreducible sets in domain theory and non-Hausdorff topology has expanded. Notably, Zhao and Ho (2009) developed the core of domain theory directly in the context of T0 spaces by choosing the irreducible sets as the topological substitute for directed sets. Just as the existence of suprema of directed subsets is featured prominently in domain theory (and hence the notion of a dcpo – a poset in which all directed suprema exist), so too is that of irreducible subsets in the topological domain theory developed by Zhao and Ho. The topological counterpart of a dcpo is thus this: A T0 space is said to be strongly complete if the suprema of all irreducible subsets exist. In this paper, we show that the category, scTop+, of strongly complete To spaces forms a reflective subcategory of a certain lluf subcategory, Top+, of To spaces.
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