John Frith

Comprehension Tests in Mathematics.

An alternative way for testing a students understanding of theory in a tertiary mathematics course is presented. Two sample questions are provided and advantages and disadvantages of the method are discussed. We argue that the method is an acceptable and flexible means of testing students, and can be adapted to be used in other contexts as well.

Quasi-nearnesses on biframes and their completions

Abstract We introduce quasi-nearness biframes. They provide a generalization of both nearness frames [5] and quasi-uniform biframes [12]. Quasi-nearness biframes are regular; they are normal if and only if the standard binary quasi-nearness is in fact a quasi-uniformity. Further, we introduce a notion of completeness, and provide a construction of a completion in which we make use of the new notion of a part-preserving right adjoint of a biframe map.

Quasi-nearness biframes: unique completions and related covering properties

Abstract Quasi-completeness was considered in [16], where a quasi-completion was constructed for any quasi-nearness biframe. In this paper, we compare the familiar notions of compactness and total boundedness with quasi-completeness. We investigate the uniqueness of quasi-completions, using the full and balanced biframes of [17]. In the case of quasi-uniform biframes it makes sense to compare this quasi-completion with the completion of [13] and [14] – surprisingly, they need not coincide.

Completion of Quasi-Uniform Frames

This paper presents the completion of a quasi-uniform frame. When the procedure outlined herein is applied to a uniform frame, the completion yields the (unique) completion of the uniform frame.

Coverages Give Free Constructions for Partial Frames

Defining objects using generators and relations has seen substantial application in the theory of frames. It is the aim of this paper to establish such a technique for partial frames, thus making it available in a variety of contexts. A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets in question are specified by means of a so-called selection function. The theory is general enough to include, as examples, bounded distributive lattices, σ-frames, κ-frames and indeed frames, but a small collection of elementary axioms suffices to describe the selection functions and thus the designated subsets. In this paper we are concerned with establishing techniques for constructing objects given certain generators and the relations that they should satisfy. Our method involves embedding the generators in an appropriate meet-semilattice, moving to the free partial frame over that meet-semilattice, and then using the relations to form a quotient with the required joins. We use a modification of Johnstone’s coverages on meet-semilattices [12] to construct partial frames freely generated by sites. We conclude with a number of applications, including the construction of coproducts for partial frames and a general method for freely adjoining complements.

The Stone-Čech compactification of a partial frame via ideals and cozero elements

Abstract A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames. We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-Č ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames. A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.

Asymmetric filter convergence and completeness

Abstract Completeness for metric spaces is traditionally presented in terms of convergence of Cauchy sequences, and for uniform spaces in terms of Cauchy filters. Somewhat more abstractly, a uniform space is complete if and only if it is closed in every uniform space in which it is embedded, and so isomorphic to any space in which it is densely embedded. This is the approach to completeness used in the point-free setting, that is, for uniform and nearness frames: a nearness frame is said to be complete if every strict surjection onto it is an isomorphism. Quasi-uniformities and quasi-nearnesses on biframes provide appropriate structures with which to investigate uniform and nearness ideas in the asymmetric context. In [9] a notion of completeness (called “quasi-completeness”) was presented for quasi-nearness biframes in terms of suitable strict surjections being isomorphisms, and a quasi-completion was constructed for any quasi-nearness biframe. In this paper we show that quasi-completeness can indeed be viewed in terms of the convergence of certain filters, namely, the regular Cauchy bifilters. We use the notion of a T -valued bifilter, which generalizes the characteristic function of a filter. An important tool is an appropriate composition for such bifilters. We show that the right adjoint of the quasi-completion is the universal regular Cauchy bifilter and use it to prove this characterization of quasi-completeness. We also construct the so-called Cauchy filter quotient for a biframe using a quotient of the downset biframe that involves only the Cauchy, and not the regularity, condition. Like the quasi-completion, this provides a universal Cauchy bifilter; unlike the quasi-completion, this construction is functorial.

Meet-Semilattice Congruences on a Frame

The congruence lattice of a frame has long been an object of considerable interest, not least because it turns out to be a frame itself. Perhaps more surprisingly congruence lattices of, for instance,

Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

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