Jeffrey T. Denniston

Functorial relationships between lattice-valued topology and topological systems

This paper investigates functorial relationships between lattice-valued topology (arising from fuzzy sets and fuzzy logic) and topological systems (arising from topological and localic aspects of domains and finite observational logic in computer science). Two such relationships are embeddings from TopSys into Loc-Top, both having two fold significance: for computer science the significance is that TopSys is not topological over Set × Loc, yet Loc-Top is topological over Set × Loc; hence these embeddings can be used to construct in Loc-Top the unique initial [final] lifts of all forgetful functor structured sources [sinks] in TopSys; and for topology, the significance is that both embeddings generate anti-stratified topological spaces from ordinary topological spaces and spatial locales rewritten as topological systems, thus justifying the current structural axioms of Loc-Top and lattice-valued topology (which include all anti-stratified, non-stratified, and stratified spaces).

Lattice-valued preordered sets as lattice-valued topological systems

This paper provides variable-basis lattice-valued analogues of the well-known results that the construct Prost of preordered sets, firstly, is concretely isomorphic to a full concretely coreflective subcategory of the category Top of topological spaces (which employs the concept of the dual of the specialization preorder), and, secondly, is (non-concretely) isomorphic to a full coreflective subcategory of the category TopSys of topological systems of S. Vickers (which employs the spatialization procedure for topological systems). Dualizing these results, one arrives at the similar properties of quasi-pseudo-metric spaces built over locales.

Topological systems as a framework for institutions

Recently, J.T. Denniston, A. Melton, and S.E. Rodabaugh introduced a lattice-valued analogue of the concept of institution of J.A. Goguen and R.M. Burstall, comparing it, moreover, with the (lattice-valued version of the) notion of topological system of S. Vickers. In this paper, we show that a suitable generalization of topological systems provides a convenient framework for certain kinds of (lattice-valued) institutions.

Formal Contexts, Formal Concept Analysis, and Galois Connections.

Formal concept analysis (FCA) is built on a special type of Galois connections called polarities. We present new results in formal concept analysis and in Galois connections by presenting new Galois connection results and then applying these to formal concept analysis. We also approach FCA from the perspective of collections of formal contexts. Usually, when doing FCA, a formal context is fixed. We are interested in comparing formal contexts and asking what criteria should be used when determining when one formal context is better than another formal context. Interestingly, we address this issue by studying sets of polarities.

Embedding TopSys into a topological category

The purpose of this paper is to make a case for the value of many-valued mathematics, often called fuzzy mathematics. We believe there may be a difference between many-valued mathematics and fuzziness, as used by those who work with fuzzy logic and fuzzy set theory and applications thereof. We think that most, if not all, fuzzy mathematics is many-valued. However, for this paper, the difference between many-valued mathematics and fuzzy mathematics, if a difference exists, is not important. We are, in this paper, content to show that many-valued mathematics can contribute to mathematics. We do understand that for those mathematicians who feel that many-valued mathematics does not have a place in mathematics this paper will not cause them to embrace many-valued mathematics, but we would like them to consider that many-valued mathematics might be able to contribute to mathematics. In this paper, we give an example of a mathematical construction which was created and defined in part to help computer scientists understand and be able to use topological ideas and concepts in their work as computer scientists. Thus, one would think that this construction, called topological systems, would be topological (as defined later). However, it seems that topological systems are clearly not topological. Thus, an interesting question is can topological systems be made topological, or said more mathematically, can topological systems be embedded into something which is topological. We answer this question in the affirmative, and we do it by embedding topological systems into something which is many-valued. It may be the case that someone(s) can some day show that topological systems are topological though this seems unlikely. Or it may be the case that someone(s) can embed topological systems into something which is topological but not many-valued. However, our point is that by using something which is many-valued we have added to mathematics, and thus, we have shown a mathematical use of many-valued mathematics. We should also say that the mathematical results in this paper are not new. We do present some ideas, including the motivation for the “topological” embeddings from topological systems, in new, and we think, illuminating ways, but the mathematical results are not new.

Sierpinski object for affine systems

Abstract Motivated by the concept of Sierpinski object for topological systems of S. Vickers, presented recently by R. Noor and A.K. Srivastava, this paper introduces the Sierpinski object for many-valued topological systems and shows that it has three important properties of the crisp Sierpinski space of general topology.