L-ordered Limit Spaces And The Categorical Properties Of Its Corresponding Category

In this paper, based on a complete residuated lattice L, the author establish-es several subcategories of the category of L-ordered convergence spaces with the tool of ordered filters, e.g.. the categories of L-ordered Kent convergence s-paces, of L-ordered limit spaces and so on. With the theory of adjoint functors, this category and its subcategories compose reflective chain. And these subcat-egories are topological and well-fibre. In addition, the category of L-ordered convergence spaces is Cartesian-closed on the duly lattice. Finally, the author studies several subcategories of the category of L-ordered convergence spaces in symmetric sense, establishes a symmetric condition (OS), and introduces the category of L-ordered convergence spaces in symmetric sense. Moreover, the re-flective relations which exist between different types of categories are established from a categorical point of view.This paper is organized as follow:The first section is preface. The author introduces the background of the theory of L-ordered convergence structures, the origin of the question that is studied and the primary work throughout the paper. The author simply intro-duces some basic concepts and reviews some foundation knowledge about lattice theory and category.In the second section, the author mainly studies the L-ordered convergence spaces. With the tool of the category theory, the author proves that the category of L-ordered convergence spaces is topological, well-fiber, and Cartesian closed on the duly Heyting algebra.In the third section, some critical subcategories of the L-ordered conver- gence spaces are studied. Firstly, with the tool of L-ordered filters, the author puts forward two notions of L-ordered Kent convergence spaces and L-ordered limit spaces. and establishes corresponding categories. Secondly, the reflective relations of L-ordered Kent convergence spaces, L-ordered limit spaces, pre-topological L-ordered convergence spaces, and strong L-topological L-ordered convergence spaces. are established. Finally, the author perfects the theory of L-ordered convergence spaces in asymmetric sense.In the fourth section, the author constructs an L-ordered convergence sys-tem in symmetric sense. From a categorical point of view, the author mainly studies these subcategories and their symmetric subcategories. Moreover, the relations which exist between different types of categories are established.