Philosophiae Doctor - PhD (Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10566/19489

Browse

Recent Submissions

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item type: Item ,
    Topogenous orders and their applications on lattices
    (University of the Western Cape, 2024) Iragi, Bakulikira Claude
    In his influential book, [Cs´a63] ´A. Cs´asz´ar developed the well-known theory of syntopogenous structures on a set. His intention was to create a comprehensive framework that simultaneously encompasses the study of topological, proximal, and uniform structures. In the same monograph, he demonstrated independently, along with Pervin [Per62], that every topological space possesses a compatible quasi-uniformity. A similar observation was noted for a uniform space, provided the topological space is completely regular. On the other hand, Herrlich in [Her74a] introduced the concept of “nearness” with the aim of unifying various topological structures. This Ph.D. thesis aims to investigate topogenous orders and their generalizations, such as quasi-uniformities, syntopogenous structures, on complete lattices which extend and generalize existing literature in this field. We explore the study of quasi-uniformities through the lens of syntopogenous structures, and establish a Galois connection between these two constructs. Furthermore, we provide conditions under which certain Cs´asz´ar structures are order isomorphic to quasi-uniformities on a complete lattice.As Cs´asz´ar structures are deeply rooted in pointfree topology, our research naturally extends into the realm of frames. We establish a correspondence between pre-nearness and Cs´asz´ar structures. In line with these ideas, we also delve into the relationship between pre-uniformities and entourage quasi-uniformities in the context of frames.
  • Thumbnail Image
    Item type: Item ,
    The Smarandache vertices of the annihilation graphs of commutator posets and lattices with respect to an element and an ideal
    (University of the Western Cape, 2024) Siame Happy
    A vertex a in a simple graph G is said to be a Smarandache vertex (or S-vertex for short) provided that there exist three distinct vertices x, y, and b (all different from a) in G such that x—a, a—b, and b—y are edges in G, but there is no edge between x and y. In this interdisciplinary subject, we investigate the interplay between the algebraic properties of the commutator posets and lattices and their associated annihilation graphs with respect to an element [resp. an ideal] using the notion of the Smarandache vertices. Actually, AGz(L) (the annihilation graph of the commutator poset [lattice] L with respect to an element z ∈ L) [resp. AGI(L) (the annihilation graph of the commutator poset [lattice] L with respect to an ideal I ⊆ L, where AGI(L) is an extension of AGz(L) from an element to an ideal of L)] is a widely generalized context for the study of the zerodivisor type (annihilating-ideal) graphs, where the vertices of the graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set [lattice] (imitating the lattice of ideals of a ring), equipped with a commutative (not necessarily associative) binary operation (imitating the product of ideals of a ring). We discuss when AGz(L) [resp. AGI(L)] is a complete r-partite graph together with some of its other graph-theoretic properties. We investigate the interplay between some (order-) lattice-theoretic properties of L and graphtheoretic properties of its associated graph AGz(L) [resp. AGI(L)]. We provide some examples to show that some conditions are not superfluous assumptions. We prove and show by an example that the class of lower sets of a commutator poset L is properly contained in the class of m-ideals of L [i.e. multiplicatively absorptive ideals (sets) of L that are defined by commutator operation].