<p> This thesis gives characterizations for all the injective and projective objects in the categories of compact, totally disconnected, distributive topological lattices, and compact, distributive topological lattices. It also contains known results concerning distributive lattices and Hausdorff topological spaces.</p> / Thesis / Master of Science (MSc)

<p> This thesis deals with the construction of Free Lattices on various sets of generators, ranging from a set of unordered generators to a countable set of chains. It also considers sublattices of the structures presented.</p> / Thesis / Master of Science (MSc)

Linear optimization aims at maximizing, or minimizing, a linear objective function over a feasible region defined by a finite number of linear constrains. For several well-studied problems such as maxcut, all the vertices of the feasible region are integral, that is, with integer-valued coordinates. The diameter of the feasible region is the diameter of the edge-graph formed by the vertices and the edges of the feasible region. This diameter is a lower bound for the worst-case behaviour for the widely used pivot-based simplex methods to solve linear optimization instances. A lattice (d,k)-polytope is the convex hull of a set of points whose coordinates are integer ranging from 0 to k. This dissertation provides new insights into the determination of the largest possible diameter δ(d,k) over all possible lattice (d,k)-polytopes. An enhanced algorithm to determine δ(d,k) is introduced to compute previously intractable instances. The key improvements are achieved by introducing a novel branching that exploits convexity and combinatorial properties, and by using a linear optimization formulation to significantly reduce the search space. In particular we determine the value for δ(3,7). / Thesis / Doctor of Philosophy (PhD)

No description available.

The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(Σ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and Lewis.

The thesis analyzes some aspects of the theory of distributive lattices, particularly two representation theorems: Birkhoff's representation theorem for finite distributive lattices and Stone's representation theorem for infinite distributive lattices.

No description available.

A lattice L is said to be dually semimodular if for all elements a and b in L, a ∨ b covers b implies that a covers a ∧ b. L is consistent if for every join-irreducible j and every element x in L, the element x ∨ j is a join-irreducible in the upper interval [x,l]. In this paper, finite dually semimodular consistent lattices are investigated. Examples of these lattices are the lattices of subnormal subgroups of a finite group. In 1954, R. P. Dilworth proved that in a finite modular lattice, the number of elements covering exactly k elements is equal to the number of elements covered by exactly k elements. Here, it is established that if a finite dually semimodular consistent lattice has the same number of join-irreducibles as meet-irreducibles, then it is modular. Hence, a converse of Dilworth's theorem, in the case when k equals 1, is obtained for finite dually semimodular consistent lattices. Several combinatorial results are shown for finite consistent lattices similar to those already established for finite geometric lattices. The reach of an element x in a lattice L is the difference between the rank of x*, the join of x and all the elements covering x, and the rank of x; the maximum reach of all elements in L is the reach of L. Sharp lower bounds for the total number of elements and the number of elements of a given reach in a semimodular consistent lattice given the rank, the reach, and the number of join-irreducibles are found. Extremal lattices attaining these bounds are described. Similar results are then obtained for finite dually semimodular consistent lattices.

No description available.

Thesis--Ph. D., University of Hong Kong, 1983.