Algebra Commons^™

Quantum Dimension Polynomials: A Networked-Numbers Game Approach, Nicholas Gaubatz

Honors College Theses

The Networked-Numbers Game--a mathematical "game'' played on a simple graph--is incredibly accessible and yet surprisingly rich in content. The Game is known to contain deep connections to the finite-dimensional simple Lie algebras over the complex numbers. On the other hand, Quantum Dimension Polynomials (QDPs)--enumerative expressions traditionally understood through root systems--corresponding to the above Lie algebras are complicated to derive and often inaccessible to undergraduates. In this thesis, the Networked-Numbers Game is defined and some known properties are presented. Next, the significance of the QDPs as a method to count combinatorially interesting structures is relayed. Ultimately, a novel closed-form expression of …

On The Chromatic Numbers Of Subgroup Lattices, Jacob C. Miles

MSU Graduate Theses

In this thesis we investigate the chromatic number of the Hasse diagram of a subgroup lattice. We combine results of Bollobás and Tůma to show that there exist infnite groups whose subgroup lattices have arbitarily high chromatic numbers. We show that fnite supersolvable groups have bipartite subgroup lattices but that CLT and non-solvable groups may not have bipartite subgroup lattices. Lastly, we give a preliminary argument suggesting that there are an infnite number of non-solvable groups whose subgroup lattices are bipartite.

(R1466) Ideals And Filters On A Lattice In Neutrosophic Setting, Lemnaouar Zedam, Soheyb Milles, Abdelhamid Bennoui

Applications and Applied Mathematics: An International Journal (AAM)

The notions of ideals and filters have studied in many algebraic (crisp) fuzzy structures and used to study their various properties, representations and characterizations. In addition to their theoretical roles, they have used in some areas of applied mathematics. In a recent paper, Arockiarani and Antony Crispin Sweety have generalized and studied these notions with respect to the concept of neutrosophic sets introduced by Smarandache to represent imprecise, incomplete and inconsistent information. In this article, we aim to deepen the study of these important notions on a given lattice in the neutrosophic setting. We show their various properties and characterizations, …

On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece

MSU Graduate Theses

In this paper we discuss the Hamiltonicity of the subgroup lattices of

different classes of groups. We provide sufficient conditions for the

Hamiltonicity of the subgroup lattices of cube-free abelian groups. We also

prove the non-Hamiltonicity of the subgroup lattices of dihedral and

dicyclic groups. We disprove a conjecture on non-abelian p-groups by

producing an infinite family of non-abelian p-groups with Hamiltonian

subgroup lattices. Finally, we provide a list of the Hamiltonicity of the

subgroup lattices of every finite group up to order 35 barring two groups.

Groups Of Divisibility, Seth J. Gerberding

Honors Thesis

In this thesis, we examine a part of abstract algebra known as Groups of Divisibility. We construct these special groups from basic concepts. We begin with partially-ordered sets, then build our way into groups, rings, and even structures akin to rings of polynomials. In particular, we explore how elementary algebra evolves when an ordering is included with the operations. Our results follow the work done by Anderson and Feil, however we include more explicit proofs and constructions. Our primary results include proving that a group of divisibility can be provided with an order to make it a partially-ordered group; that …

Basis Reduction In Lattice Cryptography, Raj Kane

Honors Theses

We develop an understanding of lattices and their use in cryptography. We examine how reducing lattice bases can yield solutions to the Shortest Vector Problem and the Closest Vector Problem.

An Introduction To Boolean Algebras, Amy Schardijn

Electronic Theses, Projects, and Dissertations

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures.

We …

On Lattice Structure Of The Probability Functions On L*, Mashaallah Mashinchi, Ghader Khaledi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the set of all probability functions on L* is studied, where L* is the lattice of bothvalued fuzzy sets or intuitionistic fuzzy sets. It is shown that the set of all probability functions on L* endowed with two appropriate operations has a monoid structure which is also a distributive complete lattice. Also the lattice structure of the set of all probability functions on L* induced by an appropriate function on [0, 1] to itself is studied. Some lattice (dual) isomorphisms are discussed that suggests probabilities on L* could be considered in the framework of theories modeling imprecision.