Introduction to Lattice Algebra

With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks

Gerhard X. Ritter, Gonzalo Urcid

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000412604
• Edition: 1
• Topic: Matemáticas
• Subtopic: Álgebra abstracta

Lattice theory extends into virtually every branch of mathematics, ranging from measure theory and convex geometry to probability theory and topology. A more recent development has been the rapid escalation of employing lattice theory for various applications outside the domain of pure mathematics. These applications range from electronic communication theory and gate array devices that implement Boolean logic to artificial intelligence and computer science in general.

Introduction to Lattice Algebra: With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks lays emphasis on two subjects, the first being lattice algebra and the second the practical applications of that algebra. This textbook is intended to be used for a special topics course in artificial intelligence with a focus on pattern recognition, multispectral image analysis, and biomimetic artificial neural networks. The book is self-contained and – depending on the student's major – can be used for a senior undergraduate level or first-year graduate level course. The book is also an ideal self-study guide for researchers and professionals in the above-mentioned disciplines.

Features

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• Filled with instructive examples and exercises to help build understanding
• Suitable for researchers, professionals and students, both in mathematics and computer science
• Contains numerous exercises.

CHAPTER 1

Elements of Algebra

DOI: 10.1201/9781003154242-1

To gain a thorough understanding of the subject of this book, the reader will have to learn the language used in it. We will try to keep the number of technical terms as small as possible, but there is a certain minimum vocabulary that is essential. Much of the language used is taken from the theory of abstract algebra, geometry, and topology, subjects which are pertinent but not the focus of this book. These subjects are, indeed, independent branches of mathematics. The aim of this chapter is to present the mathematical notation as well as pertinent facts from algebra that are used throughout this study of lattice algebra and its applications in various areas of artificial intelligence.

The initial focus is on the foundation of abstract algebra, namely operations on sets and the concepts of groups, rings, and fields. These concepts play a major role in lattice algebra. The chapter concludes with a brief discourse of linear algebra with emphasis on vector spaces. Although the reader of this manuscript is most likely very familiar with linear algebra, the addition of vector spaces will provide for a quick comparison with the concept of vector spaces based on lattice algebra (Chapter 4).

1.1Sets, Functions, and Notation

Throu...