Applied Abstract Algebra

In 1854, George Boole (1815–1864) introduced an important class of algebraic structures in connection with his research in mathematical logic. His goal was to find a mathematical model for human reasoning. In his honor these structures have been called Boolean algebras. They are special types of lattices. It was E. Schröder, who about 1890 considered the lattice concept in today’s sense. At approximately the same time, R. Dedekind developed a similar concept in his work on groups and ideals. Dedekind defined, in modern terminology, modular and distributive lattices, which are types of lattices of importance in applications. The rapid development of lattice theory proper started around 1930, when G. Birkhoff made major contributions to the. theory.

One of the most important applications of lattice theory and also one of the oldest applications of modern algebra is the use of Boolean algebras in modeling and simplifying switching or relay circuits. This application will be described in §7. It should be noted that the algebra of switching circuits is presented here not only because of its importance today but also for historical reasons and because of its elegant mathematical formulation. The same theory will also describe other systems, e.g., plumbing systems, road systems with traffic lights, etc. Then we consider propositional logic and indicate connections to probability theory.

Finite fields give rise to particularly useful and, in our view, beautiful examples of the applicability of algebra. Such applications arise both within mathematics and in other areas; for example, in communication theory, in computing, and in statistics. In this chapter we present the basic properties of finite fields, with special emphasis on polynomials over these fields. The simplest finite field is the field 𝔽₂ consisting of 0 and 1, with binary addition and imultiplication as operations. Many of the results for 𝔽₂ can be extended to more general finite fields.

In many ways, coding theory or the theory of error-correcting codes represents a beautiful example of the applicability of abstract algebra. Applications of codes range from enabling the clear transmission of pictures from distant planets to securing the enjoyment of listening to noise-free CDs. A variety of algebraic concepts can be used to describe codes and their properties, including matrices, polynomials and their roots, linear shift registers, and discrete Fourier transforms. The theory is still relatively young, having started in 1948 with an influential paper by Claude Shannon. This chapter provides the reader with an introduction to the basic concepts of (block) codes, beginning in §16 with general background, §17 deals with properties of linear codes, §18 introduces cyclic codes, and §19 and §20 contain material on special cyclic codes.

The word cryptology stems from the Greek kryptos, “hidden,” and logos, “word.” Cryptology is the science of secure communications. Cryptology comprises cryptography and cryptanalysis, the former deals with methods to ensure the security, secrecy, or authenticity; the latter is concerned with methods of breaking secret messages or forging signals that will be accepted as authentic. In this chapter we will be concentrating mainly on those aspects of cryptography that rely on mathematical, in particular algebraic, techniques and tools.

We now turn to some applications of group theory. The first application makes use of the observation that computing in ℤ can be replaced by computing in ℤ_n, if n is sufficiently large; ℤ_n can be decomposed into a direct product of groups with prime power order, so we can do the computations in parallel in the smaller components. In §25, we look at permutation groups and apply these to combinatorial problems of finding the number of “essentially different” configurations, where configurations are considered as “essentially equal” if the second one can be obtained from the first one, e.g., by a rotation or reflection.

This chapter contains several topics from various areas in which algebra can be applied. The material is mainly selected to give a brief indication of some further applications of algebraic concepts, mainly groups, semigroups, rings, and fields, without being able to go into much depth. Many results are given without proof. We refer the interested reader to the special literature on these topics in the Bibliography.