Applied Abstract Algebra

1 Lattices.- § 1. Properties of Lattices.- §2. Boolean Algebras.- §3. Minimal Forms of Boolean Polynomials.- Notes.- 2 Applications of Lattices.- §1. Switching Circuits.- §2. Propositional Logic.- §3. Further Applications.- Notes.- 3 Finite Fields and Polynomials.- §1. Rings and Fields.- §2. Finite Fields.- §3. Irreducible Polynomials over Finite Fields.- §4. Factorization of Polynomials over Finite Fields.- §5. The Nullspace of a Matrix (Appendix to §4).- Notes.- 4 Coding Theory.- §1. Linear Codes.- §2. Cyclic Codes.- §3. Special Cyclic Codes.- Notes.- 5 Further Applications of Fields and Groups.- §1. Combinatorial Applications.- §2. Algebraic Cryptography.- §3. Linear Recurring Sequences.- §4. Fast Adding.- §5. Polya’s Theory of Enumeration.- Notes.- 6 Automata.- § 1. Semiautomata and Automata.- §2. Description of Automata; Examples.- §3. Semigroups.- §4. Input Sequences.- §5. The Monoid of a (Semi-) Automaton and the (Semi-) Automaton of a Monoid.- §6. Composition and Decomposition.- §7. Minimal Automata.- Notes.- 7 Further Applications of Semigroups.- §1. Formal Languages.- §2. Semigroups in Biology.- §3. Semigroups in Sociology.- 8 Solutions to the Exercises.- 1.- 2.- 3.- 4.- 5.- 6.- 7.- A. Some Fundamental Concepts.- B. Computer Programs.- Author Index.