Applied Abstract Algebra - Softcover

Inhaltsangabe

1 Lattices.- § 1. Properties of Lattices.- A. Lattice Definitions.- B. Modular and Distributive Lattices.- §2. Boolean Algebras.- A. Basic Properties.- B. Boolean Polynomials, Ideals.- §3. Minimal Forms of Boolean Polynomials.- Notes.- 2 Applications of Lattices.- §1. Switching Circuits.- A. Basic Definitions.- B. Applications of Switching Circuits.- §2. Propositional Logic.- §3. Further Applications.- Notes.- 3 Finite Fields and Polynomials.- §1. Rings and Fields.- A. Rings, Ideals, Homomorphisms.- B. Polynomials.- C. Fields.- D. Algebraic Extensions.- §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.- A. Hadamard Matrices.- B. Balanced Incomplete Block Designs.- C. Steiner Systems, Difference Sets and Latin Squares.- §2. Algebraic Cryptography.- A. Single Key (Symmetric) Cryptosystems.- B. Public-Key Cryptosystems.- §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.- A. Fundamental Concepts.- B. Subsemigroups, Homomorphisms.- C. Free Semigroups.- §4. Input Sequences.- §5. The Monoid of a (Semi-) Automaton and the (Semi-) Automaton of a Monoid.- §6. Composition and Decomposition.- A. Elementary Constructions of Automata.- B. Cascades.- §7. Minimal Automata.- Notes.- 7 Further Applications of Semigroups.- §1. Formal Languages.- A. Approach via Grammar.- B. Approach via Automata and Semigroups.- C. Connections Between the Different Approaches.- §2. Semigroups in Biology.- §3. Semigroups in Sociology.- A. Kinship Systems.- B. Social Networks 400 Notes.- 8 Solutions to the Exercises.- 1.- 2.- 3.- 4.- 5.- 6.- 7.- A. Some Fundamental Concepts.- B. Computer Programs.- Author Index.

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