Algebraic Systems

Paperback (Softcover reprint of the original 1st ed. 1973)
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Product Details

  • ISBN-13: 9783642653766
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 7/31/2012
  • Series: Grundlehren der mathematischen Wissenschaften Series , #192
  • Edition description: Softcover reprint of the original 1st ed. 1973
  • Edition number: 1
  • Pages: 336
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.70 (d)

Table of Contents

I General Concepts.- 1. Relations and Mappings.- 1.1 Sets.- 1.2 Relations.- 1.3 Mappings.- 1.4 Equivalence Relations.- 1.5 Partial and Linear Orderings.- 1.6 Multivalued and Partial Mappings.- 1.7 Cardinal and Ordinal Numbers.- Problems and Complements.- 2. Models and Algebras.- 2.1 n-ary Relations and Functions.- 2.2 Algebraic Systems.- 2.3 Subsystems. Generating Sets.- 2.4 Congruences.- 2.5 Direct Products.- 2 6 Operations on Cardinal and Ordinal Numbers.- Problems and Complements.- II Classical Algebras.- 3. Groupoids and Groups.- 3.1 Groupoids and Semigroups.- 3.2 Quasigroups and Loops.- 3.3 Groups.- Problems and Complements.- 4. Rings and Fields.- 4.1 Rings.- 4.2 Algebraically Closed Fields.- 4.3 Alternative Skew Fields.- 4.4 Linear Algebras.- Problems and Complements.- 5. Lattices (Structures).- 5.1 Lattices.- 5.2 Modular and Distributive Lattices. Boolean Algebras.- III First and Second-Order Languages.- 6. Syntax and Semantics.- 6.1 Terms.- 6.2 Formulas.- 6.3 Second-Order Properties.- 6.4 Elementary Theories and Axiomatic Classes.- Problems and Complements.- 7. Classification of Formulas.- 7.1—-Formulas and—-Formulas.- 7.2 Universall Axiomatic Subclasses.- 7.3—?-Formulas and—?-Formulas.- 7.4 Positive Formulas.- 7.5 Multiplicatively Stable Formulas.- IV Products and Complete Classes.- 8. Filters and Filtered Products.- 8.1 Filters and Ultrafilters.- 8.2 Ultraproducts.- 8.3 Applications of Ultraproducts.- 8.4 Conditionally Filtered Formulas.- 8.5 Cardinality of Ultraproducts.- 8.6 Regular Products.- Problems and Complements.- 9. Indistinguishability and Elementary Embedding.- 9 1 Elementary Embedding.- 9.2 Elementary Subsystems.- 10. Completeness and Model Completeness.- 10.1 Complete Systems of Formulas.- 10.2 Model Completeness.- Problems and Complements.- V Quasivarieties.- 11. General Properties.- 11.1 Characteristic Properties.- 11.2 Defining Relations.- 11 3 Replicas.- Problems and Complements.- 12. Free Systems and Free Compositions.- 12.1 Free Compositions.- 12 2 Independent Elements and Free Systems.- 12.3 Amalgamated Compositions.- Problems and Complements.- VI Varieties.- 13. General Properties.- 13 1 Structural Characteristics.- 13.2 Ranks of Varieties.- 13.3 Varieties of Unoids.- Problems and Complements.- 14. Primitive Closures.- 14 1 Generating Systems.- 14.2 The Lattice of Varieties.- 14.3 Minimal Varieties and Quasivarieties.- Problems and Complements.- Name Index.
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