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The theory of abstract algebras with infinitary operations

Seria
Rozprawy Matematyczne tom/nr w serii: 18 wydano: 1959
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CONTENTS
Introduction............................................................................................................... 3
CHAPTER I. Operations
§ 1. Definition of operation........................................................................................ 6
§ 2. Homomorphisms of operations.......................................................................... 6
§ 3. Congruences of operations................................................................................ 8
§ 4. Direct product of operations............................................................................... 7
CHAPTER II. Abstract algebras
§ 1. Definition of algebra......................................................................................... 10
§ 2. Subalgebras and sets of generators................................................................ 10
§ 3. Borel-classes of elements in algebras............................................................. 12
§ 4. Powers of subalgebras.................................................................................... 13
§ 5. Homomorphisms and congruences of algebras.............................................. 15
§ 6. Direct product of algebras................................................................................ 18
§ 7. $\[\mathfrak{A}\]$-free algebras....................................................................... 20
CHAPTER III. Equationally definable classes of algebras
§ 1. Absolutely free algebra..................................................................................... 21
§ 2. Terms and equations........................................................................................ 25
§ 3. Validity of an equation...................................................................................... 27
§ 4. Validity in subalgebras..................................................................................... 33
§ 5. Validity and homomorphisms........................................................................... 34
§ 6. Validity in direct, products of algebras.............................................................. 35
§ 7. Definition of an equationally definable class of algebras.................................. 36
§ 8. Free algebras in equationally definable classes of algebras............................ 37
§ 9. The characterizations of equationally definable classes of algebras............... 40
APPENDIX TO CHAPTER III. Functionally free algebras....................................... 46
CHAPTER IV. Gödel’s theorem for O-systems
§ 1. O-formulae....................................................................................................... 49
§ 2. The operations of consequence....................................................................... 50
§ 3. Validity.............................................................................................................. 54
§ 4. Lindenbaum model........................................................................................... 58
§ 5. Gödel’s theorem............................................................................................... 59
References.............................................................................................................. 66
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 18
Liczba stron
67
Liczba rozdzia³ów
Opis fizyczny
Rozprawy Matematyczne, Tom XVIII
Daty
wydano
1959
Twórcy
  • Uniwersytet Mikołaja Kopernika w Toruniu
Bibliografia
  • [1] G. Вirkhoff, On the combination of subalgebras, Proc. Cambridge Phil. Soc. 29 (1933), p. 441-404.
  • [2] G. Вirkhoff, On the structure of abstract algebras, Proc. Cambridge, Phil. Soc. 31 (1935), p. 433-454.
  • [3] G. Вirkhoff, Subdirect unions in, universal algebras, Bull. Amer. Math. Soc. 50 (1944), p. 764-768.
  • [4] C. C. Chang, Some general theorems on direct products and their applications in the theory of models, Proc. Kon. Ned. Akad. v. Wetenschappen A 57 (1954), p. 592-598.
  • [5] J. Łoś, The algebraic treatment of the methodology of elementary deductive systems, Studia Logica 2 (1955), p. 151-212.
  • [6] J. Łoś, Quelques remarques, théorèmes et problèmes sur classes définissables d'algèbres, Mathematical Interpretation of Formal Systems, Studies in Logic, Amsterdam 1955, p. 98-113.
  • [7] W. Sierpiński, Hypothèse du continu, Warszawa-Lwów 1934.
  • [8] R. Sikorski, Products of abstract algebras, Fund. Math. 39 (1952), p. 212-228.
  • [9] J. Słomiński, Theory of models with infinitary operations and relations, Bull. Acad. Polon. Sci., Cl. III, 6 (1958), p. 449-456.
  • [10] A. Tarski, A remark on functionally free algebras. Annals of Math. 47 (1946), p. 163-165.
  • [11] A. Tarski, Contributions to the theory of models I, II, III, Indag. Math. 16 (1954), p. 572-588, 582-588. and 17 (1955), p. 56-64.
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