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A theory of extensions of quasi-algebras to algebras

Seria
Rozprawy Matematyczne tom/nr w serii: 40 wydano: 1964
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Abstrakty
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CONTENTS
INTRODUCTION...................................................................................................................................................................... 3
1. TERMS NOTATION AND LEMMAS.................................................................................................................................. 4
A. Quasi-algebras and algebras....................................................................................................................................................................... 4
B. Subquasi-algebras and sets of generators............................................................................................................................................... 4
C. Homomorphisms of quasi-algebras.......................................................................................................................................................... 5
D. Direct. products quasi-algebras and homomorphisms.......................................................................................................................... 10
E. Congruences of quasi-algebras and homomorphisms.......................................................................................................................... 10
F. Terms and equations...................................................................................................................................................................................... 14
G. Analytical operations defined by terms in quasi-algebras...................................................................................................................... 15
H. Tensor product of quasi-algebras................................................................................................................................................................ 16
I. The general transposition law of operations and algebras of homomorphisms and of bilinears of quasi-algebras.................. 17
J. The form of congruences determined by terms......................................................................................................................................... 20
§ 2. ON EXTENDING QUASI-ALGEBRAS TO ALGEBRAS............................................................................................................................ 21
§ 3. ON THE COMMON EXTENSION OF QUASI-ALGEBRAS TO ALGEBRAS.......................................................................................... 30
§ 4. A THEORY OF EXTENSIONS OF MAP-SYSTEMS IN EQUATIONALLY DEFINABLE CLASSES OF ALGEBRAS........................ 33
A. Map-systems in equationally definable clauses of algebras.................................................................................................................. 33
B. Quasi-ideals and ideals in A-map-systems...................................................................................................................... 35
C. On dividing map-systems by ideals............................................................................................................................................................ 41
D. Operator-systems in equationally definable classes of algebras......................................................................................................... 43
E. The equivalence of the notions of quasi-ideals and ideals for A-operator-systems over R..................................... 45
§ 5. ALGEBRAS WITH DIFFERENTIAL OPERATORS................................................................................................................................... 52
A. Algebras with differential operators over commutative, algebras R...................................................................................................... 53
B. Algebras with differential operators in the classes in which the general transposition law of operations holds........................ 54
References............................................................................................................................................................................................................ 61
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 40
Liczba stron
63
Liczba rozdzia³ów
Opis fizyczny
Rozprawy Matematyczne, Tom XL
Daty
wydano
1964
Twórcy
  • Department of Mathematics, N. Copernicus University, Toruń
  • Katedra Matematyki, Uniwersytet M. Kopernika, Toruń
Bibliografia
  • |1| G. Birkhoff, Subdirect unions universal algebra, Bulletin of the Amer. Mathem. Soc. 50 (1944), p.764
  • [2] I. M. Gelfand, G. E. Šilov, Quelques applications de la théorie des fonctions qénéralisées, J. Math. Pures Appl. 35 (1956), p. 383.
  • [3] А. Мальцев, О включении ассоциативных систем в группы, Мат. сб. (Rec. Math) 6 (48) (1939), pp. 331-336.
  • [4] А. Мальцев, О включении ассоциативных систем в группы, Мат. сб. (Rec. Math) 8 (60) (1940), pp. 251-264.
  • [5] J. G. Mikusiński, Sur les fondaments du calcul opératoire, Studia Math. 11 (1950), pp. 41-70.
  • [6] A. Mostowski, Logika matematyczna, Warszawa-Wrocław 1948.
  • [7] В. Птак, О включении семигрупп, Чехословацкий мат. журнал 2 (77) (1952), pp. 247-271.
  • |8] L. Schwartz, Théorie des distributions I, Paris 1950.
  • [9] R. Sikorski, A definition of the notion of distribution, Bull. Acad. Polon. Sci., Cl. III, 2 (1954). pp. 209-211.
  • [10] R. Sikorski, Products of abstract algebras, Fund. Math. 39 (1952), pp. 211-228.
  • [11] R. Sikorski, Funkcje rzeczywiste, tom I, Warszawa 1958.
  • [12] J. Słomiński, The theory of abstract algebras with infinitary operations, Rozprawy Matem. 18 (1959), pp. 1-67.
  • [13] J. Słomiński, On the determining of the form of congruences in abstract algebras with equationally definable constant elements, Fund. Math. 48 (1960), pp. 325-341.
  • [14] J. Słomiński, On the embedding of abstract quasi algebras into equationally definable classes of abstract algebras. Bull. Acad. Polon. Sci., Cl. III, 8 (1960), pp. 11-17.
  • [15] J. Słomiński, On the common embedding of abstract quasi algebras into equationally definable classes of abstract algebras, Bull. Acad. Polon. Sci., Cl. III, 8 (1960), pp. 277-282.
  • [16] J. Słomiński, A theory of extensions of map-systems into equationally definable classes of abstract algebras, Bull. Acad. Polon. Sci., Cl. III, 12 (1962), pp. 621-626.
  • [17] J. Słomiński, On the solving of systems of equations over quasi algebras and algebras, Bull. Acad. Polon. Sci., Cl. III, 12 (1962), pp. 627-635.
  • [18] J. Słomiński, On the extending of models III. Extensions in equationally definable classes of algebras. Fund. Math. 43 (1956), pp. 69-76.
  • [19] J. Łoś and R. Suszko, On the extending of models V. Embedding theorems for relational models, Fund. Math. 48 (1960), pp. 113-121.
  • [20] W. Słowikowski, On the theory of operator systems I and II. Bull. Acad. Polon. Sci., Cl. III, 3 (1955), pp. 137- 142, and 6 (1958), pp. 383-386.
  • [21] W. Słowikowski, On regular embeddings of a group with differential operators. Bull. Acad. Polon. Sci., Cl. III, 7 (1959), pp. 119-124.
  • [22] W. Słowikowski, Categories of regular embeddings of groups with differential operators. Bull. Acad. Polon. Sci., Cl. III, 7 (1959), pp. 125-130.
  • [23] W. Słowikowski, A theory of extensions of map-systems I, Fund. Math. 46 (1959), pp. 243-275.
  • [24] S. L. Sobolev, Sur quelques évaluations concernant les families des fonctions ayant des dérivées à carré intégrable, C.R. Acad. Sci. URSS 1 (1936), pp. 279-282.
  • [25] S. L. Sobolev, Sur un théorème de l'analyse fonctionelle, C. R. Acad. Sci. URSS 20 (1938), pp. 5-9.
  • [26] A. Tarski, Contributions to the theory of models. Proceedings Kon. Nederl. Akad. v. Wetenschappen A 57 (1954), pp. 572-588.
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