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A theory of extensions of quasi-algebras to algebras

100%
Słomiński J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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
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The theory of abstract algebras with infinitary operations

100%
Słomiński J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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
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Peano-algebras and quasi-algebras

100%
Słomiński J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction.................................................................................................................... 5 § 1. Fundamental concepts for quasi-algebras..................................................... 5 § 2. Peano-algebras.................................................................................................... 13 § 3. Peano-algebras and free quasi-algebras....................................................... 25 § 4. Theorems concerning free sums of quasi-algebras..................................... 37 § 5. The concepts of validity of equations for quasi-algebras.............................. 39 References.................................................................................................................... 57
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On the determining of the form of congruences in abstract algebras with equationally definable constant elements

60%
Słomiński J.
Fundamenta Mathematicae
|
1959-1960
|
tom 48
|
nr 3
325-341
5
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On extending of models V. Embedding theorems for relational models

37%
Łoś J., Słomiński J., Suszko R.
Fundamenta Mathematicae
|
1959-1960
|
tom 48
|
nr 2
113-121
6
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On mappings between quasi-algebras

30%
Słomiński J.
Colloquium Mathematicum
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1966
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tom 15
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nr 1
25-44
7
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On systems of mappings between models

25%
Słomiński J.
Colloquium Mathematicum
|
1967
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tom 17
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nr 2
247-268
8
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On the extending of models (III)

25%
Słomiński J.
Fundamenta Mathematicae
|
1956
|
tom 43
|
nr 1
69-76
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