Distributive lattices / Raymond Balbes and Philip Dwinger.
Editor: Columbia : University of Missouri Press, c1974Descripción: xiii, 294 p. : il. ; 27 cmISBN: 0826201636Tema(s): Lattices, DistributiveOtra clasificación: 06DxxContents Preface, ix Acknowledgments, xiii I. Preliminaries, [1] Part 1. Foundations 1. Basic definitions and terminology, [1] 2. Partially ordered sets, [3] 3. Ordinals and cardinals, [4] Part 2. Universal Algebra 4. Algebras and homomorphisms, [6] 5. Direct products, [8] 6. Congruence relations, [9] 7. Classes of algebras, [10] 8. The duality principles, [12] 9. Subdirect products, [13] 10. Free algebras, [14] 11. Polynomials and identities, [15] 12. Birkhoff's characterization of equational classes, [17] Part 3. Categories 13. Definition of a category, [20] 14. Special morphisms, [21] 15. Products and coproducts. Limits and colimits, [23] 16. Injectives and projectives, [24] 17. Functors, [25] 18. Reflective subcategories, [27] 19. Categories of algebras, [30] 20. Equational categories, [31] II. Lattices 1. Partially ordered sets (continued), [39] 2. Definition of a lattice, [42] 3. Lattices as algebras and categories, [43] 4. Complete lattices and closure operators, [45] 5. The distributive law, [48] 6. Complements, Boolean algebras, [51] 7. Relatively complemented distributive lattices, [54] 8. 77»e various categories, [56] 9. Ideals and congruence relations, [57] 10. Subdirect product representation for distributive lattices, [63] III. The Prime Ideal Theorem 1. Irreducible elements, [65] 2. The descending chain condition, [66] 3. Prime ideals and maximal ideals, [67] 4. The prime ideal theorem, [70] 5. Representation by sets, [71] 6. Some corollaries of the prime ideal theorem, [72] IV. Topological Representations 1. Stone spaces, [75] 2. Topological representation theory for D01 [79] 3. Topological representation theory for B, [81] V. Extension Properties 1. Subalgebras generated by sets, [85] 2. Extending functions to homomorphisms, [86] 3. Free algebras, [88] 4. Free Boolean extensions, [97] 5. Embedding of relatively complemented distributive lattices in Boolean algebras, [101] 6. Free relatively complemented distributive extensions, [102] 7. Boolean algebras generated by a chain. Countable Boolean algebras, [105] 8. Monomorphisms and epimorphisms, [109] 9. Injectlves in D, D01 and B, [112] 10. Weak projectives in D and D01, [114] VI. Subdirect Products of Chains 1. Introduction, [119] 2. Definitions, some basic theorems, and an example, [120] 3. Basic K-chains, [121] 4. The classes Cn for n ≥ 3. Locally separated n-chains, [122] 5. The class C3, [125] 6. A structure theorem for C3, [127] 7. Maximal ideals and Nachbin's theorem, [129] 8. Subdirect products of infinite chains, [130] VII. Coproducts and Colimits 1. Existence and characterization of coproducts in D, D01 and B,132 2. The coproduct convention, [135] 3. The center of the coproduct in D01, [136] 4. Structure theorems for coproducts of Boolean algebras and chains, [138] 5. Rigid chains, [142] 6. Uniqueness of representation of coproducts of Boolean algebras and chains, [144] 7. The representation space of coproducts in D01 and B,147 8. Colimits, [149] VIII. Pseudocomplemented Distributive Lattices 1. Definitions and examples, [151] 2. Basic properties, [153] 3. The class Bw, [155] 4. Glivenko's theorem and some implications, [156] 5. Subdirectly irreducible algebras in Bw, [159] 6. The equationalsubclasses of Bw, [161] 7. Stone algebras, [164] 8. Injective Stone algebras, [168] 9. Coproducts of Stone algebras. Free Stone algebras, [171] IX. Heyting Algebras 1. Introduction, [173] 2. Definitions, [173] 3. Examples, [177] 4. The class H, [177] 5. Some fundamental theorems, [179] 6. Free Heyting algebras, [182] 7. Finite weakly projective Heyting algebras, [185] X. Post Algebras 1. Introduction, [191] 2. Definitions and basic properties, [192] 3. The category of Post algebras, [195] 4. Representation theory, [197] 5. The partially ordered set ofprime ideals of a Post algebra, [198] 6. Post algebras as equational classes of algebras, [200] 7. Coproducts in &n and free Post algebras, [202] 8. Generalized Post algebras, [203] 9. Stone algebras of order n, [205] XI. De Morgan Algebras and Lukasiewicz Algebras 1. Introduction, [211] 2. De Morgan algebras. Basic properties and representation theory, [212] 3. Equational subclasses of ML, [215] 4. Coproducts of de Morgan algebras. Free de Morgan algebras, [216] 5. Lukasiewicz algebras. Definitions and basic theorems, [218] 6. Post algebras as special cases of Lukasiewicz algebras, [221] 7. A representation theory for Lukasiewicz algebras, [223] 8. 77ze partially ordered set ofprime ideals of a Lukasiewicz algebra, [224] 9. Embedding of Lukasiewicz algebras in Post algebras. Injective objects in [224] 10. Free Lukasiewicz algebras, [226] XU. Complete and a-CompIete Distributive Lattices 1. Introduction. Definitions and some theorems, [228] 2. Normal completion of lattices. The special case of distributive lattices, [233] 3. normal completion of a Boolean algebra, [240] 4. Complete, completely distributive lattices, [243] 5. Representation of distributive, a-complete lattices, [250] Bibliography, [261] Index of Authors and Terms, [284] Index of Symbols, [292]
| Item type | Home library | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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| 06 Ab148 Boolean rings / | 06 An545 Lattice-ordered groups : | 06 An545 Lattice-ordered groups : | 06 B172 Distributive lattices / | 06 B241 Ordre et classification : | 06 B241 Ordre et classification : | 06 B619 Lattice theory / |
Incluye referencias bibliográficas (p. 261-283) e índices.
Contents --
Preface, ix --
Acknowledgments, xiii --
I. Preliminaries, [1] --
Part 1. Foundations --
1. Basic definitions and terminology, [1] --
2. Partially ordered sets, [3] --
3. Ordinals and cardinals, [4] --
Part 2. Universal Algebra --
4. Algebras and homomorphisms, [6] --
5. Direct products, [8] --
6. Congruence relations, [9] --
7. Classes of algebras, [10] --
8. The duality principles, [12] --
9. Subdirect products, [13] --
10. Free algebras, [14] --
11. Polynomials and identities, [15] --
12. Birkhoff's characterization of equational classes, [17] --
Part 3. Categories --
13. Definition of a category, [20] --
14. Special morphisms, [21] --
15. Products and coproducts. Limits and colimits, [23] --
16. Injectives and projectives, [24] --
17. Functors, [25] --
18. Reflective subcategories, [27] --
19. Categories of algebras, [30] --
20. Equational categories, [31] --
II. Lattices --
1. Partially ordered sets (continued), [39] --
2. Definition of a lattice, [42] --
3. Lattices as algebras and categories, [43] --
4. Complete lattices and closure operators, [45] --
5. The distributive law, [48] --
6. Complements, Boolean algebras, [51] --
7. Relatively complemented distributive lattices, [54] --
8. 77»e various categories, [56] --
9. Ideals and congruence relations, [57] --
10. Subdirect product representation for distributive lattices, [63] --
III. The Prime Ideal Theorem --
1. Irreducible elements, [65] --
2. The descending chain condition, [66] --
3. Prime ideals and maximal ideals, [67] --
4. The prime ideal theorem, [70] --
5. Representation by sets, [71] --
6. Some corollaries of the prime ideal theorem, [72] --
IV. Topological Representations --
1. Stone spaces, [75] --
2. Topological representation theory for D01 [79] --
3. Topological representation theory for B, [81] --
V. Extension Properties --
1. Subalgebras generated by sets, [85] --
2. Extending functions to homomorphisms, [86] --
3. Free algebras, [88] --
4. Free Boolean extensions, [97] --
5. Embedding of relatively complemented distributive lattices in Boolean algebras, [101] --
6. Free relatively complemented distributive extensions, [102] --
7. Boolean algebras generated by a chain. Countable Boolean algebras, [105] --
8. Monomorphisms and epimorphisms, [109] --
9. Injectlves in D, D01 and B, [112] --
10. Weak projectives in D and D01, [114] --
VI. Subdirect Products of Chains --
1. Introduction, [119] --
2. Definitions, some basic theorems, and an example, [120] --
3. Basic K-chains, [121] --
4. The classes Cn for n ≥ 3. Locally separated n-chains, [122] --
5. The class C3, [125] --
6. A structure theorem for C3, [127] --
7. Maximal ideals and Nachbin's theorem, [129] --
8. Subdirect products of infinite chains, [130] --
VII. Coproducts and Colimits --
1. Existence and characterization of coproducts in D, D01 and B,132 --
2. The coproduct convention, [135] --
3. The center of the coproduct in D01, [136] --
4. Structure theorems for coproducts of Boolean algebras and chains, [138] --
5. Rigid chains, [142] --
6. Uniqueness of representation of coproducts of Boolean algebras and chains, [144] --
7. The representation space of coproducts in D01 and B,147 --
8. Colimits, [149] --
VIII. Pseudocomplemented Distributive Lattices --
1. Definitions and examples, [151] --
2. Basic properties, [153] --
3. The class Bw, [155] --
4. Glivenko's theorem and some implications, [156] --
5. Subdirectly irreducible algebras in Bw, [159] --
6. The equationalsubclasses of Bw, [161] --
7. Stone algebras, [164] --
8. Injective Stone algebras, [168] --
9. Coproducts of Stone algebras. Free Stone algebras, [171] --
IX. Heyting Algebras --
1. Introduction, [173] --
2. Definitions, [173] --
3. Examples, [177] --
4. The class H, [177] --
5. Some fundamental theorems, [179] --
6. Free Heyting algebras, [182] --
7. Finite weakly projective Heyting algebras, [185] --
X. Post Algebras --
1. Introduction, [191] --
2. Definitions and basic properties, [192] --
3. The category of Post algebras, [195] --
4. Representation theory, [197] --
5. The partially ordered set ofprime ideals of a Post algebra, [198] --
6. Post algebras as equational classes of algebras, [200] --
7. Coproducts in &n and free Post algebras, [202] --
8. Generalized Post algebras, [203] --
9. Stone algebras of order n, [205] --
XI. De Morgan Algebras and Lukasiewicz Algebras --
1. Introduction, [211] --
2. De Morgan algebras. Basic properties and representation theory, [212] --
3. Equational subclasses of ML, [215] --
4. Coproducts of de Morgan algebras. Free de Morgan algebras, [216] --
5. Lukasiewicz algebras. Definitions and basic theorems, [218] --
6. Post algebras as special cases of Lukasiewicz algebras, [221] --
7. A representation theory for Lukasiewicz algebras, [223] --
8. 77ze partially ordered set ofprime ideals of a Lukasiewicz algebra, [224] --
9. Embedding of Lukasiewicz algebras in Post algebras. Injective objects in [224] --
10. Free Lukasiewicz algebras, [226] --
XU. Complete and a-CompIete Distributive Lattices --
1. Introduction. Definitions and some theorems, [228] --
2. Normal completion of lattices. The special case of distributive lattices, [233] --
3. normal completion of a Boolean algebra, [240] --
4. Complete, completely distributive lattices, [243] --
5. Representation of distributive, a-complete lattices, [250] --
Bibliography, [261] --
Index of Authors and Terms, [284] --
Index of Symbols, [292] --
MR, 51 #10185
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