Dualities for structures of applied logics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
College Publications
[2015]
|
| Series: | Studies in logic
56 : Mathematical logic and foundations |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xii, 381 Seiten 24 cm |
| ISBN: | 9781848901810 |
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Record in the Search Index
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| adam_text | Contents 1 Preliminaries 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1 Introduction...................................................................................... Set-theoretical notations................................................................ Binary relations................................................................................ Functions......................................................................................... Ordered sets...................................................................................... Lattices............................................................................................ Galois connections.......................................................................... Operators determined by binary relations.................................. Frames............................................................................................... Correspondence Theory................................................................ Topological spaces.......................................................................... Categories......................................................................................... Coalgebras ...................................................................................... 25 2 Foundations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 1 1 2 4 5 7 13 14 17 17 19 21 22 Introduction...................................................................................... Discrete duality and duality via truth ........................................ Discrete duality for
Boolean algebras........................................... Duality via truth for Boolean algebras........................................ Discrete duality for bounded distributive lattices..................... Discrete duality for bounded distributive lattice homomorphisms Duality via truth for bounded distributivelattices ................... Discrete representation for general lattices.................................. Duality via truth for general lattices........................................... 25 26 29 31 32 35 37 39 43 I Boolean algebraswithoperators 47 3 Boolean algebras with modal operators 49 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Introduction....................................................................................... Possibility Boolean algebras........................................................... Sufficiency Boolean algebras........................................................... Context algebras in formal concept analysis............................... Boolean algebras with possibility and sufficiency operators ... Mixed algebras................................................................................ An axiomatic extension of mixed algebras.................................. ix 49 50 55 58 61 62 67
x CONTENTS Preference algebras......................................................................... Event algebras ................................................................................ 68 70 4 Boolean algebras with information operators 4.1 Introduction...................................................................................... 4.2 Information systems....................................................................... 4.3 Approximation operations............................................................. 4.4 Knowledge operator ....................................................................... 4.5 Boolean algebras with a knowledge operator.............................. 4.6 Diversity algebras............................................................................. 4.7 Algebras of weak similarity........................................................... 4.8 Algebras of strong right orthogonality........................................ 75 75 76 78 79 82 85 86 88 5 Boolean algebras with relational operators 5.1 Introduction....................................................................................... 5.2 Relation algebras............................................................................. 5.3 Cylindric algebras of finite dimension........................................... 91 91 92 97 3.8 3.9 6 Multirelational structures 103 6.1 Introduction....................................................................................... 103 6.2
Multirelations.................................................................................... 104 6.3 Boolean algebras with a monotone operator............................... 108 6.4 Boolean algebras with an antifone operator............................... 110 7 Boolean algebras with relations 113 7.1 Introduction....................................................................................... 113 7.2 Proximity algebras........................................................................... 114 7.3 Axiomatic extensions of proximity algebras............................... 116 7.4 Boolean contact algebras.................................................................. 118 7.5 Boolean contact algebras represented withclans........................... 120 7.6 Syllogistic V-algebras........................................................................ 126 7.7 V3-algebras....................................................................................... 129 II Distributive lattices with operators 135 8 Bounded distributive lattices with modal operators 137 8.1 Introduction....................................................................................... 137 8.2 Possibility distributive lattices....................................................... 138 8.3 Necessity distributive lattices....................................................... 140 8.4 Sufficiency distributive lattices.................................................... 142 8.5 Dual sufficiency distributive lattices ........................................... 144 8.6 Positive modal
algebras................................................................. 146 8.7 Axiomatic extensions of positive modal algebras........................ 149 8.8 Negative modal algebras................................................................. 152 8.9 Galois algebras................................................................................ 155 8.10 Dual Galois algebras....................................................................... 157 8.11 Residuated algebras ....................................................................... 158
CONTENTS 9 Distributive lattices with negations 161 Introduction....................................................................................... 161 Relatively pseudocomplemented lattices...................................... 162 RP-lattices with seminegation........................................................ 164 RP-lattices with contrapositional negation.................................. 166 RP-lattices with minimal negation............................................... 168 Lattices with regular negation........................................................ 170 RP-lattices with pre-De Morgan negation.................................. 172 Lattices with De Morgan negation............................................... 176 De Morgan lattices with possibility andsufficiency operators . 179 Pseudocomplemented algebras........................................................ 181 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 10 Stone algebras 10.1 10.2 10.3 10.4 Introduction....................................................................................... Double Stone algebras..................................................................... Regular double Stone algebras........................................................ Rough relation algebras................................................................. 11 Heyting algebras 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Introduction....................................................................................... Heyting algebras..............................................................................
Heyting algebras with modal operators......................................... Axiomatic extensions of HM-algebras............................................ Frontal Heyting algebras................................................................. Symmetric Heyting algebras........................................................... Heyting-Brouwer algebras.............................................................. Apartness algebras........................................................................... 12 Distributive residuated lattices 12.1 Introduction....................................................................................... 12.2 Distributive residuated lattices..................................................... 12.3 Bounded integral residuated lattices............................................ 12.4 Commutative residuated lattices .................................................. 12.5 MTL-algebras.................................................................................... 12.6 Axiomatic extensions of MTL-algebras........................................ 12.7 SMTL-algebras................................................................................. 12.8 IMTL-algebras................................................................................. 12.9 CMTL-algebras................................................................................. 12.10 n-potent MTL-algebras ..................................................... 12.112 -potent BL-algebras............................................................ III xi General lattices
with operators 13 General lattices with modal operators 13.1 Introduction....................................................................................... 13.2 Possibility lattices ........................................................................... 13.3 Axiomatic extensions of possibility lattices.................................. 185 185 186 191 192 201 201 202 204 209 215 217 222 223 229 229 230 234 235 236 238 239 241 242 243 245 249 251 251 252 257
xii CONTENTS 13.4 13.5 13.6 13.7 13.8 Necessity lattices............................................................................... Axiomatic extensions of necessity lattices................................... Lattices with possibility and necessity operators......................... Sufficiency lattices............................................................................ Dual sufficiency lattices.................................................................. 14 General lattices with negations 14.1 14.2 14.3 14.4 14.5 Introduction........................................................................................ General lattices with De Morgan negation................................. General lattices with a weak pseudocomplement........................ General lattices with an orthonegation........................................ Pseudocomplemented general lattices............................................ 15 Residuated and doubly residuated general lattices 15.1 Introduction........................................................................................ 15.2 Residuated general lattices............................................................... 15.3 Doubly residuated general lattices ............................................... 16 General lattices with relational operators 16.1 16.2 16.3 16.4 Introduction........................................................................................ Residuated general lattices with involution................................... Doubly residuated general lattices with involution
................... Lattice-based relation algebras ..................................................... 17 Algebras of substructural logics 17.1 17.2 17.3 17.4 17.5 17.6 Introduction....................................................................................... FL֊algebras....................................................................................... FLe-algebras .................................................................................... FLc-algebras .................................................................................... FLw-algebras.................................................................................... An axiomatic extension of FLew-algebras.................................. 18 Beyond Urquhart representation 18.1 Introduction....................................................................................... 18.2 Exhaustive representation of lattices........................................... 18.3 Non-disjoint representation of lattices ........................................ 18.4 Non-exhaustive representation of lattices..................................... 18.5 Representation of lattices with two-sorted frames...................... 18.6 Dedekind-MacNeille representations ........................................... 18.7 Canonical Extensions....................................................................... 18.8 Dedekind-MacNeille-Wille representation of lattices................... 18.9 Ortholattices.................................................................................... 18.10 Representation of
ortholattices with two-sorted frames............ 258 260 261 265 269 273 273 273 276 279 281 283 283 283 295 299 299 299 304 306 309 309 310 311 312 313 314 317 317 318 321 324 326 330 331 333 335 339 Bibliography 343 Index 361
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| adam_txt |
Contents 1 Preliminaries 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1 Introduction. Set-theoretical notations. Binary relations. Functions. Ordered sets. Lattices. Galois connections. Operators determined by binary relations. Frames. Correspondence Theory. Topological spaces. Categories. Coalgebras . 25 2 Foundations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 1 1 2 4 5 7 13 14 17 17 19 21 22 Introduction. Discrete duality and duality via truth . Discrete duality for
Boolean algebras. Duality via truth for Boolean algebras. Discrete duality for bounded distributive lattices. Discrete duality for bounded distributive lattice homomorphisms Duality via truth for bounded distributivelattices . Discrete representation for general lattices. Duality via truth for general lattices. 25 26 29 31 32 35 37 39 43 I Boolean algebraswithoperators 47 3 Boolean algebras with modal operators 49 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Introduction. Possibility Boolean algebras. Sufficiency Boolean algebras. Context algebras in formal concept analysis. Boolean algebras with possibility and sufficiency operators . Mixed algebras. An axiomatic extension of mixed algebras. ix 49 50 55 58 61 62 67
x CONTENTS Preference algebras. Event algebras . 68 70 4 Boolean algebras with information operators 4.1 Introduction. 4.2 Information systems. 4.3 Approximation operations. 4.4 Knowledge operator . 4.5 Boolean algebras with a knowledge operator. 4.6 Diversity algebras. 4.7 Algebras of weak similarity. 4.8 Algebras of strong right orthogonality. 75 75 76 78 79 82 85 86 88 5 Boolean algebras with relational operators 5.1 Introduction. 5.2 Relation algebras. 5.3 Cylindric algebras of finite dimension. 91 91 92 97 3.8 3.9 6 Multirelational structures 103 6.1 Introduction. 103 6.2
Multirelations. 104 6.3 Boolean algebras with a monotone operator. 108 6.4 Boolean algebras with an antifone operator. 110 7 Boolean algebras with relations 113 7.1 Introduction. 113 7.2 Proximity algebras. 114 7.3 Axiomatic extensions of proximity algebras. 116 7.4 Boolean contact algebras. 118 7.5 Boolean contact algebras represented withclans. 120 7.6 Syllogistic V-algebras. 126 7.7 V3-algebras. 129 II Distributive lattices with operators 135 8 Bounded distributive lattices with modal operators 137 8.1 Introduction. 137 8.2 Possibility distributive lattices. 138 8.3 Necessity distributive lattices. 140 8.4 Sufficiency distributive lattices. 142 8.5 Dual sufficiency distributive lattices . 144 8.6 Positive modal
algebras. 146 8.7 Axiomatic extensions of positive modal algebras. 149 8.8 Negative modal algebras. 152 8.9 Galois algebras. 155 8.10 Dual Galois algebras. 157 8.11 Residuated algebras . 158
CONTENTS 9 Distributive lattices with negations 161 Introduction. 161 Relatively pseudocomplemented lattices. 162 RP-lattices with seminegation. 164 RP-lattices with contrapositional negation. 166 RP-lattices with minimal negation. 168 Lattices with regular negation. 170 RP-lattices with pre-De Morgan negation. 172 Lattices with De Morgan negation. 176 De Morgan lattices with possibility andsufficiency operators . 179 Pseudocomplemented algebras. 181 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 10 Stone algebras 10.1 10.2 10.3 10.4 Introduction. Double Stone algebras. Regular double Stone algebras. Rough relation algebras. 11 Heyting algebras 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Introduction. Heyting algebras.
Heyting algebras with modal operators. Axiomatic extensions of HM-algebras. Frontal Heyting algebras. Symmetric Heyting algebras. Heyting-Brouwer algebras. Apartness algebras. 12 Distributive residuated lattices 12.1 Introduction. 12.2 Distributive residuated lattices. 12.3 Bounded integral residuated lattices. 12.4 Commutative residuated lattices . 12.5 MTL-algebras. 12.6 Axiomatic extensions of MTL-algebras. 12.7 SMTL-algebras. 12.8 IMTL-algebras. 12.9 CMTL-algebras. 12.10 n-potent MTL-algebras . 12.112 -potent BL-algebras. III xi General lattices
with operators 13 General lattices with modal operators 13.1 Introduction. 13.2 Possibility lattices . 13.3 Axiomatic extensions of possibility lattices. 185 185 186 191 192 201 201 202 204 209 215 217 222 223 229 229 230 234 235 236 238 239 241 242 243 245 249 251 251 252 257
xii CONTENTS 13.4 13.5 13.6 13.7 13.8 Necessity lattices. Axiomatic extensions of necessity lattices. Lattices with possibility and necessity operators. Sufficiency lattices. Dual sufficiency lattices. 14 General lattices with negations 14.1 14.2 14.3 14.4 14.5 Introduction. General lattices with De Morgan negation. General lattices with a weak pseudocomplement. General lattices with an orthonegation. Pseudocomplemented general lattices. 15 Residuated and doubly residuated general lattices 15.1 Introduction. 15.2 Residuated general lattices. 15.3 Doubly residuated general lattices . 16 General lattices with relational operators 16.1 16.2 16.3 16.4 Introduction. Residuated general lattices with involution. Doubly residuated general lattices with involution
. Lattice-based relation algebras . 17 Algebras of substructural logics 17.1 17.2 17.3 17.4 17.5 17.6 Introduction. FL֊algebras. FLe-algebras . FLc-algebras . FLw-algebras. An axiomatic extension of FLew-algebras. 18 Beyond Urquhart representation 18.1 Introduction. 18.2 Exhaustive representation of lattices. 18.3 Non-disjoint representation of lattices . 18.4 Non-exhaustive representation of lattices. 18.5 Representation of lattices with two-sorted frames. 18.6 Dedekind-MacNeille representations . 18.7 Canonical Extensions. 18.8 Dedekind-MacNeille-Wille representation of lattices. 18.9 Ortholattices. 18.10 Representation of
ortholattices with two-sorted frames. 258 260 261 265 269 273 273 273 276 279 281 283 283 283 295 299 299 299 304 306 309 309 310 311 312 313 314 317 317 318 321 324 326 330 331 333 335 339 Bibliography 343 Index 361 |
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| spelling | Orłowska, Ewa 1935- Verfasser (DE-588)122556100 aut Dualities for structures of applied logics Ewa Orłowska, Anna Maria Radzikowska and Ingrid Rewitzky London College Publications [2015] © 2015 xii, 381 Seiten 24 cm txt rdacontent n rdamedia nc rdacarrier Studies in logic 56 Mathematical logic and foundations Numerisches Gitter (DE-588)4286867-1 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Dualität (DE-588)4013161-0 gnd rswk-swf Logic, Symbolic and mathematical Algebra, Boolean Dualität (DE-588)4013161-0 s Numerisches Gitter (DE-588)4286867-1 s Algebra (DE-588)4001156-2 s DE-604 Radzikowska, Anna Maria Sonstige (DE-588)1274001633 oth Rewitzky, Ingrid 1968- Sonstige (DE-588)1274001773 oth Studies in logic 56 : Mathematical logic and foundations (DE-604)BV022429273 56 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033933335&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Orłowska, Ewa 1935- Dualities for structures of applied logics Studies in logic Numerisches Gitter (DE-588)4286867-1 gnd Algebra (DE-588)4001156-2 gnd Dualität (DE-588)4013161-0 gnd |
| subject_GND | (DE-588)4286867-1 (DE-588)4001156-2 (DE-588)4013161-0 |
| title | Dualities for structures of applied logics |
| title_auth | Dualities for structures of applied logics |
| title_exact_search | Dualities for structures of applied logics |
| title_exact_search_txtP | Dualities for structures of applied logics |
| title_full | Dualities for structures of applied logics Ewa Orłowska, Anna Maria Radzikowska and Ingrid Rewitzky |
| title_fullStr | Dualities for structures of applied logics Ewa Orłowska, Anna Maria Radzikowska and Ingrid Rewitzky |
| title_full_unstemmed | Dualities for structures of applied logics Ewa Orłowska, Anna Maria Radzikowska and Ingrid Rewitzky |
| title_short | Dualities for structures of applied logics |
| title_sort | dualities for structures of applied logics |
| topic | Numerisches Gitter (DE-588)4286867-1 gnd Algebra (DE-588)4001156-2 gnd Dualität (DE-588)4013161-0 gnd |
| topic_facet | Numerisches Gitter Algebra Dualität |
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