Function algebras on finite sets: a basic course on many-valued logic and clone theory ; with 46 tables
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2006
|
| Schriftenreihe: | Springer monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 639 - 654 |
| Beschreibung: | XIV, 668 S. graph. Darst. |
| ISBN: | 9783540360223 3540360220 |
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MARC
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| 100 | 1 | |a Lau, Dietlinde |d 1950-2018 |e Verfasser |0 (DE-588)140287086 |4 aut | |
| 245 | 1 | 0 | |a Function algebras on finite sets |b a basic course on many-valued logic and clone theory ; with 46 tables |c Dietlinde Lau |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XIV, 668 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer monographs in mathematics | |
| 500 | |a Literaturverz. S. 639 - 654 | ||
| 650 | 4 | |a Clones (Algebra) | |
| 650 | 4 | |a Function algebras | |
| 650 | 4 | |a Many-valued logic | |
| 650 | 0 | 7 | |a Funktionenalgebra |0 (DE-588)4155686-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Funktionenalgebra |0 (DE-588)4155686-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=2842207&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-015620094 | |
Datensatz im Suchindex
| _version_ | 1805088947501531136 |
|---|---|
| adam_text |
DIETLINDE LAU
FUNCTION ALGEBRAS
ON FINITE SETS
A BASIC COURSE ON MANY-VALUED LOGIC
AN
D CLONE THEOR
Y
WITH 42 FIGURES AN
D 46 TABLES
4U SPRINGER
CONTENTS
INTRODUCTION 1
PRELIMINARIES 17
PART I UNIVERSAL ALGEBRA
1 BASIC CONCEPTS OF UNIVERSAL ALGEBRA
25
1.1
UNIVERSAL ALGEBRAS 25
1.2 EXAMPLES OF UNIVERSAL ALGEBRAS 27
1.2.1 GRUPPOIDS 27
1.2.2 SEMIGROUPS 28
1.2.3 MONOIDS 28
1.2.4 GROUPS 28
1.2.5 SEMIRINGS 28
1.2.6 RINGS 28
1.2.7 FIELDS 29
1.2.8 MODULES 29
1.2.9 VECTOR SPACES 29
1.2.10 SEMILATTICES 29
1.2.11 LATTICES 30
1.2.12 BOOLEAN ALGEBRAS 30
1.2.13 FUNCTION ALGEBRAS 30
1.3 SUBALGEBRAS 31
2 LATTICE
S
35
2.1 TWO DEFINITIONS OF A LATTICE 35
2.2 EXAMPLES FOR LATTICES 39
2.3 ISOMORPHIC LATTICES AND SUBLATTICES 39
' 2.4 COMPLETE LATTICES AND EQUIVALENCE RELATIONS 41
X CONTENTS
3 HUL
L SYSTEM
S AN
D CLOSUR
E OPERATOR
S
45
3.1 BASIC CONCEPTS 45
3.2 SOME PROPERTIES OF HULL SYSTEMS AND CLOSURE OPERATORS 46
4 HOMOMORPHISMS
, CONGRUENCES
, AN
D GALOI
S CONNECTION
S
.
. 51
4.1 HOMOMORPHISMS AND ISOMORPHISMS 51
4.2 CONGRUENCE RELATIONS AND FACTOR ALGEBRAS OF ALGEBRAS 52
4.3 EXAMPLES FOR CONGRUENCE RELATIONS AND SOME
HOMOMORPHISM THEOREMS 56
4.3.1 CONGRUENCES ON GROUPS 56
4.3.2 CONGRUENCES ON RINGS 58
4.4 GALOIS CONNECTIONS 59
5 DIREC
T AN
D SUBDIREC
T PRODUCT
S
61
5.1 DIRECT PRODUCT
S 61
5.2 SUBDIRECT PRODUCT
S 66
6 VARIETIES
, EQUATIONA
L CLASSES
, AN
D FRE
E ALGEBRA
S
71
6.1 VARIETIES 71
6.2 TERMS, TERM ALGEBRAS, AND TERM FUNCTIONS 73
6.3 EQUATIONS AND EQUATIONAL CLASSES 76
6.4 FREE ALGEBRAS . 78
6.5 CONNECTIONS BETWEEN VARIETIES AND EQUATIONAL DEFINED CLASSES 81
6.6 DEDUCTIVE CLOSURE OF EQUATION SETS AND EQUATIONAL THEORY .
. 82
6.7 FINITE AXIOMATIZABILITY OF ALGEBRAS 84
PAR
T I
I FUNCTIO
N ALGEBRA
S
1 BASI
C CONCEPTS
, NOTATIONS
, AN
D FIRS
T PROPERTIE
S
91
1.1 FUNCTIONS ON FINITE SETS 91
1.2 OPERATIONS ON
PA,
FUNCTION ALGEBRAS 94
1.3 SUPERPOSITIONS, SUBCLASSES, AND CLONES 96
1.4 GENERATING SYSTEMS FOR
P
A
98
1.5 SOME APPLICATIONS OF TH
E FUNCTION ALGEBRAS 104
1.5.1 CLASSIFICATION OF UNIVERSAL ALGEBRAS . . 104
1.5.2 PROPOSITIONAL LOGIC AND FIRST ORDER LOGIC 105
1.5.3 MANY-VALUED LOGICS 115
1.5.4 INFORMATION TRANSFORMER 116
1.5.5 CLASSIFICATION OF COMBINATORIAL PROBLEMS 118
2 TH
E GALOIS-CONNECTIO
N BETWEE
N FUNCTION
- AN
D
RELATION-ALGEBRA
S
125
2.1 RELATIONS 125
2.2 DIAGONAL RELATIONS 126
CONTENTS XI
2.3 ELEMENTAR
Y OPERATION
S ON
RK . '.
127
2.4 RELATIO
N ALGEBRAS
, CO-CLONES, AN
D DERIVATIO
N OF RELATION
S .
. 127
2.5 SOME OPERATION
S ON
RK
DERIVABLE FROM TH
E ELEMENTAR
Y
OPERATION
S 128
2.6 TH
E PRESERVIN
G OF RELATIONS
; POL
, IN
V 130
2.7 TH
E RELATION
S XN AN
D
G
N
132
2.8 TH
E OPERATO
R
R
A
134
2.9 TH
E GALOIS THEOR
Y FOR FUNCTION
- AN
D RELATION-ALGEBRA
S 135
2.10 SOME MODIFICATIONS OF TH
E
POL-INV-CONNECTION
137
2.10.1 GALOIS THEOR
Y FOR FINIT
E MONOIDS AN
D FINIT
E GROUP
S . . . 137
2.10.2 GALOIS THEOR
Y FOR ITERATIV
E FUNCTIO
N ALGEBRA
S 139
2.11 SOME CONNECTION
S BETWEEN TH
E RELATIO
N OPERATION
S 142
TH
E SUBCLASSE
S O
F
P
2
145
3.1 DEFINITIONS OF TH
E SUBCLASSES OF
PI
AN
D POST'
S THEORE
M 145
3.2 APROO
F FOR POST'
S THEORE
M 149
3.2.1 TH
E SUBCLASSES
A
OF
P
2
WIT
H
A % L
AN
D
A
S
149
3.2.2 TH
E SUBCLASSES OF
L
154
3.2.3 TH
E SUBCLASSES OF 5
, WHIC
H AR
E NOT SUBSET
S OF
L
155
3.2.4 A COMPLETENES
S CRITERIO
N FOR
PI
156
TH
E SUBCLASSE
S O
F
P
K
WHIC
H CONTAI
N
P
159
TH
E MAXIMA
L CLASSE
S O
F
P
K
163
5.1 INTRODUCTION
, A ROUG
H DESCRIPTIO
N OF TH
E MAXIMA
L CLASSES .
. 163
5.2 DEFINITIONS OF TH
E MAXIMA
L CLASSES OF
PK
165
5.2.1 MAXIMA
L CLASSES OF TYP
E 9JT (MAXIMA
L CLASSES OF
MONOTON
E FUNCTIONS
) 165
5.2.2 MAXIMA
L CLASSES OF TYP
E 6 (MAXIMA
L CLASSES OF
AUTODUA
L FUNCTIONS
) 167
5.2.3 MAXIMA
L CLASSES OF TYP
E I
L (MAXIMA
L CLASSES OF
FUNCTIONS
, WHIC
H PRESERV
E NON-TRIVIA
L EQUIVALENC
E
RELATIONS
) 170
5.2.4 MAXIMA
L CLASSES OF TYP
E (MAXIMA
L CLASSES OF
QUASI-LINEA
R FUNCTIONS
) 171
5.2.5 MAXIMA
L CLASSES OF TYP
E
(MAXIMA
L CLASSES OF
FUNCTIONS
, WHIC
H PRESERV
E CENTRA
L RELATIONS
) 173
5.2.6 MAXIMA
L CLASSES OF TYP
E 03 (MAXIMA
L CLASSES OF
FUNCTIONS
, WHIC
H PRESERV
E /I-UNIVERSAL RELATIONS
) 174
5.3 PROOF OF TH
E MAXIMALIT
Y OF TH
E CLASSES DEFINED IN SECTION 5.2 .17
9
5.4 TH
E NUMBE
R OF TH
E MAXIMA
L CLASSES OF
P
K
183
5.5 REMARK
S T
O TH
E MAXIMA
L CLASSES OF
PK(L)
188
ROSENBERG'
S COMPLETENES
S CRITERIO
N FOR
P
K
191
6.1 PROOF OF COMPLETENES
S CRITERIO
N 191
XII CONTENTS
7 FURTHER COMPLETENES
S CRITERIA
211
7.1 A CRITERION FOR SHEFFER-FUNCTIONS 211
7.2 A COMPLETENESS CRITERION FOR SURJECTIVE FUNCTIONS 216
7.3 FUNDAMENTAL SETS 217
8 SOM
E PROPERTIE
S OF TH
E LATTIC
E
LFE 219
8.1 CARDINALITY STATEMENTS 219
8.2 ON THE CARDINALITIES OF MAXIMAL SUBLATTICES OF LFC 224
8.3 SOME STRATEGIES FOR THE DETERMINATION OF SUBLATTICES OF L^.
. 229
9 CONGRUENCE
S AN
D AUTOMORPHISM
S O
N FUNCTIO
N ALGEBRA
S
. .
. 233
9.1 SOME BASIC CONCEPTS AND FIRST PROPERTIES . 234
9.2 CONGRUENCES ON TH
E SUBCLASSES OF
P
2
235
9.3 CHARACTERIZATION OF THE NON-ARITY CONGRUENCES . . . . :
. / 238
9.4 ABOUT TH
E NUMBER OF TH
E CONGRUENCES ON A SUBCLASS OF
PK YY YY YY
243
9.5 A CRITERION FOR THE PROOF OF THE COUNTABILITY OF
CON A
FOR
CERTAIN
A
C P
FC
248
9.6 CONGRUENCES ON SOME CLASSES OF LINEAR FUNCTIONS 250
9.7 CONGRUENCES ON THE MAXIMAL CLASSES OF PFC 256
9.8 CONGRUENCES ON SUBCLASSES OF [P^] : 265
9.9 CONGRUENCES ON SOME SUBCLASSES OF PFC^ 273
9.10 SOME FURTHER GENERAL PROPERTIES OF THE CONGRUENCES AND THE
/-CLASSES 278
9.11 THE CONNECTION BETWEEN CLONE CONGRUENCES AND FULLY
INVARIANT CONGRUENCES 282
9.12 AUTOMORPHISMS OF FUNCTION ALGEBRAS 285
10 TH
E RELATIO
N DEGRE
E AND TH
E DIMENSIO
N OF SUBCLASSE
S
OF P
FC
291
10.1 THE DEFINITION OF THE RELATION DEGREE AND OF THE DIMENSION
OF A SUBCLASS OF P
FC
291
10.2 THE DIMENSIONS AND RELATION DEGREES OF POST'S CLASSES 293
10.3 FURTHER EXAMPLES OF THE DIMENSION AND RELATION DEGREE OF
CLASSES 301
11 ON GENERATIN
G SYSTEM
S AN
D ORDERS OF TH
E SUBCLASSE
S
OF P
FE
307
11.1 SOME GENERAL PROPERTIES OF GENERATING SYSTEMS AND BASES .
. 308
11.2 THE ORDERS AND SHEFFER-FUNCTIONS OF THE CLASSES OF TYPE C
1
,
S OR I
L 310
11.3 ORDERS OF THE CLASSES OF TYPE
, C,
B 314
11.4 THE ORDER OF
POL
K
G
FOR
G
YY
M
K
AND
K 7
319
11.5 A MAXIMAL CLONE OF MONOTONE FUNCTIONS THA
T IS NOT
FINITELY GENERATED 324
11.6 CLASSIFICATIONS AND BASIS ENUMERATIONS IN PFC 332
CONTENTS
XIII
12 SUBCLASSE
S OF P
FCI2
335
12.1 NOTATIONS 336
12.2 SOME PROPERTIES OF THE INVERSE IMAGES 337
12.3 ON THE NUMBER OF THE B-PROJECTABLE SUBCLASSES
OF P
FC
,2,
B
C P
2
342
12.4 THE P;-PROJECTABLE AND THE
POK
{A}-PROJECTABLE SUBCLASSES
OF PFC,; 350
12.5 THE MAXIMAL AND THE SUBMAXIMAL CLASSES OF PFC
J2
354
12.6 THE CLASSES
A
WITH
M
N T
O
N
T
X
C
PRA
OR
L
N T
O
N 5 C
PRA
OR
PRA
= MFL
S 361
13 CLASSES OF LINEAR FUNCTION
S
383
13.1 SOME PROPERTIES OF THE SUBCLASSES OF
UD
THA
T CONTAIN R
J 384
13.2 THE SUBCLASSES OF LINEAR FUNCTIONS OF P
FC
WITH FCEP 387
13.3 A SURVEY OF FURTHER RESULTS ON LINEAR FUNCTIONS 390
14 SUBMAXIMAL CLASSES OF P3
399
14.1 A SURVEY OF THE SUBMAXIMAL CLASSES OF P
3
400
14.2 SOME DECLARATIONS AND LEMMAS FOR SECTIONS 14.3-14.9 408
14.3 PROOF OF THEOREM 14.1.2 410
14.4 PROOF OF THEOREM 14.1.3 412
14.5 PROOF OF THEOREM 14.1.4 415
14.6 PROOF OF THEOREM 14.1.5 418
14.7 PROOF OF THEOREM 14.1.7 418
14.8 PROOF OF THEOREM 14.1.8 421
14.9 PROOF OF THEOREM 14.1.9 424
' 14.10ON THE CARDINALITY OF L^-A) FOR SUBMAXIMAL CLONES A 425
15 FINIT
E AND COUNTABL
Y INFINITE SUBLATTICE
S OF DEPT
H 1 OR 2
OF
L
3
433
15.1 THE LATTICE OF SUBCLASSES OF P3 OF LINEAR FUNCTIONS 433
15.2 THE SUBSEMIGROUPS OF (P
3
X
; *) 434
15.3 CLASSES OF QUASILINEAR FUNCTIONS OF P
3
456
15.3.1 SOME NOTATIONS 456
15.3.2 SUBCLASSES OF
0
,
I
'.
457
15.3.3 THE SUBCLASSES OF
0
,
I U
0
,
2
THA
T ARE NOT SUBCLASSES
OF
0
,
I OR
0
,2 461
15.3.4 THE REMAINING SUBCLASSES OF 463
15.4 THE SUBCLASSES OF
[O
1
U
{MAX}}
464
15.4.1 SOME DESCRIPTIONS OF THE CLASS M 464
15.4.2 SOME LEMMAS AND A ROUGH PARTITION OF THE SUBCLASSES
OF
M
465
15.4.3 THE SUBCLASSES OF [M
1
] 470
15.4.4 THE SUBCLASSES OF
R
471
15.4.5 THE SUBCLASSES OF
M
N POZ
3
{(0,2)} 482
XIV CONTENTS
15.4.6 THE REMAINING SUBCLASSES OF
M
488
16 TH
E MAXIMA
L CLASSES OF F|
O
6Q
P
OLK{A}
FOR
Q
C
E
K
499
16.1 NOTATIONS 499
16.2 RESULTS OF CHAPTER 16 501
16.3 SOME LEMMAS 502
16.4 PROOF OF THEOREM 16.2.1 513
17 MAXIMA
L CLASSES OF
POL
K
E
T
FOR 2
I
K
515
17.1 NOTATIONS, DEFINITIONS, AND SOME LEMMAS 515
17.2 RESULTS OF CHAPTER 17 519
17.3 MAXIMALITY PROOFS 520
17.4 SOME LEMMAS 528
17.5 NOT THROUGH RELATIONS OF
R
M
AX{PI)
U
R
MAX
{P
K
)
DESCRIBABLE
CLASSES 529
17.6 CLASSES DESCRIBABLE BY RELATIONS OF
R
M
AX(PI)
U
R
MA
X(PK)
549
18 FURTHER SUBMAXIMA
L CLASSES OF PFC
555
18.1 THE MAXIMAL CLASSES OF
POLKQS
FOR
G
S
& &
K
555
18.2 SOME MAXIMAL CLASSES OF A MAXIMAL CLASS OF TYPE I
L 561
18.3 THE MAXIMAL CLASSES OF PO/
FE
(^.
1
U {(FE-1, FC-1)}) . 573
18.3.1 DEFINITIONS OF THE [/-MAXIMAL CLASSES 573
18.3.2 PROOF OF THE [/-MAXIMALITY OF THE CLASSES DEFINED IN
18.3.1 576
18.3.3 PROOF OF THE COMPLETENESS CRITERION FOR
U
584
19 MINIMA
L CLASSES AND MINIMA
L CLONES OF P
FC
589
19.1 MINIMAL CLASSES 589
19.2 THE FIVE TYPES OF MINIMAL CLONES 590
20 PARTIA
L FUNCTION ALGEBRAS
597
20.1 BASIC CONCEPTS 598
20.2 ONE-POINT EXTENSION 600
20.3 DESCRIPTION OF PARTIAL CLONES BY RELATIONS 604
20.4 THE MAXIMAL PARTIA
L CLASSES OF
PG
AND P
3
606
20.5 THE COMPLETENESS CRITERION FOR P
FE
^ 614
20.6 SOME PROPERTIES OF THE MAXIMAL PARTIAL CLONES OF P^ 616
20.7 INTERVALS OF PARTIA
L CLONES THA
T CONTAIN A MAXIMAL CLONE .
. 619
20.8 INTERVALS OF BOOLEAN PARTIAL CLASSES 627
20.9 ON CONGRUENCES OF PARTIAL CLONES . 628
REFERENCE
S
639
GLOSSARY
655
INDE
X
663 |
| adam_txt |
DIETLINDE LAU
FUNCTION ALGEBRAS
ON FINITE SETS
A BASIC COURSE ON MANY-VALUED LOGIC
AN
D CLONE THEOR
Y
WITH 42 FIGURES AN
D 46 TABLES
4U SPRINGER
CONTENTS
INTRODUCTION 1
PRELIMINARIES 17
PART I UNIVERSAL ALGEBRA
1 BASIC CONCEPTS OF UNIVERSAL ALGEBRA
25
1.1
UNIVERSAL ALGEBRAS 25
1.2 EXAMPLES OF UNIVERSAL ALGEBRAS 27
1.2.1 GRUPPOIDS 27
1.2.2 SEMIGROUPS 28
1.2.3 MONOIDS 28
1.2.4 GROUPS 28
1.2.5 SEMIRINGS 28
1.2.6 RINGS 28
1.2.7 FIELDS 29
1.2.8 MODULES 29
1.2.9 VECTOR SPACES 29
1.2.10 SEMILATTICES 29
1.2.11 LATTICES 30
1.2.12 BOOLEAN ALGEBRAS 30
1.2.13 FUNCTION ALGEBRAS 30
1.3 SUBALGEBRAS 31
2 LATTICE
S
35
2.1 TWO DEFINITIONS OF A LATTICE 35
2.2 EXAMPLES FOR LATTICES 39
2.3 ISOMORPHIC LATTICES AND SUBLATTICES 39
' 2.4 COMPLETE LATTICES AND EQUIVALENCE RELATIONS 41
X CONTENTS
3 HUL
L SYSTEM
S AN
D CLOSUR
E OPERATOR
S
45
3.1 BASIC CONCEPTS 45
3.2 SOME PROPERTIES OF HULL SYSTEMS AND CLOSURE OPERATORS 46
4 HOMOMORPHISMS
, CONGRUENCES
, AN
D GALOI
S CONNECTION
S
.
. 51
4.1 HOMOMORPHISMS AND ISOMORPHISMS 51
4.2 CONGRUENCE RELATIONS AND FACTOR ALGEBRAS OF ALGEBRAS 52
4.3 EXAMPLES FOR CONGRUENCE RELATIONS AND SOME
HOMOMORPHISM THEOREMS 56
4.3.1 CONGRUENCES ON GROUPS 56
4.3.2 CONGRUENCES ON RINGS 58
4.4 GALOIS CONNECTIONS 59
5 DIREC
T AN
D SUBDIREC
T PRODUCT
S
61
5.1 DIRECT PRODUCT
S 61
5.2 SUBDIRECT PRODUCT
S 66
6 VARIETIES
, EQUATIONA
L CLASSES
, AN
D FRE
E ALGEBRA
S
71
6.1 VARIETIES 71
6.2 TERMS, TERM ALGEBRAS, AND TERM FUNCTIONS 73
6.3 EQUATIONS AND EQUATIONAL CLASSES 76
6.4 FREE ALGEBRAS . 78
6.5 CONNECTIONS BETWEEN VARIETIES AND EQUATIONAL DEFINED CLASSES 81
6.6 DEDUCTIVE CLOSURE OF EQUATION SETS AND EQUATIONAL THEORY .
. 82
6.7 FINITE AXIOMATIZABILITY OF ALGEBRAS 84
PAR
T I
I FUNCTIO
N ALGEBRA
S
1 BASI
C CONCEPTS
, NOTATIONS
, AN
D FIRS
T PROPERTIE
S
91
1.1 FUNCTIONS ON FINITE SETS 91
1.2 OPERATIONS ON
PA,
FUNCTION ALGEBRAS 94
1.3 SUPERPOSITIONS, SUBCLASSES, AND CLONES 96
1.4 GENERATING SYSTEMS FOR
P
A
98
1.5 SOME APPLICATIONS OF TH
E FUNCTION ALGEBRAS 104
1.5.1 CLASSIFICATION OF UNIVERSAL ALGEBRAS . . 104
1.5.2 PROPOSITIONAL LOGIC AND FIRST ORDER LOGIC 105
1.5.3 MANY-VALUED LOGICS 115
1.5.4 INFORMATION TRANSFORMER 116
1.5.5 CLASSIFICATION OF COMBINATORIAL PROBLEMS 118
2 TH
E GALOIS-CONNECTIO
N BETWEE
N FUNCTION
- AN
D
RELATION-ALGEBRA
S
125
2.1 RELATIONS 125
2.2 DIAGONAL RELATIONS 126
CONTENTS XI
2.3 ELEMENTAR
Y OPERATION
S ON
RK . '.
127
2.4 RELATIO
N ALGEBRAS
, CO-CLONES, AN
D DERIVATIO
N OF RELATION
S .
. 127
2.5 SOME OPERATION
S ON
RK
DERIVABLE FROM TH
E ELEMENTAR
Y
OPERATION
S 128
2.6 TH
E PRESERVIN
G OF RELATIONS
; POL
, IN
V 130
2.7 TH
E RELATION
S XN AN
D
G
N
132
2.8 TH
E OPERATO
R
R
A
134
2.9 TH
E GALOIS THEOR
Y FOR FUNCTION
- AN
D RELATION-ALGEBRA
S 135
2.10 SOME MODIFICATIONS OF TH
E
POL-INV-CONNECTION
137
2.10.1 GALOIS THEOR
Y FOR FINIT
E MONOIDS AN
D FINIT
E GROUP
S . . . 137
2.10.2 GALOIS THEOR
Y FOR ITERATIV
E FUNCTIO
N ALGEBRA
S 139
2.11 SOME CONNECTION
S BETWEEN TH
E RELATIO
N OPERATION
S 142
TH
E SUBCLASSE
S O
F
P
2
145
3.1 DEFINITIONS OF TH
E SUBCLASSES OF
PI
AN
D POST'
S THEORE
M 145
3.2 APROO
F FOR POST'
S THEORE
M 149
3.2.1 TH
E SUBCLASSES
A
OF
P
2
WIT
H
A % L
AN
D
A
S
149
3.2.2 TH
E SUBCLASSES OF
L
154
3.2.3 TH
E SUBCLASSES OF 5
, WHIC
H AR
E NOT SUBSET
S OF
L
155
3.2.4 A COMPLETENES
S CRITERIO
N FOR
PI
156
TH
E SUBCLASSE
S O
F
P
K
WHIC
H CONTAI
N
P
159
TH
E MAXIMA
L CLASSE
S O
F
P
K
163
5.1 INTRODUCTION
, A ROUG
H DESCRIPTIO
N OF TH
E MAXIMA
L CLASSES .
. 163
5.2 DEFINITIONS OF TH
E MAXIMA
L CLASSES OF
PK
165
5.2.1 MAXIMA
L CLASSES OF TYP
E 9JT (MAXIMA
L CLASSES OF
MONOTON
E FUNCTIONS
) 165
5.2.2 MAXIMA
L CLASSES OF TYP
E 6 (MAXIMA
L CLASSES OF
AUTODUA
L FUNCTIONS
) 167
5.2.3 MAXIMA
L CLASSES OF TYP
E I
L (MAXIMA
L CLASSES OF
FUNCTIONS
, WHIC
H PRESERV
E NON-TRIVIA
L EQUIVALENC
E
RELATIONS
) 170
5.2.4 MAXIMA
L CLASSES OF TYP
E (MAXIMA
L CLASSES OF
QUASI-LINEA
R FUNCTIONS
) 171
5.2.5 MAXIMA
L CLASSES OF TYP
E
(MAXIMA
L CLASSES OF
FUNCTIONS
, WHIC
H PRESERV
E CENTRA
L RELATIONS
) 173
5.2.6 MAXIMA
L CLASSES OF TYP
E 03 (MAXIMA
L CLASSES OF
FUNCTIONS
, WHIC
H PRESERV
E /I-UNIVERSAL RELATIONS
) 174
5.3 PROOF OF TH
E MAXIMALIT
Y OF TH
E CLASSES DEFINED IN SECTION 5.2 .17
9
5.4 TH
E NUMBE
R OF TH
E MAXIMA
L CLASSES OF
P
K
183
5.5 REMARK
S T
O TH
E MAXIMA
L CLASSES OF
PK(L)
188
ROSENBERG'
S COMPLETENES
S CRITERIO
N FOR
P
K
191
6.1 PROOF OF COMPLETENES
S CRITERIO
N 191
XII CONTENTS
7 FURTHER COMPLETENES
S CRITERIA
211
7.1 A CRITERION FOR SHEFFER-FUNCTIONS 211
7.2 A COMPLETENESS CRITERION FOR SURJECTIVE FUNCTIONS 216
7.3 FUNDAMENTAL SETS 217
8 SOM
E PROPERTIE
S OF TH
E LATTIC
E
LFE 219
8.1 CARDINALITY STATEMENTS 219
8.2 ON THE CARDINALITIES OF MAXIMAL SUBLATTICES OF LFC 224
8.3 SOME STRATEGIES FOR THE DETERMINATION OF SUBLATTICES OF L^.
. 229
9 CONGRUENCE
S AN
D AUTOMORPHISM
S O
N FUNCTIO
N ALGEBRA
S
. .
. 233
9.1 SOME BASIC CONCEPTS AND FIRST PROPERTIES . 234
9.2 CONGRUENCES ON TH
E SUBCLASSES OF
P
2
235
9.3 CHARACTERIZATION OF THE NON-ARITY CONGRUENCES . . . . :
. / 238
9.4 ABOUT TH
E NUMBER OF TH
E CONGRUENCES ON A SUBCLASS OF
PK YY YY YY
243
9.5 A CRITERION FOR THE PROOF OF THE COUNTABILITY OF
CON A
FOR
CERTAIN
A
C P
FC
248
9.6 CONGRUENCES ON SOME CLASSES OF LINEAR FUNCTIONS 250
9.7 CONGRUENCES ON THE MAXIMAL CLASSES OF PFC 256
9.8 CONGRUENCES ON SUBCLASSES OF [P^] : 265
9.9 CONGRUENCES ON SOME SUBCLASSES OF PFC^ 273
9.10 SOME FURTHER GENERAL PROPERTIES OF THE CONGRUENCES AND THE
/-CLASSES 278
9.11 THE CONNECTION BETWEEN CLONE CONGRUENCES AND FULLY
INVARIANT CONGRUENCES 282
9.12 AUTOMORPHISMS OF FUNCTION ALGEBRAS 285
10 TH
E RELATIO
N DEGRE
E AND TH
E DIMENSIO
N OF SUBCLASSE
S
OF P
FC
291
10.1 THE DEFINITION OF THE RELATION DEGREE AND OF THE DIMENSION
OF A SUBCLASS OF P
FC
291
10.2 THE DIMENSIONS AND RELATION DEGREES OF POST'S CLASSES 293
10.3 FURTHER EXAMPLES OF THE DIMENSION AND RELATION DEGREE OF
CLASSES 301
11 ON GENERATIN
G SYSTEM
S AN
D ORDERS OF TH
E SUBCLASSE
S
OF P
FE
307
11.1 SOME GENERAL PROPERTIES OF GENERATING SYSTEMS AND BASES .
. 308
11.2 THE ORDERS AND SHEFFER-FUNCTIONS OF THE CLASSES OF TYPE C
1
,
S OR I
L 310
11.3 ORDERS OF THE CLASSES OF TYPE
, C,
B 314
11.4 THE ORDER OF
POL
K
G
FOR
G
YY
M
K
AND
K 7
319
11.5 A MAXIMAL CLONE OF MONOTONE FUNCTIONS THA
T IS NOT
FINITELY GENERATED 324
11.6 CLASSIFICATIONS AND BASIS ENUMERATIONS IN PFC 332
CONTENTS
XIII
12 SUBCLASSE
S OF P
FCI2
335
12.1 NOTATIONS 336
12.2 SOME PROPERTIES OF THE INVERSE IMAGES 337
12.3 ON THE NUMBER OF THE B-PROJECTABLE SUBCLASSES
OF P
FC
,2,
B
C P
2
342
12.4 THE P;-PROJECTABLE AND THE
POK
{A}-PROJECTABLE SUBCLASSES
OF PFC,; 350
12.5 THE MAXIMAL AND THE SUBMAXIMAL CLASSES OF PFC
J2
354
12.6 THE CLASSES
A
WITH
M
N T
O
N
T
X
C
PRA
OR
L
N T
O
N 5 C
PRA
OR
PRA
= MFL
S 361
13 CLASSES OF LINEAR FUNCTION
S
383
13.1 SOME PROPERTIES OF THE SUBCLASSES OF
UD
THA
T CONTAIN R
J 384
13.2 THE SUBCLASSES OF LINEAR FUNCTIONS OF P
FC
WITH FCEP 387
13.3 A SURVEY OF FURTHER RESULTS ON LINEAR FUNCTIONS 390
14 SUBMAXIMAL CLASSES OF P3
399
14.1 A SURVEY OF THE SUBMAXIMAL CLASSES OF P
3
400
14.2 SOME DECLARATIONS AND LEMMAS FOR SECTIONS 14.3-14.9 408
14.3 PROOF OF THEOREM 14.1.2 410
14.4 PROOF OF THEOREM 14.1.3 412
14.5 PROOF OF THEOREM 14.1.4 415
14.6 PROOF OF THEOREM 14.1.5 418
14.7 PROOF OF THEOREM 14.1.7 418
14.8 PROOF OF THEOREM 14.1.8 421
14.9 PROOF OF THEOREM 14.1.9 424
' 14.10ON THE CARDINALITY OF L^-A) FOR SUBMAXIMAL CLONES A 425
15 FINIT
E AND COUNTABL
Y INFINITE SUBLATTICE
S OF DEPT
H 1 OR 2
OF
L
3
433
15.1 THE LATTICE OF SUBCLASSES OF P3 OF LINEAR FUNCTIONS 433
15.2 THE SUBSEMIGROUPS OF (P
3
X
; *) 434
15.3 CLASSES OF QUASILINEAR FUNCTIONS OF P
3
456
15.3.1 SOME NOTATIONS 456
15.3.2 SUBCLASSES OF
0
,
I
'.
457
15.3.3 THE SUBCLASSES OF
0
,
I U
0
,
2
THA
T ARE NOT SUBCLASSES
OF
0
,
I OR
0
,2 461
15.3.4 THE REMAINING SUBCLASSES OF 463
15.4 THE SUBCLASSES OF
[O
1
U
{MAX}}
464
15.4.1 SOME DESCRIPTIONS OF THE CLASS M 464
15.4.2 SOME LEMMAS AND A ROUGH PARTITION OF THE SUBCLASSES
OF
M
465
15.4.3 THE SUBCLASSES OF [M
1
] 470
15.4.4 THE SUBCLASSES OF
R
471
15.4.5 THE SUBCLASSES OF
M
N POZ
3
{(0,2)} 482
XIV CONTENTS
15.4.6 THE REMAINING SUBCLASSES OF
M
488
16 TH
E MAXIMA
L CLASSES OF F|
O
6Q
P
OLK{A}
FOR
Q
C
E
K
499
16.1 NOTATIONS 499
16.2 RESULTS OF CHAPTER 16 501
16.3 SOME LEMMAS 502
16.4 PROOF OF THEOREM 16.2.1 513
17 MAXIMA
L CLASSES OF
POL
K
E
T
FOR 2
I
K
515
17.1 NOTATIONS, DEFINITIONS, AND SOME LEMMAS 515
17.2 RESULTS OF CHAPTER 17 519
17.3 MAXIMALITY PROOFS 520
17.4 SOME LEMMAS 528
17.5 NOT THROUGH RELATIONS OF
R
M
AX{PI)
U
R
MAX
{P
K
)
DESCRIBABLE
CLASSES 529
17.6 CLASSES DESCRIBABLE BY RELATIONS OF
R
M
AX(PI)
U
R
MA
X(PK)
549
18 FURTHER SUBMAXIMA
L CLASSES OF PFC
555
18.1 THE MAXIMAL CLASSES OF
POLKQS
FOR
G
S
& &
K
555
18.2 SOME MAXIMAL CLASSES OF A MAXIMAL CLASS OF TYPE I
L 561
18.3 THE MAXIMAL CLASSES OF PO/
FE
(^.
1
U {(FE-1, FC-1)}) . 573
18.3.1 DEFINITIONS OF THE [/-MAXIMAL CLASSES 573
18.3.2 PROOF OF THE [/-MAXIMALITY OF THE CLASSES DEFINED IN
18.3.1 576
18.3.3 PROOF OF THE COMPLETENESS CRITERION FOR
U
584
19 MINIMA
L CLASSES AND MINIMA
L CLONES OF P
FC
589
19.1 MINIMAL CLASSES 589
19.2 THE FIVE TYPES OF MINIMAL CLONES 590
20 PARTIA
L FUNCTION ALGEBRAS
597
20.1 BASIC CONCEPTS 598
20.2 ONE-POINT EXTENSION 600
20.3 DESCRIPTION OF PARTIAL CLONES BY RELATIONS 604
20.4 THE MAXIMAL PARTIA
L CLASSES OF
PG
AND P
3
606
20.5 THE COMPLETENESS CRITERION FOR P
FE
^ 614
20.6 SOME PROPERTIES OF THE MAXIMAL PARTIAL CLONES OF P^ 616
20.7 INTERVALS OF PARTIA
L CLONES THA
T CONTAIN A MAXIMAL CLONE .
. 619
20.8 INTERVALS OF BOOLEAN PARTIAL CLASSES 627
20.9 ON CONGRUENCES OF PARTIAL CLONES . 628
REFERENCE
S
639
GLOSSARY
655
INDE
X
663 |
| any_adam_object | 1 |
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| author | Lau, Dietlinde 1950-2018 |
| author_GND | (DE-588)140287086 |
| author_facet | Lau, Dietlinde 1950-2018 |
| author_role | aut |
| author_sort | Lau, Dietlinde 1950-2018 |
| author_variant | d l dl |
| building | Verbundindex |
| bvnumber | BV022411609 |
| classification_rvk | SK 130 SK 240 SK 600 |
| classification_tum | MAT 050f |
| ctrlnum | (OCoLC)180941590 (DE-599)BVBBV022411609 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022411609 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:22:19Z |
| indexdate | 2024-07-20T09:16:06Z |
| institution | BVB |
| isbn | 9783540360223 3540360220 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015620094 |
| oclc_num | 180941590 |
| open_access_boolean | |
| owner | DE-824 DE-703 DE-91G DE-BY-TUM DE-11 DE-188 |
| owner_facet | DE-824 DE-703 DE-91G DE-BY-TUM DE-11 DE-188 |
| physical | XIV, 668 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Lau, Dietlinde 1950-2018 Verfasser (DE-588)140287086 aut Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables Dietlinde Lau Berlin [u.a.] Springer 2006 XIV, 668 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Literaturverz. S. 639 - 654 Clones (Algebra) Function algebras Many-valued logic Funktionenalgebra (DE-588)4155686-0 gnd rswk-swf Funktionenalgebra (DE-588)4155686-0 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2842207&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015620094&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lau, Dietlinde 1950-2018 Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables Clones (Algebra) Function algebras Many-valued logic Funktionenalgebra (DE-588)4155686-0 gnd |
| subject_GND | (DE-588)4155686-0 |
| title | Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables |
| title_auth | Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables |
| title_exact_search | Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables |
| title_exact_search_txtP | Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables |
| title_full | Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables Dietlinde Lau |
| title_fullStr | Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables Dietlinde Lau |
| title_full_unstemmed | Function algebras on finite sets a basic course on many-valued logic and clone theory ; with 46 tables Dietlinde Lau |
| title_short | Function algebras on finite sets |
| title_sort | function algebras on finite sets a basic course on many valued logic and clone theory with 46 tables |
| title_sub | a basic course on many-valued logic and clone theory ; with 46 tables |
| topic | Clones (Algebra) Function algebras Many-valued logic Funktionenalgebra (DE-588)4155686-0 gnd |
| topic_facet | Clones (Algebra) Function algebras Many-valued logic Funktionenalgebra |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2842207&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015620094&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT laudietlinde functionalgebrasonfinitesetsabasiccourseonmanyvaluedlogicandclonetheorywith46tables |