This book discusses the ways in which the algebras in a locally finite quasivariety determine its lattice of subquasivarieties. The book starts with a clear and comprehensive presentation of the basic structure theory of quasivariety lattices, and then develops new methods and algorithms for their analysis. Particular attention is paid to the role of quasicritical algebras. The methods are illustrated by applying them to quasivarieties of abelian groups, modular lattices, unary algebras and pure relational structures. An appendix gives an overview of the theory of quasivarieties. Extensive references to the literature are provided throughout.
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Chapter 1 Introduction and Background |
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1 | (10) |
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1 | (3) |
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4 | (4) |
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1.3 Review of Complete Lattices |
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8 | (3) |
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Chapter 2 Structure of Lattices of Subquasivarieties |
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11 | (26) |
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2.1 Completely Join Irreducible Quasivarieties |
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11 | (7) |
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18 | (5) |
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2.3 Completely Meet Irreducible Quasivarieties |
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23 | (3) |
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2.4 Quasivarieties of Modular Lattices |
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26 | (6) |
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2.5 Quasivarieties of Abelian Groups |
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32 | (1) |
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2.6 Quasivarieties of Infinite Type |
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33 | (4) |
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Chapter 3 Omission and Bases for Quasivarieties |
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37 | (12) |
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3.1 Characterizing Quasivarieties by Excluded Subalgebras |
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37 | (6) |
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43 | (6) |
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Chapter 4 Analyzing Lq(K) |
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49 | (16) |
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49 | (3) |
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52 | (4) |
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4.3 Equational Quasivarieties |
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56 | (1) |
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57 | (2) |
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59 | (4) |
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63 | (2) |
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Chapter 5 Unary Algebras with 2-Element Range |
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65 | (38) |
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5.1 The Variety Generated by M |
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65 | (3) |
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5.2 Illustrating the Algorithms: 1-Generated Algebras |
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68 | (13) |
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5.3 Illustrating the Algorithms: 2-Generated Algebras |
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81 | (9) |
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5.4 Uncountably Many Subquasivarieties |
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90 | (8) |
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98 | (5) |
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Chapter 6 1-Unary Algebras |
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103 | (18) |
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6.1 1-Unary Algebras With and Without 0 |
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103 | (11) |
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6.2 Some Quasivarieties N0r,s |
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114 | (3) |
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6.3 Some Quasivarieties Nr,s |
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117 | (4) |
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Chapter 7 Pure Unary Relational Structures |
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121 | (12) |
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7.1 Pure Unary Relational Quasivarieties |
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121 | (1) |
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122 | (2) |
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124 | (1) |
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125 | (5) |
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130 | (3) |
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133 | (2) |
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Appendix A Properties of Lattices of Subquasivarieties |
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135 | (10) |
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A.1 Representations of Quasivariety Lattices |
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135 | (1) |
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A.2 Representations of Quasivariety Lattices |
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135 | (1) |
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A.3 Basic Consequences of the Representations |
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136 | (2) |
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A.4 Equaclosure Operators |
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138 | (2) |
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A.5 Examples of Quasivariety Lattices |
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140 | (1) |
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A.6 The Quasivariety Generated by a Quasiprimal Algebra |
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140 | (3) |
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143 | (2) |
| Bibliography |
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145 | (10) |
| Symbol Index |
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155 | (2) |
| Author Index |
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157 | (2) |
| Subject Index |
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159 | |
Jennifer Hyndman was a founding faculty member of the University of Northern British Columbia. There she honed her passion for teaching that led to her winning the Canadian Mathematical Society Excellence in Teaching Award. When not engrossed in research on natural duality theory or quasi-equational theory she can be found in a dance studio learning jazz, modern, and ballet choreography.J. B. Nation is professor emeritus at the University of Hawaii. His research interests include lattice theory, universal algebra, coding theory and bio-informatics. He enjoys running, refereeing soccer, and playing jazz flugelhorn.
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