组合半群的若干课题(英文版)

1 Basic Concepts and Notations 1 1.1 Semigroups (Monoids) 1 1.2 Free Semigroups (Monoids) and Languages 8 1.3 Disjunctive (Regular) and Dense Subsets of Semigroups 13 1.4 Periods and Primitive Words 18 1.5 Infinite Words 22 2 Some Common-Used Codes 27 2.1 Methods for Defining Codes 27 2.2 Maximal Codes and Complete Codes 34 2.3 Convex Codes 40 2.4 Semaphore Codes 48 2.5 Solid Codes 54 2.6 Congruences Coming From Solid Codes 66 3 Regular Languages 77 3.1 Automata 77 3.2 Recognizability of Regular Languages 82 3.3 Rationality of Regular Languages 86 3.4 A Decomposition of Regular Languages 90 3.5 Restricted Burnside Problem and Regular Languages 96 4 Disjunctive Languages 103 4.1 Disjunctive Languages Over One-Letter Alphabets 103 4.2 Disjunctive Languages and Dense Languages 105 4.3 Disjunctive Domains 108 4.4 Infinite Disjunctive Decompositions of Dense Languages 113 4.4.1 Disjunctive Splittability of Dense Languages 113 4.4.2 DP-Splittability 114 4.4.3 Semi-DP-Splittability 117 4.5 Finite Disjunctive Decompositions of Dense Languages 122 4.6 A Constructing Method of Decomposition Components 125 5 F-Disjunctive Languages 131 5.1 F-Disjunctive Languages 131 5.1.1 Definition,Elementary Properties and Examples 131 5.1.2 F-Disjunctive Pairs 134 5.1.3 Operations on F-Disjunctive Languages 136 5.1.4 Special Classes of F-Disjunctive Languages 138 5.2 F-Disjunctive Domains 141 5.2.1 Examples and Basic Properties 141 5.2.2 On the Uniform Density of F-Disjunctive Domains 144 5.3 Syntactic Semigroups of F-Disjunctive Languages 152 5.3.1 Semigroups with I-Quasi Length 153 5.3.2 Syntactic I-Ql-Semigroups 159 5.3.3 Ql-Semigroups and Free Semigroups 164 5.3.4 I-Ql-Monoids 170 6 Relatively Disjunctive Languages and Relatively Regular Languages 173 6.1 Relatively Disjunctive Languages 173 6.2 Relatively Regular Languages 177 6.3 The Hierarchies of Relatively Disjunctive Languages (Ⅰ) 180 6.4 The Hierarchies of Relatively Disjunctive Languages (Ⅱ) 189 7 Generalized Disjunctive Languages 197 7.1 Language Classes and Monoid Classes 197 7.2 Quasi-Disjunctive Languages 204 7.2.1 Quasi-Disjunctivity and Auto-disjunctivity 204 7.2.2 Regular Quasi-Disjunctive and Auto-disjunctive Languages 208 7.3 Qf-Disjunctive Languages 210 7.3.1 Definitions and Elementary Properties 210 7.3.2 Operations on Qf-Disjunctive Languages 214 7.3.3 Qf-Disjunctive Languages and Infix Codes 218 7.4 C-Disjunctive Languages 225 7.5 Generalized Disjunctive Hierarchy 231 8 PS-Regular Languages 237 8.1 Permissible Sets and Prefix-Suffix-Sets 237 8.2 Some Characterizations of -Regular Languages and Regular Languages 240 8.3 On the Class of PS-Regular Languages 254 8.4 Conclusions 260 References 261 Index 267