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Full Description
The aim of this monograph is to present some recent research work on the combinatorial aspects of the theory of semigroups which are of great inter est for both algebra and theoretical computer science. This research mainly concerns that part of combinatorics of finite and infinite words over a finite alphabet which is usually called the theory of "unavoidable" regularities. The unavoidable regularities ofsufficiently large words over a finite alpha bet are very important in the study of finiteness conditions for semigroups. This problem consists in considering conditions which are satisfied by a fi nite semigroup and are such as to assure that a semigroup satisfying them is finite. The most natural requirement is that the semigroup is finitely gener ated. Ifone supposes that the semigroup is also periodic the study offiniteness conditions for these semigroups (or groups) is called the Burnside problem for semigroups (or groups). There exists an important relationship with the theory of finite automata because, as is well known, a language L over a fi nite alphabet is regular (that is, recognizable by a finite automaton) if and only if its syntactic monoid S(L) is finite. Hence, in principle, any finite ness condition for semigroups can be translated into a regularity condition for languages. The study of finiteness conditions for periodic languages (Le. , such that the syntactic semigroup is periodic) has been called the Burnside problem for languages.
Contents
1. Combinatorics on Words.- 1.1 Preliminaries.- 1.2 Infinite words.- 1.3 Metric and topology.- 1.4 Periodicity and conjugacy.- 1.5 Lyndon words.- 1.6 Factorial languages and subword complexity.- 2. Unavoidable Regularities.- 2.1 Ramsey's theorem.- 2.2 Van der Waerden's theorem.- 2.3 Uniformly recurrent words.- 2.4 Shirshov's theorem.- 2.5 Bounded languages.- 2.6 Power-free words.- 2.7 Bi-ideal sequences.- 3. Finiteness Conditions for Semigroups.- 3.1 Preliminaries on semigroups.- 3.2 Finitely generated semigroups.- 3.3 The Burnside problem.- 3.4 Permutation property.- 3.5 Partial commutations.- 3.6 Chain conditions.- 3.7 Iteration property.- 3.8 Permutation and iteration property.- 3.9 Repetitivity.- 4. Finitely Recognizable Semigroups.- 4.1 The Myhill-Nerode theorem.- 4.2 Finitely recognizable semigroups.- 4.3 The factor semigroup.- 4.4 Rewriting systems.- 4.5 The word problem.- 4.6 On a conjecture of Brzozowski.- 4.7 On a conjecture of Brown.- 5. Regularity Conditions.- 5.1 Uniform conditions.- 5.2 Pumping properties.- 5.3 Permutative property.- 6. Well Quasi-orders and Regularity.- 6.1 Well quasi-orders.- 6.2 Higman's theorem.- 6.3 The generalized Myhill theorem.- 6.4 Quasi-orders and rewriting systems.- 6.5 A regularity condition for permutable languages.- 6.6 Almost-commutative languages.- 6.7 Copying systems.- References.
