Abstract
This chapter introduces the main characters that will appear in this book: sets, words, graphs, automata, rewriting systems, various kinds of (universal) algebras, varieties, free algebras (including free semigroups and groups) and subshifts. We also introduce the main properties of algebras that we are interested in: the Burnside property, the finite basis property, properties of the growth function and the growth series, etc.
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- 1.
This long instruction is reminiscent of some recipes from a cook book. These usually end with something like “cook for 16 minutes at 350 degrees, flipping after every 4 minutes”. That is not what you want to do with an automaton.
- 2.
Note that one needs to distinguish here \(a^{n} =\mathop{\underbrace{ a \cdot \ldots \cdot a}}\limits _{n}\) from \(a^{t} = t^{-1}at\).
- 3.
We do not give a definition of algorithm because everybody knows what it is, but nobody can define it precisely.
- 4.
Warning: Some knowledge of elementary topology is required to read this subsection.
- 5.
Note that we are using Zorn’s lemma here. The partially ordered set to which Zorn’s lemma applied is the set of all subsystems of (D, T), so that (D′, T) ≤ (D″, T) if and only if \(D' \supseteq D''\).
- 6.
The set {z∣ | z | = 1} that separates X a from X b is of course the ping-pong net.
- 7.
Moreover, an infinite linear combination of elements from A + is invertible if and only if the coefficient of 1 is not 0.
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Sapir, M.V. (2014). Main Definitions and Basic Facts. In: Combinatorial Algebra: Syntax and Semantics. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-08031-4_1
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