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Main Definitions and Basic Facts

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Combinatorial Algebra: Syntax and Semantics

Part of the book series: Springer Monographs in Mathematics ((SMM))

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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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Notes

  1. 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. 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. 3.

    We do not give a definition of algorithm because everybody knows what it is, but nobody can define it precisely.

  4. 4.

    Warning: Some knowledge of elementary topology is required to read this subsection.

  5. 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. 6.

    The set {z∣ | z |  = 1} that separates X a from X b is of course the ping-pong net.

  7. 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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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Mark V. Sapir

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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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