Automatic semigroup

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In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator.

Formally, let S be a semigroup and A be a finite set of generators. Then an automatic structure for S with respect to A consists of a regular language L over A such that every element of S has at least one representative in L and such that for each a \in A \cup \{\varepsilon\}, the relation consisting of pairs (u,v) with ua = v is regular.

The concept of an automatic semigroup was generalized from automatic groups by Campbell et al. (2001)

Unlike automatic groups (see Epstein et al. 1992), a semigroup may have an automatic structure with respect to one generating set, but not with respect to another. However, if an automatic semigroup has an identity, then it has an automatic structure with respect to any generating set (Duncan et al. 1999).

Like automatic groups, automatic semigroups have word problem solvable in quadratic time.

Automatic structures for groups have an elegant geometric characterization called the fellow traveller property (Epstein et al. 1992, ch. 2). Automatic structures for semigroups possess the fellow traveller property but are not in general characterized by it (Campbell et al. 2001). However, the characterization can be generalized to certain 'group-like' classes of semigroups, notably completely simple semigroups (Campbell et al. 2002) and group-embeddable semigroups (Cain et al. 2006).

[edit] Examples of automatic semigroups

[edit] References

  • Campbell, Colin M.; Robertson, Edmund F.; Ruskuc, Nik & Thomas, Richard M. (2001), “Automatic semigroups”, Theoretical Computer Science 250 (1–2): 365–391, DOI 10.1016/S0304-3975(99)00151-6 .
  • Campbell, Colin M.; Robertson, Edmund F.; Ruskuc, Nik & Thomas, Richard M. (2002), “Automatic completely simple semigroups”, Acta Mathematica Hungarica 95 (3): 201–215 .
  • Duncan, A. J.; Robertson, E. F. & Ruskuc, N. (1999), “Automatic monoids and change of generators”, Mathematical Proceedings of the Cambridge Philosophical Society 127 (3): 403–409, DOI 10.1017/S0305004199003722 .