Automatic semigroup acts

Dombi, Erzsébet R. and Hartmann, Miklós (2015) Automatic semigroup acts. Journal of Algebra, 435. 286–307. ISSN 0021-8693

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    To give a general framework for the theory of automatic groups and semigroups, we introduce the notion of automaticity for semigroup acts. We investigate their basic properties and discuss how the property of being automatic behaves under changing the generators of the acting semigroup and under changing the generators of the semigroup act. In particular, we prove that under some conditions on the acting semigroup, the automaticity of the act is invariant under changing the generators. Since automatic semigroups can be seen as a special case of automatic semigroup acts, our result generalizes and extends the corresponding result on automatic semigroups, where the semigroup S satisfies S=SSS=SS. We also give a geometric approach in terms of the fellow traveller property and discuss the solvability of the equality problem in automatic semigroup acts. Our notion gives rise to a variety of definitions of automaticity depending on the set chosen as a semigroup act and we discuss future research directions.