This overview highlights recent results on algebraic structures, including semigroups and groupoids, suggesting their potential in understanding partial symmetries.
The paper is an overview of recent results on algebraic structures (semigroups, groupoids, algebras, inverse semigroups, and groups) associated with objects with a rich set of partial symmetries. We discuss etale groupoids and inverse semigroups, Steinberg algebras, C*-algebras, topological full groups etc.. In particular, we describe in detail the main definitions and constructions related to contracting self-similar inverse semigroups. They are illustrated by several examples, including the semigroup generated by the golden mean rotation and the tiling semigroup of the Penrose tiling.
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Volodymyr Nekrashevych (2025) studied this question.
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