In this thesis we study and classify all isolated and completely isolatedsubsemigroups and congruences in classical finite transformationsemigroups.

One of the most important mathematicians of all time was Euclid. Even though his books laid thegroundwork for plane geometry, they did have some limitations. In this paper we show why some importantconstructions are impossible, as well as giving Swedish mathematics teachers some ideas on how to implementthis in their lessons.

We use the theory of Tambara modules to extend and generalize the reconstruction theorem for module categories over a rigid monoidal category to the non-rigid case. We show a biequivalence between the 2-category of cyclic module categories over a monoidal category C and the bicategory of algebra and bimodule objects in the category of Tambara modules on C . Using it, we prove that a cyclic module category can be reconstructed as the category of certain free module objects in the category of Tambara modules on C , and give a sufficient condition for its reconstructability as module objects in C . To that end, we extend the definition of the Cayley functor to the non-closed case, and show that Tambara modules give a proarrow equipment for C-module categories, in which C-module functors are characterized as 1-morphisms admitting a right adjoint. Finally, we show that the 2-category of all C -module categories embeds into the 2-category of categories enriched in Tambara modules on C , giving an “action via enrichment” result.

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