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In this thesis we aim to study the Lefschetz properties ofmonomial algebras. First, we present the necessary concepts and resultsfrom commutative algebra, in particular we build up to the Hilbert-Serre theorem regarding the rationality of Hilbert series. We then reviewsome important results from the literature on the Lefschetz properties(whereof many provide drastic computational shortcuts under certainconditions) and provide some examples of these. The second half isdevoted to the study of Artinian quadratic monomial algebras of theform A(Δ) = K[x1, . . . , xn]/JΔ, where JΔ = (x21, . . . , x2n) + IΔ and IΔis the ideal obtained from some (abstract) simplicial complex Δ viathe Stanley-Reisner correspondence. In particular, we review a recentarticle [2] by H. Dao and R. Nair, provide examples and refine some ofthe formulations and results.

Let Q be a Dynkin quiver of type A and K be an algebraically closedfield. We start with the preprojective algebra Λ associated to Q andpresent a 2-cluster tilting module of Λ constructed by Geiss, Leclerc, andSchr¨oer.After that we move on to something more general. Let A be a drepresentationfinite algebra, its (d+1)-preprojective algebra has a (d+1)-cluster tilting module given by Iyama and Opperman, we shall comparethis with the case d = 1. Then we will compute an example where d = 2. In both cases we will investigate the endomorphism algebra of thecorresponding d-cluster tilting module.

Exceptional algebraic groups are divided into five types, namely G2, F4,E6, E7 and E8. In this thesis we discuss G2, F4 and E6. We discuss the exceptionalalgebraic groups via octonion algebras and Jordan algebras. We firstconsider the groups of type G2. Groups of type G2 are automorphism groupsof octonion algebras, a form of composition algebras. We take the algebra of Zorn vector matrices and find the possible values of automorphisms of thisalgebra with the help of U-operators. We also discuss the product of two andthree U-operators. Then we discuss Albert algebras, since groups of type E6and F4 are related to these algebras. The Albert algebras are a form of Jordanalgebras. We also study the U-operators in Albert algebras. In this thesis wework over algebraically closed fields of characteristic zero.

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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