A Prelude and Fugue on a Theme of Factorization
North, Marissa Emiko
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The past decade has seen much research on factorizations of elements in multiplicative monoids and integral domains. In particular, the use of graphical representations such as irreducible divisor graphs, compressed irreducible divisor graphs, and irreducible divisor simplicial complexes has yielded many results. This thesis continues this research by introducing the compressed irreducible divisor simplicial complex. We show that many results obtained by using the aforementioned constructions will also hold in the compressed irreducible divisor simplicial complex. Additionally, we obtain new results by comparing all of these constructions and expand on some ideas already existing in the literature. Lastly, we characterize the prime elements in a monoid by considering irreducible divisor graphs and irreducible divisor simplicial complexes.