This thesis falls within algebra, and it covers Riesz spaces, positive operators and f-algebras. The thesis aims to introduce f-algebras and prove the thesis's main theorem, which states that every Archimedean f-algebra is commutative. In the beginning, some crucial terms are defined so the definitions of partially ordered vector space and Riesz space can follow. Some useful identities regarding the lattice operations infimum and supremum are proved. The Archimedean property of Riesz spaces is introduced, and it later turns out it is the Archimedean property that is crucial for an f-algebra to be commutative. After studying Riesz spaces, the disjointness of two elements is defined, and two particular types of subspaces of Riesz spaces, called ideals and bands, are discussed. The second part of the thesis covers operators, i. e. linear mappings between Riesz spaces, and their properties. Modulus of an operator and its null ideal are defined. Some equivalent characterizations of lattice homomorphisms, i. e. operators that preserve lattice operations infimum and supremum, are given. Orthomorphisms are introduced. These are order bounded operators that are also band preserving. They are a powerful tool used in the proof of the main theorem of this thesis. The latter follows after the introduction of f-algebras and some of their properties. In the end, the theorem about the uniqueness of a multiplication unit in f-algebras is proved.
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