In this thesis we study profinite semigroups. We introduce the notion of projective limits of topological spaces and topological semigroups, and prove their existence. A profinite semigroup is defined as a projective limit of finite semigroups. We also prove a theorem giving five equivalent characterizations of a compact topological semigroup: profiniteness, residual finiteness, being a closed subdirect product of finite semigroups, total disconnectedness, and zero-dimensionality. In the proof we use Hunter's lemma on syntactic congruences. We also focus on monogenic semigroups. We describe the structure of finite monogenic semigroups and homomorphisms between them. We prove that a projective limit of finite monogenic semigroups, in which the homomorphisms map generators to generators, has exactly one idempotent and that the subsemigroup generated by the limit of the generators is dense. We also prove a characterization of monogenic profinite semigroups: a profinite semigroup is monogenic if and only if it is isomorphic to a projective limit of finite monogenic semigroups with surjective homomorphisms that preserve generators.
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