Functor Property And Maximum Compactification Of Some Topological Algebra Systems

According to the classical view of Bourbaki, pure mathematics mainly considers three abstract structures which are topology, algebra, order. In a specific object of study, the three structures reveal deep connections. So the study of the relevance between them is particularly important, and it has developed the following three branches:topological algebra, ordered topology, ordered algebra. This thesis discusses some classical structures of topological algebra. It mainly explores some commutative diagrams of the category of topological spaces, some basic internal structures and commutative diagrams of topological semigroups, topological groups, topological semigroup actions.Using topological tool and functor theory of category, this thesis gives the basic structures of the following aspects:â‘ In the category of T2topological spaces, induced by locally compact space the covariant product functor and covariant exponential functor under compact open topology are topological adjoint functors; â‘¡The structures of subsystems and products of topological semigroups, topological groups and topological semigroup actions. In particular, under topological semigroup has a compact neighborhood basis, the structures of quotient system of topological semigroup actions are given;â‘¢The structures of tensor product and tensor product functor of the topological semigroup actions;â‘£The concept of concrete topological categrories is introduced. Some special functors are discussed, and the important relations between them and maximum compactification of topological algebra are given. This thesis mainly proves that existences and specific structures of maximum compactifications of topological semigroup, topological group, topological semigroup action, under certain separable conditions.