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Monadic convergence structures revisited. (English) Zbl 1464.54003

Summary: For \(L\)-power enriched monad \((\mathbb{T}, \tau)\), the notions of pretopological and topological \(\mathbb{T} \)-convergence structures are introduced from the viewpoint of topology. It is shown that the pretopological \(\mathbb{T} \)-convergence structures are exactly the reflexive and unitary \((\mathbb{T}, L)\)-relations and the topological pretopological \(\mathbb{T} \)-convergence structures are precisely \((\mathbb{T}, L)\)-algebras. When \((\mathbb{T}, \tau)\) is the stratified \(L\)-filter monad, the \(L\)-ordered filter monad, or the \(L\)-concave filter monad, the general setting produces known results on stratified \(L\)-convergence structures, \(L\)-ordered convergence structures and \(L\)-concave structures.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
06F07 Quantales
18C20 Eilenberg-Moore and Kleisli constructions for monads
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