Extending binary operations to funtor-spaces

Given a continuous monadic functor T in the category of Tychonov spaces for each discrete topological semigroup X we extend the semigroup operation of X to a right-topological semigroup operation on TX whose topological center contains the dense subsemigroup of all elements of TX that have finite support.

💡 Research Summary

The paper investigates how to lift a binary operation defined on a discrete topological semigroup (X) to a semigroup operation on the functor‑space (TX) when a continuous monadic functor (T) acts on the category of Tychonoff spaces. The authors start by recalling the essential features of a monadic functor: a unit natural transformation (\eta_X : X \to TX) and a multiplication natural transformation (\mu_X : TTX \to TX), both continuous, satisfying the usual monad axioms (associativity and unit laws). These structures guarantee that (T) behaves like a “space‑valued” construction that can be iterated without losing coherence.

Given a discrete semigroup ((X,\cdot)), the binary operation is first embedded into the functor space via the unit: (\eta_X(x)) becomes a point of (TX). The authors then apply the functor to the original operation, obtaining a map (T(\cdot) : TX \times TX \to T X). Finally, the multiplication (\mu_X) collapses the double application of (T) back to a single one, yielding a new binary operation
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