Hypergroup Deformations of Semigroups and Ramsey Hypergroups

We view the well-known example of the dual of a countable compact hypergroup, motivated by the orbit space of p-adic integers by Dunkl and Ramirez (1975), as hypergroup deformation of the max semigroup structure on the linearly ordered set $\mathbb{Z}_+$ of the non-negative integers along the diagonal. This works as motivation for us to study hypergroups or semi convolution spaces arising from general commutative semigroups via hypergroup deformation on idempotents. Another motivation for this came from Ramsey theory for finite sums, which we initiate in the context of hypergroups somewhat on the lines of Willson (2014).