De Finetti theorems for a Boolean analogue of easy quantum groups

We show an organized form of quantum de Finetti theorem for Boolean independence. We define a Boolean analogue of easy quantum groups for the categories of interval partitions, which is a family of sequences of quantum semigroups. We construct the Haar states on those quantum semigroups. The proof of our de Finetti theorem is based on the analysis of the Haar states. [Modified]Definition of the Boolean quantum semigroups on categories of interval partitions [Delete]Classification of categories of interval partitions [Add]Proof of the positiveness of the Haar functionals (in particular they are Haar states)