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Boolean spaces with internal semigroups generalize profinite semigroups andare pertinent for the recognition of not-necessarily regular languages. Viarecognition, the study of existential quantification in logic on words amountsto the study of certain spans of Boolean spaces with internal semigroups. Inturn, these can be understood as the superposition of a span of Boolean spacesand a span of semigroups. In this note, we first study these separately. Moreprecisely, we identify the conditions under which each of these spans givesrise to a morphism into the respective power or Vietoris construction of thecorresponding structure. Combining these characterizations, we obtain such acharacterization for spans of Boolean spaces with internal semigroups which weuse to describe the topo-algebraic counterpart of monadic second-orderexistential quantification. This is closely related to a part of the earlierwork on existential quantification in first-order logic on words by Gehrke,Petri\c san and Reggio. The observation that certain morphisms liftcontravariantly to the appropriate power structures makes our analysis verysimple.

A note on powers of Boolean spaces with internal semigroups