Recognizability, hypergraph operations, and logical types

We study several algebras of graphs and hypergraphs and the corresponding notions of equational sets and recognizable sets. We generalize and unify several existing results which compare the associated equational and recognizable sets. The basic algebra on relational structures is based on disjoint union and quantifier-free definable operations. We expand it to an equivalent one by adding operations definable with “few quantifiers”, i.e., operations that take into account local information about elements or tuples. We also consider monadic second-order transductions and we prove that the inverse image of a recognizable set under such a transduction is recognizable.