Abstract
We present a framework for the composition of overlapping objects based on semigroup theory. To do so, we develop a language theory of (labelled) birooted trees, that is, subsets of (extension of) free inverse monoids. In the underlying setting of partial algebras, we define a suitable notion of a syntactic congruence such that (i) having a syntactic congruence of finite index captures \(\mathrm{MSO}\)-definability; (ii) a certain order-bisimulation refinement of the syntactic congruence captures (so called) quasi-recognisability in the same way.
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