  In this paper, we study the geospatial ontologies that we are interested in
together as a geospatial ontology system, consisting of a set of the geospatial
ontologies and a set of geospatial ontology operations, without any internal
details of the geospatial ontologies and their operations being needed,
algebraically. A homomorphism between two geospatial ontology systems is a
function between two sets of geospatial ontologies in the systems, which
preserves the geospatial ontology operations. We view clustering a set of the
ontologies as partitioning the set or defining an equivalence relation on the
set or forming a quotient set of the set or obtaining the surjective image of
the set. Each geospatial ontology system homomorphism can be factored as a
surjective clustering to a quotient space, followed by an embedding. Geospatial
ontology merging systems, natural partial orders on the systems, and geospatial
ontology merging closures in the systems are then transformed under geospatial
ontology system homomorphisms that are given by quotients and embeddings.
