Merging Ontologies Algebraically

Ontology operations, e.g., aligning and merging, were studied and implemented extensively in different settings, such as, categorical operations, relation algebras, typed graph grammars, with different concerns. However, aligning and merging operations in the settings share some generic properties, e.g., idempotence, commutativity, associativity, and representativity, labeled by (I), (C), (A), and (R), respectively, which are defined on an ontology merging system (mathfrakO,~,merge), where mathfrakO is a set of the ontologies concerned, ~ is a binary relation on mathfrakO modeling ontology aligning and merge is a partial binary operation on mathfrakO modeling ontology merging. Given an ontology repository, a finite set mathbbOsubseteq mathfrakO, its merging closure widehatmathbbO is the smallest set of ontologies, which contains the repository and is closed with respect to merging. If (I), (C), (A), and (R) are satisfied, then both mathfrakO and widehatmathbbO are partially ordered naturally by merging, widehatmathbbO is finite and can be computed efficiently, including sorting, selecting, and querying some specific elements, e.g., maximal ontologies and minimal ontologies. We also show that the ontology merging system, given by ontology V-alignment pairs and pushouts, satisfies the properties: (I), (C), (A), and (R) so that the merging system is partially ordered and the merging closure of a given repository with respect to pushouts can be computed efficiently.