The definitions listed below come from "Ontology Mapping: The State of the Art"
by Yannis Kalfoglou and Marco Schorlemmer
(if we adopt an algebraic approach and present ontologies as logical theories,)
An ontology is then a pair O=(S, A), where S is the (ontologica) signature - desribing the vocabulary - and A is a set of (ontological) axioms - specifying the intended interpretatoin of the vocabulary in some domain of discourse.
- Ongological signature morphisms:
We understand ontology mapping as the task of relating the vocabulary of two ontoloiges that share the same domain of discourse in such a way that the mathematical structure of ontological sugnatures and their intended interpretations, as specified by the ontological axioms, are respected.
structure-preserving mappings between mathematical structures are call morphisms.
the morphisms of ontological signatures is to determine which concept and relation symbols of one ontology are mapped to concept and relation symbols of the other.
total ontology mapping:
A total ontology mapping from O1=(S1,A1) to O2=(S2,A2) is a morphism f: S1->S2 of ontological signatures, such that, A2 = f(A1), i.e. all interpretations that satisfy O2's axioms also satisfy O1's translated axioms. (see also theroy morphism)
partial ontology mapping:
a prtial ontology mapping from O1=(S1, A1) to O2=(S2, A2) if there exists a sub-ontology O'1 =(S'1, A'1) (S'1 belong to S1, and A'1 belong to A1) such that there is a total mapping from O'1 to O2.
- Populated ontoloies:
In a populated ontoloies, classes of an ontological signature come equipped with their respective instances.
- Ontology morphisms,
is a more ambitious and practically necessary approach than ontological signature morphisms. It takes into account how particular ontological axioms are mapped as well. Formally, this would require ontology mappings to be defined in terms of morphisms of ontologies, i.e., signature+ axioms, instead of morphisms of signatures only.
- Ontology alignment, articulation, merging, translation and integration.
Ontology mapping only constitutes a fragment of a more ambitious task concerning the alignment, articulation and merging of ontologies.
An ontology mapping is a morphism, which usually will consist of a collection of functions assigning the symbols used in one vocabulary to the symbols of the other. But two ontologies may be related in a more general fashion, namely by means of relations instead of functions. Hence, ontology alignment is the task of establishing a collection of binary relations betwwen the vocabularies of two ontoloiges.
ontology mapping denotes the process of defining a collection of functions that specify which concepts and relations correspond to which other concepts and relations, wile the ontology translation is the application of the mapping functions to acturally translate the sentences that use the one ontology into the other.
the composition of ontologies to build new ones, but its respective vocabulary are usually not interpreted in the same domain of discourse.
- mapping between the schemas: a set of expressions that specify how the data in one source corresponds to the data in the other.
- marching is typically the firt phase in generating schema mappings.