Section 3.2 of the document you indicated says:
Indeed, it is possible to translate this rule into OWL 2, but this is due to the transformation, which we call the transformation: the birth of the concept of A is the (new) role of R A defined by the axiom A & equiv; & Exists; R <sub> Asub> .Self. Now we can now express rule (1) by the axiom ...
OWL2 does not support the expression of an axiom such as Elephant (x) & wedge; Mouse (y) -> largeThan (x, y) directly. As I understand it, you manually use the folding process, which is described in the article, to create a new axiom that can be directly expressed in OWL2.
Rolling
As for the specific process, if you want to express something like Elephant (x) & wedge; Mouse (y) -> bigThan (x, y), you first advertise the Elephant and Mouse classes. This means that you are introducing new roles (properties) of R Elephant and R Mouse (but you are not deleting the Elephant and Mouse classes). These new roles are such that R Elephant (x, x) if and only if Elephant (x). This is ensured by the addition of axioms
Elephant & equiv; & Exist, R <sub> slonsub. .Self
Mouse & equiv; & Exist, R <sub> mouse sub. .Self
each of which is expressed in OWL2. With these two axioms in hand, you finally add the axiom of a chain of subcategories
R Elephant & bullet; topObjectProperty & bullet; R Mouse & sqsubseteq; biggerThan
which is also expressed in OWL2. Since for any elephant e and any mouse m we have that
R <sub> slonsub> (e, e)
topObjectProperty (e, t)
R <sub> mouse sub> (m, m)
then by the axiom of the chain of subspaces we have that
biggerThan (e, t)
what we wanted to express.
Axiom Syntax
In the input syntax adopted by Protege, these axioms are written as follows.
Mouse Equivalent To R_Mouse Some Self
R?
In Protege they look as follows.
In N3:
@prefix : <http://www.example.org/rolification#> . @prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#> . @prefix owl: <http://www.w3.org/2002/07/owl#> . @prefix xsd: <http://www.w3.org/2001/XMLSchema#> . @prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> . :Elephant a owl:Class ; owl:equivalentClass [ a owl:Restriction ; owl:hasSelf "true"^^xsd:boolean ; owl:onProperty :R_Elephant ] . :R_Elephant a owl:ObjectProperty . :biggerThan a owl:ObjectProperty ; owl:propertyChainAxiom (:R_Elephant owl:topObjectProperty :R_Mouse) . :Mouse a owl:Class ; owl:equivalentClass [ a owl:Restriction ; owl:hasSelf "true"^^xsd:boolean ; owl:onProperty :R_Mouse ] . <http://www.example.org/rolification> a owl:Ontology . :R_Mouse a owl:ObjectProperty .
In RDF / XML:
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:owl="http://www.w3.org/2002/07/owl#" xmlns="http://www.example.org/rolification#" xmlns:xsd="http://www.w3.org/2001/XMLSchema#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#"> <owl:Ontology rdf:about="http://www.example.org/rolification"/> <owl:Class rdf:about="http://www.example.org/rolification#Elephant"> <owl:equivalentClass> <owl:Restriction> <owl:onProperty> <owl:ObjectProperty rdf:about="http://www.example.org/rolification#R_Elephant"/> </owl:onProperty> <owl:hasSelf rdf:datatype="http://www.w3.org/2001/XMLSchema#boolean" >true</owl:hasSelf> </owl:Restriction> </owl:equivalentClass> </owl:Class> <owl:Class rdf:about="http://www.example.org/rolification#Mouse"> <owl:equivalentClass> <owl:Restriction> <owl:onProperty> <owl:ObjectProperty rdf:about="http://www.example.org/rolification#R_Mouse"/> </owl:onProperty> <owl:hasSelf rdf:datatype="http://www.w3.org/2001/XMLSchema#boolean" >true</owl:hasSelf> </owl:Restriction> </owl:equivalentClass> </owl:Class> <owl:ObjectProperty rdf:about="http://www.example.org/rolification#biggerThan"> <owl:propertyChainAxiom rdf:parseType="Collection"> <owl:ObjectProperty rdf:about="http://www.example.org/rolification#R_Elephant"/> <rdf:Description rdf:about="http://www.w3.org/2002/07/owl#topObjectProperty"/> <owl:ObjectProperty rdf:about="http://www.example.org/rolification#R_Mouse"/> </owl:propertyChainAxiom> </owl:ObjectProperty> </rdf:RDF>