Object Property Axioms

Object properties link individuals to individuals.

A set of instances connected to the property is called a property extension.

SubObjectPropertyOf axiom

This axiom says that a property 'p1' is a subproperty 'sp1' of another property 'p2'. It also means that the instances of the subproperty are subsets to the property extension of the second property.

Equivalent object property axiom

Equivalent property axioms exist, when two properties have the same property extension.

Inverse object properties axiom

Properties have a direction defined with domain to range, the 'inverseof' term is used to mirror the direction.

This Axiom can equally be used on object properties and data properties.

Disjoint object properties axiom

Two properties are disjoint when they don't have individuals in common.

Functional object properties axiom

A functional property is a property which can only have one value.

For an individual 'x', there can be only one definite individual 'y' such that 'x' is connected by the object property expression 'OPE' to 'y'.

For example:
FunctionalObjectProperty( a:hasHusband ) 	                Each object can have at most one husband.
ObjectPropertyAssertion( a:hasHusband a:Nicole a:Steve ) 	Steve is Nicole's husband.

Inverse functional object properties axiom

For an individual 'x', there can be at most one individual 'y' such that 'y' is connected by the object property expression 'OPE' with 'x'.

InverseFunctionalObjectProperty( a:hasHusband ) 	        Each object can have at most one husband.
ObjectPropertyAssertion( a:hasHusband a:Steve a:Nicole ) 	Steve is Nicole's husband.

Transitive object property axiom

Transitive means that if a property contains the pair(x,y)and the pair (y,z) then we can conclude that the pair(x,z) is also part of der property.

$\begin{align}
            &\forall a,b,c \in A\colon\\
            &(a,b) \in R \,\land\, (b,c) \in R\\
            &\Rightarrow\; (a,c) \in R
        \end{align}$

Symmetric object property axiom

If a property is symmetric it means that if a pair(x,y) exists, then there is also a pair (y,x).

$\begin{align}
            &\forall a,b \in A\colon\\
            &(a,b) \in R \;\Rightarrow\; (b,a) \in R
        \end{align}$

Asymmetric object property axiom

A asymmetric property has a pair(x,y) but never a pair(y,x).

$\begin{align}
            &\forall a,b \in A\colon\\
            & (a,b) \in R \;\Rightarrow\; (b,a) \notin R
        \end{align}$

Reflexive object property axiom

Reflexive means that a property relates to itself.

$\forall a \in A\colon\; (a,a) \in R$

Irreflexive object property axiom

Subsequently irreflexive means that no individual relates to itself.

$\forall a \in A\colon\; (a,a) \notin R$

Object property domain axiom

This axiom states that the domain of the object property expression is the class expression.

Object property range axiom

This axiom states that the range of the object property expression is the class expression.

SubPropertyChainOf axiom

These axioms are used to create a chain of multiple properties. E.g. two 'hasParents' properties are linked by a chain thus a 'hasGrandparents' property will be identified.

Sources

1. https://www.w3.org/TR/owl2-primer/#Object_Properties/
2. https://www.w3.org/TR/owl2-syntax/#Object_Property_Axioms