Ontology layers

The three dimensions Generalization, Aggregation and Particularization shown in Figure layers span what could be considered a kind of knowledge space. Figure layers shows the layers and dimensions of an ontology O. The set of classes is C = {C1, ..., C6}, the set of data properties is A = {A1, ..., A6} and the set of particulars is E = {P1, ..., P7). The class hierarchy (C , ≤_∧) is organized along the Generalization dimension. Particulars p ∈ E lying in the Particulars Layer (M₀ = PL) are connected to their classes in the Schema Layer (M₁ = SL) by an assertion (p, »pof, C). So, particulars are instantiated along the Particularization dimension. Be ≤_∧ and ≤_◊ non-strict, acyclic, transitive ordering relations for classes and object properties. In the following, »subClassOf and »subOpOf have the same meaning as SubClassOf and SubObjectPropertyOf in OWL, and »subModelOf will be used as a subobject property of »subClassOf. We define class and OP hierarchies (see figure o4toptl2) using principal ideals (see Ganter and Wille, [GaWi2024]) as follows: Be x ≤_∧ y ⇔ x, y ∈ C: (x, »subClassOf, y) ∨ (x, »subModelOf, y) such that for 0 < i < j ≤ n it holds C_i ≤_∧ C_j. Then the class hierarchy (C, ≤_∧) has the set of classes ∴C := {c ∈ C | c ≤_∧ C}. The maximum level of (C, ≤_∧) is the maximum distance of an element from the root of the hierarchy. Therefore, in Figure layers the class hierarchy has n = 4 levels and minLevel = 0 and maxLevel = 3. Note, that we only distinguish between two layers SL and PL for each concrete domain ontology, which will be complemented by a third abstraction layer of the O4Top blueprint schema in the next section. However, the number of levels in an inheritance hierarchy is not limited. In analogy to a class hierarchy, a hierarchy of Object Properties (OP) is defined as (R , ≤_◊) and has the set of OPs ∴R := {r ∈ R | r ≤_◊ R} with x ≤_◊ y ⇔ x, y ∈ R: (x, »subOpOf, y). In general, the extension of the OP ◊R is defined as ◊R = {(x, y) | (x, »R , y)}.

Object Properties (OP) / Binary Relationship Types represent extrinsic features of universals, i.e., how things are related to other things. This can, for example be merenomy / parthood relations or family relationships. Object Property Instantiations (OPI) can take place between particulars P and Q in the PL layer. An OPI takes the form (P, »op, Q). The blue arrow from C4 to C2 in figure layers indicates an opd = (C4, ◊op, C2) while the purple arrow indicates an opi = (P4, »op, P2). In a Object Property Definition (OPD) (C, ◊op, D) the class C plays the role of the domain class and D the role of the range class.