Theory - Definitions

Before answering any of the questions raised in the Introduction section, we need to introduce some naming conventions and graphical notations. These are mainly based on the author‘s article [Bens2014].

Let N be the set of strings containing all possible sequences of arbitrary length, including the empty sequence, that can be generated from the characters of a given character set. As our key naming convention, the name of an ontological entity always uses the index of the designating set as its prefix: N_x = {xn | n ∈ N}. To unambiguously denote the names of ontological concepts we introduce the following name sets: N_^ is the set of class names, N_. is the set of data property names, N_◊ is the set of object property Names, N_> is the set of particular names, N_» is the set of relator names, N_° is the set of process names, N_~ is the set of function names, and the union of the sets is N^* = ∪ { N_x | x ∈ { ^, ., ◊, >, », °, ~ }}. In short, the symbols are motivated to evoke associations with hierarchies, properties, processes, things in flux, and references / pointers. This means that the name sets defined in this way form the vocabulary of a meta-language for compact and expressive notations. For example, the RDFS definition of a class ^Car based on our naming conventions is: ^Car rdf:type rdfs:Class.

We represent an ontology by a Knowledge Graph (KG). A Knowledge Graph is defined as KG ⊆ N x N x N. As in RDF each element is a triple of the form (s, p, o) which we call Atomic Knowledge Expression (AKE). First, we define the inverse axiom which we think is very important for a General Ontology Theory because it concerns every triple of a KG. For every property p there exists a property q = p^-1 such that the following axiom holds:

{{definition:InvAx:InvAx = Inverse Axiom:(s, p, o) ∈ KG ⇔ (o, q, s) ∈ KG.}}

Let M be a set and ≤ be a non-strict, acyclic, transitive ordering relation. The pair (M, ≤) is a non-strictly ordered set. A principal filter and a principal ideal are special subsets of a partially ordered set (see Ganter and Wille, [GaWi2024]). We use principal filters to model hierarchies of super-elements symbolized by the ∵ icon, and to model hierarchies of sub-elements with principal ideals, symbolized by the ∴ icon:

{{definition:PF:principal filter:∵τa = {x ∈ M| a ≤_τ x} is the principal filter of τa.}}

{{definition:PI:principal ideal:∴τa = {z ∈ M| z ≤_τ a} is the principal ideal of τa.}}