GFO Part I Basic Principles
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14.5 Parthood Relation

Part-of is a basic relation between certain kinds of entities, and several relations have a similar character.

14.5.1 Abstract and Domain-specific Part-of Relations

The abstract part-of relation is denoted by $\Gp(x,y)$, while the argument-types of this relation are not specified, i.e., we allow arbitrary entities to be arguments. We assume that $\Gp(x,y)$ satisfies the condition of a partial ordering, i.e., the following axioms:

$\Gp(x,x)$, $\Gp(x,y) \wedge \Gp(y,x) \rightarrow x = y$, and $\Gp(x,y) \wedge \Gp(y,z) \rightarrow \Gp(x,z)$.

Domain-specific part-of-relations are related to a particular domain $D$, which might be the set of instances of a category. We denote these relations as $\Gpart_{D}(x,y)$. We assume that for a domain $D$, the entities of $D$ and its parts are determined. There is a large family of domain-specific part-of relations, the most general of these are related to basic categories as $\GChron(x)$, $\GTReg(x)$, $\GTop(x)$, $\GSReg(x)$, $\GMatS(x)$, $\GProc(x)$. In the following sections we provide an overview of the most important category-specific part-of relations.

14.5.2 Part-of Relation for Sets

We hold that the part-of-relation of sets is defined by the set inclusion, hence $\Gpart_{S} (x,y) =_{df} \GSet(x) \wedge \GSet(y) \wedge x \subseteq y $. If we assume the power-set axiom for sets, then the mereology of sets corresponds to the theory of Boolean algebras.

14.5.3 Part-of-Relations for Time and Space

The part-of relations of time and space are related to chronoids, time-regions, topoids, and space regions. We introduce the unary predicates $\GChron(x)$, $\GTReg(x)$, $\GTop(x)$, $\GSReg(x)$, and the binary relations $\Gtpart(x,y)$, $\Gspart(x,y)$.

Every notion of part-of allows for a non-reflexive version of the relationship, which expresses proper parthood. These are denoted by adding a ``p'' to the above predicates, e.g. $\Gpp(x,y)$ or $\Gtppart(x,y)$.

In particular, $\Gspart$ applies to spatial regions, $\Gtpart$ refers to time regions and chronoids, while $\Gcpart$ represents a relationship between situoids (or situations) and their constituents. The constituents of a situoid $s$ include, among other entities, the pertinent material structures (that participate in $s$) and the qualities that inhere in them. Further, facts and configurations are constituents of situoids. Not every part of a constituent of a situoid, however, is contained in it.

14.5.4 Part-of Relation for Material Structures

The basic relations pertaining to material structures are $\GMatS(x)$, for ``$x$ is a material structure'', and $\Gmatpart(x,y)$, which means that the material structure $x$ is a part of the material structure $y$. We assume among the basic axioms:

$\forall x y u v (\GMatS(x) \wedge \Gmatpart(y,x) \wedge
\Gocc(x,u) \wedge \Gocc(y,v) \rightarrow \Gspart(v,u))$

We stipulate that the relation $\Gmatpart(x,y)$ is a partial ordering, but additional axioms depend strongly on the domain under consideration.

14.5.5 Part-of-Relation for Processes

The part-of relation between processes is denoted by $\Gprocpart(x, y)$, meaning that the process $x$ is a processual part of the process $y$. We assume the basic axiom:

$\forall x y (\GProc(x) \wedge \Gprocpart (y,x) \wedge \Gprt(x,u) \wedge
\Gprt(y,v) \rightarrow \Gtpart(v,u)$.

$\Gprt(x,u)$ states that the process $x$ has the temporal extension $u$, or that the process $x$ is temporally projected onto $u$.

Again, we stipulate that the relation $\Gprocpart(x, y)$ is a partial ordering, but additional properties of this relation depend on a concrete domain. For example, in the processes of surgery, only certain processual parts are relevant.

14.5.6 Role-of

The role-of relationship was introduced as a close relative of part-of. It relates roles $x$ and their contexts $y$, denoted by $\Groleof(x,y)$. Thus far we have introduced role-of between processual roles and processes and between relational roles and relators.

Robert Hoehndorf 2006-10-18


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