A Unifying Approach to Boundaries and
Multidimensional Mereotopology
Yixin Sun, Michael Grüninger
Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada
Abstract
Mereotopologists generally come from either a point-set or algebraic topology perspective when de-
veloping boundary-based theories. Some involve dimensional aspects while others do not. Little work
has studied whether the existing boundary characterizations convey the same meaning regardless of
the mereotopology they build upon. To address this gap, we introduce a new mathematical theory that
can be used as a unifying framework for boundaries in both multidimensional and nondimensional
mereotopologies. With the proposed unifying framework, we are able to interpret both multidimensional-
and nondimensional- based mereotoplogies.
Keywords
mereology, dimension, boundary, topology, manifold
1. Introduction
The Winograd Schema Challenge (WSC) [1] is an improvement to the Turing test, where a
machine is asked to resolve the reference to a pronoun phrase (anaphora resolution). Many
questions within WSC refer to spatial relations among parts, parts of parts and the boundaries
between parts. For example [2]:
There is a gap in the wall. You can see the garden [through/behind] it. You can see
the garden [through/behind] what?
Answers: The gap/the wall.
To approach this specific anaphora resolution, one must understand the intended semantics
behind the spatial preposition, “through". In particular, the intended semantics can be cap-
tured through formalizing in terms of mereotopology: a theory manifesting mereological and
topological classes and relations[3].
It is also worth characterizing two-dimensional lines, paths and cycles–exemplified in the
following oil painting done by Wassily Kandinsky. Different from perfect geometric shapes,
this composite of zero-, one-, and two-dimensional entities is what appears in reality. We want
to investigate the capabilities of existing mereotopologies in characterizing such “imperfect"
spaces.
Workshop on Geospatial Ontologies 2023: 9th Joint Ontology Workshops (JOWO 2023), co-located with FOIS 2023, 19-20
July, 2023, Sherbrooke, Québec, Canada
" yixin.sun@mail.utoronto.ca (Y. Sun); gruninger@mie.utoronto.ca (M. Grüninger)
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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Figure 1: Wassily Kandinsky[4], Composition 8, 1923.
To answer the questions posed by these two motivating scenarios, we need to closely examine
the notion of boundary. Little study has successfully formalized the notion of crossing a
boundary, inclusive of all dimensions. In the WSC sentence pair, the preposition “through"[5]
indicates a movement that goes into at one side and out at another. Hence, we need to study
crossing the boundary in terms of a three-dimensional region. On the other hand, the depiction
of overlapped points, lines, and cycles in Kandinsky’s oil painting crosses both boundaries
and dimensions. This brings our attention to the representation of boundaries across one and
multiple dimensions.
The formalization of boundary has long been a topic of discussion in mereotopology.
Boundary-agnostic theories have been excluded in this work for two reasons[3]. First, ig-
noring boundaries contradict the topological distinctions between open and closed entities.
Next, in the absence of boundaries, an entity can be connected to its complement. The omission
of boundary-less approaches leaves us with two directions for boundary-based approaches.
Previous work has taken either a multi-dimensional or non-dimensional perspective. Smith’s
mereotopology[6] ignores dimension. His axiomatization arises from point-set topology in
which he uses the relation 𝑠𝑡𝑟𝑎𝑑𝑑𝑙𝑒(𝑥, 𝑦), defined by a primitive relation 𝐼𝑃 (𝑥, 𝑦) (x is an
interior part of y). He then takes the sum of all boundary points to form a complete set of bound-
ary. The advocates for multidimensional characterizations come from an algebraic topology
perspective; led by Gotts[7] and Galton[8], they take the boundary entity to be one dimension
lower than the object it is attached to. Galton’s theory defines the notion of boundary as a prim-
itive relation. Gott’s establishment in INCH Calculus is examined by Hahmann[9]. Hahmann’s
mereotopology – CODI – is definably equivalent to the corrected INCH Calculus.
1.1. Main Contributions
The various formalization regarding the notion of boundary best showcases disagreement
behind characterizing spatial entities among mereotopologies. Few studies have focused on
comparing and harmonizing different perspectives. The main problem we are solving is to
find a minimal theory to axiomatize the notion of dimension for elements of a mereology. We
introduce the notion of a multimereology, which amalgamates a mereology with an incidence
structure to partition partial orderings.
The paper is organized as follows. Section 2 introduces the root theories of multimereology.
Inspired by the notion of topological manifolds, Section 3 is initiated from a combinatorial
topology standpoint to examine the comprehensiveness of possible multimereology candidates
against certain use cases. Section 4 showcases a successful application of our proposed mul-
timereology to be logical equivalent to CODI, a multidimensional mereotopology. Section 5
returns to the motivating question of boundary classification and unification.
2. Structures for Multidimensional Mereologies
In designing an ontology, our objective is twofold – first, to prove that the models of the
ontology are actually the intended models, and second, to demonstrate that the intended models
do indeed formalize the ontological commitments. Our strategy is to first specify a class of
mathematical structures and show that the ontology axiomatizes this class of structures (that
is, there is a one-to-one correspondence between the class of models of the ontology and the
class of mathematical structures). We then specify a representation theorem for this class
of mathematical structures to demonstrate that it formalizes the ontological commitments.
The primary benefit of this strategy is that it makes explicit the modular organization of the
subtheories of the ontology, thereby highlighting how other ontologies are reused.
2.1. Multimereologies
The first step is to specify the class of structures which capture the following intuitions and
ontological commitments about multidimensional mereology:
1. Each entity has a unique dimension.
2. Dimensions are linearly ordered.
3. There is a parthood relation that is specified on entities with the same dimension.
4. Incidence corresponds to a multidimensional ordering.
To formalize these semantic requirements for a multidimensional mereology, we introduce
the notion of a multimereology as the amalgamation of a partial ordering with an incidence
structure. Incidence structures are a generalization of geometry, first introduced by Gino Fano
and later used by Hilbert in his axiomatization of Euclidean geometry.
Definition 1. A k-partite incidence structure⋃︀ is a tuple I = ⟨Ω1 , ..., Ω𝑘 , I⟩, where Ω1 , ..., Ω𝑘 are
sets such that Ω𝑖 ∩ Ω𝑗 = ∅, 𝑖 ̸= 𝑗 and I ⊆ ( 𝑖̸=𝑗 Ω𝑖 × Ω𝑗 ).
Two elements of I that are related by I are called incident. The neighbourhood of an element is
the set of elements which are incident with it: 𝑁 (x) = {y : ⟨x, y⟩ ∈ I}
Incidence structures are ideally suited to represent the dimensionality of different entities. In a
tripartite incidence structure, points can be thought of as 0-dimensional, lines as 1-dimensional,
and planes as 2-dimensional.
How is the incidence structure in a multimereology related to the partial ordering? The first
approach to this question is to consider incidence to itself be a parthood relation, and enforce
the transitivity of the incidence relation:
Definition 2. A tripartite incidence structure I = ⟨𝑃, 𝐿, 𝑄, I⟩ is transitive iff
I∘I⊆I
At first glance, this might seem unusual since the incidence relation is symmetric; we therefore
use the following notion from graph theory which allows us to associate a transitive symmetric
incidence relation with a partial ordering:
Definition 3. The comparability graph for a partial ordering Q = ⟨𝑀, ≤⟩ is a graph GQ =
⟨𝑉, E⟩ such that
(x, y) ∈ E ⇔ x ≤ y or y ≤ x
A second alternative approach is to start with the “global" partial ordering that applies to all
entities, and then use the incidence structure to specify “local" partial orderings on the sets of
points, lines, and planes. We therefore need to specify how the suborderings on each of these
sets is related to the entire partial ordering.
Definition 4. The upper set for x in a poset P, denoted by 𝑈 P (x), is
𝑈 P (x) = {y : x ≤ y}
The lower set for x in a poset P, denoted by 𝐿P (x), is
𝐿P (x) = {y : y ≤ x}
With the aid of this terminology, we can impose the following conditions. The set of points
forms a lower set in the partial ordering, so that all parts of points are points. The set of planes
forms an upper set in the partial ordering, so that all elements that contain planes as parts are
also planes. The set of lines forms an interval within the partial ordering. Alternatively, we can
think of the incidence structure as partitioning the elements of the partial ordering into the
disjoint classes of points, lines, and planes. Both of these perspectives will be used as the basis
for the formalization.
Definition 5. Q ⊕ I 1 is a multimereology iff
1
The ⊕ symbol denotes the amalgamation of structures [10].
12345
1234 2345 1234 12345 2345
123 234 345
12 123 23 234 34 345 45
12 23 34 45
1 2 3 4 5 1 2 3 4 5
(a) (b)
Figure 2: Examples of multimereologies.
1. Q = ⟨𝑃 ∪ 𝐿 ∪ 𝑄, ≤⟩ such that Q ∈ M𝑝𝑎𝑟𝑡𝑖𝑎𝑙_𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 ;
2. I = ⟨𝑃, 𝐿, 𝑄, I⟩ such that I ∈ M𝑐𝑙𝑜𝑠𝑒𝑑_𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒_𝑡𝑟𝑖𝑝𝑎𝑟𝑡𝑖𝑡𝑒 ;
3. ⟨𝑃 ∪ 𝐿 ∪ 𝑄, I⟩ ⊆ GQ ;
4. if x ∈ 𝑃 , then 𝐿Q (x) ⊆ 𝑃 ;
5. if x, y ∈ 𝐿, then 𝑈 Q (x) ∩ 𝐿Q [y] ⊆ 𝐿;
6. if x ∈ 𝑄, then 𝑈 Q (x) ⊆ 𝑄.
Examples of multimereologies can be seen in Figure 2. Figure 2(a) shows a mereology that
has been partitioned into three disjoint sets containing equidimensional elements, while Figure
2(b) is the corresponding tripartite incidence structure. In this example, the element 34 is an
equidimensional part of the element 345, and the element 2345 is an equidimensional part of
the element 12345. On the other hand, since 345 is a line and 2345 is a plane in the incidence
structure, they are related by incidence, even though in the “global" mereology they are related
by parthood.
2.1.1. Representation Theorem
Intuitively, each partitioning of a partial ordering corresponds to a unique multimereology, and
each multimereology can be used to specify a partitioning of a partial ordering. This intuition
forms the basis for the representation theorem for multimereologies. We begin by formalizing
the idea of partitioning.
Definition 6. Suppose Q, P ∈ M𝑝𝑎𝑟𝑡𝑖𝑎𝑙_𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 such that Q = ⟨𝑉1 , ≤⟩, P = ⟨𝑉2 , ⪯⟩.
A mapping 𝜇 : Q → P is a poset homomorphism iff
x ≤ y ⇒ 𝜇(x) ⪯ 𝜇(y)
For poset homomorphism, we want to explicitly identify the equivalence class of elements of
Q that all map to the same element of P:
Definition 7. Suppose the mapping 𝜇 : Q → P is a poset homomorphism.
𝑆 𝜇 (x) = {y : 𝜇(x) = 𝜇(y)}
For example, in Figure 2(a),
𝑆 𝜇 (12) = {12, 23, 34, 45, 123, 234, 345}
The idea is that all elements with the same dimension are in the same equivalence class. One of
the semantic requirements for multimereologies is that there be a linear ordering on dimensions;
if we consider the poset homomorphism to be the mapping from elements to their dimension,
then we are particularly interested in the set of poset homomorphisms between partial orderings
and finite linear orderings.
Theorem 1. Let 𝐻𝑜𝑚(M𝑝𝑎𝑟𝑡𝑖𝑎𝑙_𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 , 3) denote the set of all poset homomorphisms between
partial orderings and the finite linear ordering with 3 elements.
There is a bijection: 𝜙 : 𝐻𝑜𝑚(M𝑝𝑎𝑟𝑡𝑖𝑎𝑙_𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 , 3) → M𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜𝑙𝑜𝑔𝑦 such that for any
Q ∈ M𝑝𝑎𝑟𝑡𝑖𝑎𝑙_𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 , 𝜙(𝜇) = Q ⊕ I iff 𝜇 : Q → 3 and
𝑁 I (x) = ((𝐿Q [x] ∪ 𝑈 Q [x]) ∖ 𝑆 𝜇 (x))
In other words, each multimereology corresponds to a poset homomorphism between a
partial ordering and a linear ordering, and any poset homomorphism between a partial ordering
and a linear ordering can be used to construct a multimereology.
In Figure 2(a), all elements of the partial ordering that are points in the incidence structure are
exactly the elements that are mapped to the minimum element of the linear ordering. Similarly,
all elements of the partial ordering that are planes in the incidence structure are exactly the
elements that are mapped to the maximum element of the linear ordering.
2.1.2. Axiomatization
𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 can be found in Figure 2.1.2. The first three axioms are for partial orderings
(condition (1) in Definition 5) axioms (4)-(10) are for closed transitive tripartite incidence
structures (condition (2) in Definition 5). Axiom (11) axiomatizes condition (3) in Definition 5.
Finally, axioms (12)-(14) axiomatize conditions (4),(5), and (6) respectively in Definition 5.
Theorem 2. There exists a bijection 𝜙 : 𝑀 𝑜𝑑(𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 )2 → M3𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 such that
1. ⟨x, y⟩ ∈ partℳ4 iff x ∈ 𝐿Q [y].
2. ⟨x, y⟩ ∈ inℳ iff x ∈ 𝑁 I [y].
2
Mod(.) denotes the class of models of the given ontology.
3
M denotes a class of mathematical structures.
4
ℳ is a specific model
∀𝑥 𝑝𝑎𝑟𝑡(𝑥, 𝑥) (1)
∀𝑥, 𝑦 𝑝𝑎𝑟𝑡(𝑥, 𝑦) ∧ 𝑝𝑎𝑟𝑡(𝑦, 𝑥) ⊃ (𝑥 = 𝑦) (2)
∀𝑥, 𝑦, 𝑧 𝑝𝑎𝑟𝑡(𝑥, 𝑦) ∧ 𝑝𝑎𝑟𝑡(𝑦, 𝑧) ⊃ 𝑝𝑎𝑟𝑡(𝑥, 𝑧) (3)
∀𝑥 𝑖𝑛(𝑥, 𝑥) (4)
∀𝑥, 𝑦 𝑖𝑛(𝑥, 𝑦) ⊃ 𝑖𝑛(𝑦, 𝑥) (5)
∀𝑥, 𝑦 𝑝𝑜𝑖𝑛𝑡(𝑥) ∧ 𝑝𝑜𝑖𝑛𝑡(𝑥) ∧ 𝑖𝑛(𝑥, 𝑦) ⊃ (𝑥 = 𝑦) (6)
∀𝑥, 𝑦 𝑙𝑖𝑛𝑒(𝑥) ∧ 𝑙𝑖𝑛𝑒(𝑥) ∧ 𝑖𝑛(𝑥, 𝑦) ⊃ (𝑥 = 𝑦) (7)
∀𝑥, 𝑦 𝑝𝑙𝑎𝑛𝑒(𝑥) ∧ 𝑝𝑙𝑎𝑛𝑒(𝑦) ∧ 𝑖𝑛(𝑥, 𝑦) ⊃ (𝑥 = 𝑦) (8)
∀𝑥, 𝑦, 𝑧 𝑖𝑛(𝑥, 𝑦) ∧ 𝑖𝑛(𝑦, 𝑧) ⊃ 𝑖𝑛(𝑥, 𝑧) (9)
∀𝑥 (𝑝𝑜𝑖𝑛𝑡(𝑥) ∨ 𝑙𝑖𝑛𝑒(𝑥) ∨ 𝑝𝑙𝑎𝑛𝑒(𝑥)) (10)
∀𝑥, 𝑦 (𝑖𝑛(𝑥, 𝑦) ⊃ (𝑝𝑎𝑟𝑡(𝑥, 𝑦) ∨ 𝑝𝑎𝑟𝑡(𝑦, 𝑥))) (11)
∀𝑥, 𝑦 ((𝑝𝑜𝑖𝑛𝑡(𝑥) ∧ 𝑝𝑎𝑟𝑡(𝑦, 𝑥)) ⊃ 𝑝𝑜𝑖𝑛𝑡(𝑦)) (12)
∀𝑥, 𝑦, 𝑧 ((𝑙𝑖𝑛𝑒(𝑥) ∧ 𝑙𝑖𝑛𝑒(𝑦) ∧ 𝑝𝑎𝑟𝑡(𝑥, 𝑧) ∧ 𝑝𝑎𝑟𝑡(𝑧, 𝑦)) ⊃ 𝑙𝑖𝑛𝑒(𝑧)) (13)
∀𝑥, 𝑦 ((𝑝𝑙𝑎𝑛𝑒(𝑥) ∧ 𝑝𝑎𝑟𝑡(𝑥, 𝑦)) ⊃ 𝑝𝑙𝑎𝑛𝑒(𝑦)) (14)
∀𝑥, 𝑦 (𝑒𝑞𝑝𝑎𝑟𝑡(𝑥, 𝑦) ≡ (15)
((𝑝𝑜𝑖𝑛𝑡(𝑥) ∧ 𝑝𝑜𝑖𝑛𝑡(𝑦) ∧ 𝑝𝑎𝑟𝑡(𝑥, 𝑦))∨
(𝑙𝑖𝑛𝑒(𝑥) ∧ 𝑙𝑖𝑛𝑒(𝑦) ∧ 𝑝𝑎𝑟𝑡(𝑥, 𝑦)) ∨ (𝑝𝑙𝑎𝑛𝑒(𝑥) ∧ 𝑝𝑙𝑎𝑛𝑒(𝑦) ∧ 𝑝𝑎𝑟𝑡(𝑥, 𝑦))))
Figure 3: 𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 : Axiomatization of multimereology.
3. ⟨x, y⟩ ∈ eqpartℳ iff x ∈ 𝐿⟨𝑃,≤⟩ [y] ∪ 𝐿⟨𝐿,≤⟩ [y] ∪ 𝐿⟨𝑄,≤⟩ [y].
Theorem 2 together with Theorem 1 show that we have characterized the models of
𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 up to isomorphism as the class of multimereologies. Moreover, we can easily
see how multimereologies satisfy the semantic requirements. The incidence structure guar-
antees that elements are assigned a unique dimension and that the dimensions are linearly
ordered. By partitioning the mereology into intervals of elements with the same dimension,
we have both a parthood relation that is specified on entities with the same dimension., and a
multidimensional parthood relation.
Although any partial ordering can be amalgamated with an incidence structure to construct
a multimereology, two questions arise:
1. What is the appropriate partial ordering to represent the “global mereology"?
2. What criteria do we need to impose on the partitioning of the partial ordering into “local"
equidimensional mereologies?
To answer these questions, we turn to the field of algebraic topology and the notion of
topological manifold.
123 234
3 4
12 13 23 24 34
1 2
1 2 3 4
(a) (b)
Figure 4: Simplicial complex associated with a graph.
3. Mereology and Topological Manifolds
Mathematicians have long used the notion of manifold to formalize the topology of shape
and space. Of key interest for this paper is that topological manifolds provide an explicit
notion of dimension – an n-dimensional manifold is a topological space in which every point
has a neighbourhood that is homeomorphic to R𝑛 . In this section, we explore how different
combinatorial structures for topological manifolds are related to multimereologies, and arrive
at a suitable generalization of both that can serve as the basis for the framework that we seek
to unify the nondimensional and multidimensional approaches to mereologies.
3.1. Simplicial Complexes
One combinatorial approach to the notion of manifold can be found in the notion of an abstract
simplicial complex, which is a collection of finite sets that is closed under taking subsets:
Definition 8. A family of sets Δ is called an abstract simplicial complex if, for every set X in Δ
and every non-empty subset 𝑌 ⊆ 𝑋, we have 𝑌 ∈ Δ.
An example of an abstract simplicial complex can be found in Figure 4(b). The dimension of
an element in an abstract simplicial complex is equal to one less than the cardinality of the set
corresponding to the element. Thus, the dimension of the element 23 is one and the dimension
of the element 123 is two. A simplicial 𝑛-complex is a simplicial complex in which the maximal
dimension of an element in 𝑛.
Interestingly, we can associate a mereology with any abstract simplicial complex. Recall
that classical extensional mereology 𝑇𝑐𝑒𝑚_𝑚𝑒𝑟𝑒𝑜𝑙𝑜𝑔𝑦 is the axiomatization of classical mereology
together with the Strong Supplementation Principle5 . Let 𝑇2𝐷𝑐𝑒𝑚_𝑚𝑒𝑟𝑒𝑜𝑙𝑜𝑔𝑦 be the extension
of 𝑇𝑐𝑒𝑚_𝑚𝑒𝑟𝑒𝑜𝑙𝑜𝑔𝑦 in which all models have rank 3 (i.e. all maximal chains in the model have
cardinality 3).
Theorem 3. 𝑇2𝐷𝑐𝑒𝑚_𝑚𝑒𝑟𝑒𝑜𝑙𝑜𝑔𝑦 is logically synonymous [10] to 𝑇𝑠𝑖𝑚𝑝𝑙𝑖𝑐𝑖𝑎𝑙_2𝑐𝑜𝑚𝑝𝑙𝑒𝑥 6 .
5
https://github.com/gruninger/colore/blob/master/ontologies/mereology/cem_mereology.clif
6
https://github.com/gruninger/colore/blob/master/ontologies/tripartite_incidence/simplicial_2complex.clif
In other words, there is a one-one-correspondence between simplicial 2-complexes and models
of 𝑇2𝐷𝑐𝑒𝑚_𝑚𝑒𝑟𝑒𝑜𝑙𝑜𝑔𝑦 . In this sense, classical extensional mereology is the natural mereology for
simplicial complexes. Nevertheless, even though we can associate a mereology with simplicial
complexes, this approach falls short because it is trivial on equidimensional elements – no
chain in the mereology contains elements with the same dimension. On the other hand,
multimereologies allow models in which there is a nontrivial mereology on equidimensional
elements.
3.2. CW Complexes
An alternative approach to topological manifolds, from a combinatorial point of view, is the
notion of a CW-complex, and we can ask how it can be related to mereologies. To start, we
adopt Hatcher’s definition [11]:
Definition 9. CW complex is a space 𝑋 constructed in the following way:
1. Start with a discrete set 𝑋 0 , the 0-cell of 𝑋.
2. Inductively, form the n-skeleton 𝑋 𝑛 from 𝑋 𝑛−1 by attaching n-cells 𝑒⨆︀𝛼𝑛 via maps 𝜙𝛼 :
𝑆 𝑛−1 → 𝑋 𝑛−1 . This means that 𝑋 𝑛 is the quotient space of 𝑋 𝑛−1 𝛼 𝐷𝛼𝑛 under the
identifications 𝑥 ∼ 𝜙𝛼 (𝑥) for 𝑥 ∈ 𝛿𝐷𝛼𝑛 . The cell 𝑒𝑛𝛼 is the homeomorphic image of 𝐷𝛼𝑛 −𝛿𝐷𝛼𝑛 .
3. 𝑋 = 𝑛 𝑋 𝑛 with the weak topology: A set 𝐴 ⊂ 𝑋 is open (or closed) iff 𝐴 ∩ 𝑋 𝑛 is open
⋃︀
(or closed) in 𝑋 𝑛 for each 𝑛.
A CW complex is constructed by induction via a cell attachment process. Simply put, each
cell is attached to the existing one at its boundary in increasing order of dimensions. The
connected graph in Figure 5(a) is the 1-skeleton of a CW complex. Figure 5(b) is the mereological
representation corresponding to the construct of a CW complex. Since the 1-cells in the
construction of a CW-complex are edges, the notion of “a path in a graph", like that of 123 in
Figure 5(a), does not exist in the context of a CW complex. Nonetheless, in multimereology, we
indeed want to differentiate between an edge and a path. Even though a path is homeomorphic
to an edge, we want to treat paths and edges as two separate entities that are of the same
dimension.
In this sense, CW complexes suffer from the same drawback as simplicial complexes on
non-trivially representing equidimensional elements. Even though we need a more general
structure than CW complexes to formalize mereologies and dimensionality, we still obtain the
key insight that each 1-complex is a connected graph. Therefore, the associated mereology
should correspond to the connected induced subgraphs of the graph that is the 1-complex.
3.3. Connected Induced Subgraph Containment Orderings
In the mereotopology 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 [12], the sum of two elements exists iff they are connected.
A weaker mereotopology 𝑇𝑤𝑒𝑎𝑘_𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 alllows connected elements that do not have sums,
although elements for which sums do exist must be connected. In a model of 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 , there
is one-to-one correspondence between elements in the mereology and connected induced
1234
3 4 5
2 1 12 14 23 34 45
1 2 4 3 5
(a) (b)
Figure 5: CW complex associated with a graph
12345 12345
1234 1234
1245 1345 2345 1245 1345 2345
124 123 134 145 234 345 124 123 134 145 234 345
12 14 23 34 45 12 14 23 34 45
1 2 4 3 5 1 2 4 3 5
(a) (b)
Figure 6: CISCO mereology representation and an associated example of partitioning the mereology
subgraphs of a connected graph, and the parthood relation is isomorphic to the containment
ordering on the set of subgraphs.
Furthermore, many graph-theoretic properties are definable within the mereotopology. Paths
in the graphs correspond to minimal upper bounds in the mereology and cycles in the graph
correspond to triangles within the connection structure of the mereotopology. The identification
of paths and cycles within an underlying graph is precisely the way in which we can use 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡
as the mereology within a multimereology that extends the structure of a CW-complex. Even
though 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 axiomatizes a mereology which is nondimensional, it is able to distinguish
among elements which in the context of CW-complexes have different dimensions. On the
other hand, 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 also axiomatizes the mereology among elements with the same dimension,
such as subpaths of paths and cycles which are subgraphs of 2-complexes.
Figure 6(a) showcases CISCO structure in representation of a 2-connected graph. We can
non-trivially partition this given mereology into Figure 6(b). Element 124 is incident to 1234,
and it is an equi-dimensional part of 1245.
An application of a multimereology based on 𝑇𝑐𝑖𝑠𝑐𝑜 can be seen in the Molecular Structure
Ontology (MoSt) [13]. MoSt is an ontology to describe the shape of a molecule; it treats
functional groups, that are composed of rings and chains, to be incident to skeletons, while
Corresponding Skeletal Diagrams
Breakdown of Skeletons & Groups
s4 s4 CH 3
H2
AAACFXicdZBNS8MwGMfT+TbnW9Wjl+AQPIzRynw5Drx4nOBeoCslzdItLE1Lkgqj9Et48at48aCIV8Gb38Z0a0VFHwj883te8uTvx4xKZVkfRmVpeWV1rbpe29jc2t4xd/d6MkoEJl0csUgMfCQJo5x0FVWMDGJBUOgz0venl3m+f0uEpBG/UbOYuCEacxpQjJRGntlIh/Mhjhj7bmo1T608GlbTKsWCZNJrZZ5ZL++wrIBfxC5IHRTR8cz34SjCSUi4wgxJ6dhWrNwUCUUxI1ltmEgSIzxFY+JoyVFIpJvOF8rgkSYjGERCH67gnH7vSFEo5Sz0dWWI1ET+zuXwr5yTqODCTSmPE0U4XjwUJAyqCOYWwREVBCs20wJhQfWuEE+QQFhpI2vahPKn8H/RO2naVtO+btXbZ4UdVXAADsExsME5aIMr0AFdgMEdeABP4Nm4Nx6NF+N1UVoxip598COMt0+jQ5qh
AAACFXicdZBNS8MwGMfT+TbnW9Wjl+AQPIzRynw5Drx4nOBeoCslzdItLE1Lkgqj9Et48at48aCIV8Gb38Z0a0VFHwj883te8uTvx4xKZVkfRmVpeWV1rbpe29jc2t4xd/d6MkoEJl0csUgMfCQJo5x0FVWMDGJBUOgz0venl3m+f0uEpBG/UbOYuCEacxpQjJRGntlIh/Mhjhj7bmo1T608GlbTKsWCZNJrZZ5ZL++wrIBfxC5IHRTR8cz34SjCSUi4wgxJ6dhWrNwUCUUxI1ltmEgSIzxFY+JoyVFIpJvOF8rgkSYjGERCH67gnH7vSFEo5Sz0dWWI1ET+zuXwr5yTqODCTSmPE0U4XjwUJAyqCOYWwREVBCs20wJhQfWuEE+QQFhpI2vahPKn8H/RO2naVtO+btXbZ4UdVXAADsExsME5aIMr0AFdgMEdeABP4Nm4Nx6NF+N1UVoxip598COMt0+jQ5qh
C C
H3 C O O
skeletons
s3
AAACFXicdVBLS8NAGNz4rPUV9ehlsQgeSkh8VL0VvHisYB+QhrDZbtqlmwe7G6GE/Akv/hUvHhTxKnjz37hJ06KiAwvDzHy7344XMyqkaX5qC4tLyyurlbXq+sbm1ra+s9sRUcIxaeOIRbznIUEYDUlbUslIL+YEBR4jXW98lfvdO8IFjcJbOYmJE6BhSH2KkVSSq9fTfnGJzYeek5rGmWldNqy6aZgF5iQT7knm6rU8kQPODDhXrFKpgRItV//oDyKcBCSUmCEhbMuMpZMiLilmJKv2E0FihMdoSGxFQxQQ4aTFQhk8VMoA+hFXJ5SwUL9PpCgQYhJ4KhkgORK/vVz8y7MT6V84KQ3jRJIQTx/yEwZlBPOK4IBygiWbKIIwp2pXiEeIIyxVkVVVwuyn8H/SOTYs07BuTmvNRllHBeyDA3AELHAOmuAatEAbYHAPHsEzeNEetCftVXubRhe0cmYP/ID2/gW1Xpqs
s2 H2
CH 3
s1 g4 H2
CH 3
s3
AAACFXicdVBLS8NAGNz4rPUV9ehlsQgeSkh8VL0VvHisYB+QhrDZbtqlmwe7G6GE/Akv/hUvHhTxKnjz37hJ06KiAwvDzHy7344XMyqkaX5qC4tLyyurlbXq+sbm1ra+s9sRUcIxaeOIRbznIUEYDUlbUslIL+YEBR4jXW98lfvdO8IFjcJbOYmJE6BhSH2KkVSSq9fTfnGJzYeek5rGmWldNqy6aZgF5iQT7knm6rU8kQPODDhXrFKpgRItV//oDyKcBCSUmCEhbMuMpZMiLilmJKv2E0FihMdoSGxFQxQQ4aTFQhk8VMoA+hFXJ5SwUL9PpCgQYhJ4KhkgORK/vVz8y7MT6V84KQ3jRJIQTx/yEwZlBPOK4IBygiWbKIIwp2pXiEeIIyxVkVVVwuyn8H/SOTYs07BuTmvNRllHBeyDA3AELHAOmuAatEAbYHAPHsEzeNEetCftVXubRhe0cmYP/ID2/gW1Xpqs
s2 s1
AAACFXicbVDNS8MwHE39nPOr6tFLcAgexmhlqMeBF48T3Ad0paRZ2oWlaUlSYZT+E178V7x4UMSr4M3/xqzrRDcfBB7vvV/yy/MTRqWyrC9jZXVtfWOzslXd3tnd2zcPDrsyTgUmHRyzWPR9JAmjnHQUVYz0E0FQ5DPS88fXU793T4SkMb9Tk4S4EQo5DShGSkueWc8GxSWOCH03sxpWgfoPsUuSh14z98za3IDLZB6tgRJtz/wcDGOcRoQrzJCUjm0lys2QUBQzklcHqSQJwmMUEkdTjiIi3axYKIenWhnCIBb6cAUL9fdEhiIpJ5GvkxFSI7noTcX/PCdVwZWbUZ6kinA8eyhIGVQxnFYEh1QQrNhEE4QF1btCPEICYaWLrOoS7MUvL5PuecO2GvZts9a6KOuogGNwAs6ADS5BC9yANugADB7AE3gBr8aj8Wy8Ge+z6IpRzhyBPzA+vgF0B5qC
C C
AAACFXicbVBNS8MwGE79nPOr6tFLcQgexmiGTr0NvHic4D6gKyXN0i0sTUuSCqP0T3jxr3jxoIhXwZv/xqyrMDcfCDx5nvd98+bxY0alsu1vY2V1bX1js7RV3t7Z3ds3Dw47MkoEJm0csUj0fCQJo5y0FVWM9GJBUOgz0vXHN1O/+0CEpBG/V5OYuCEachpQjJSWPLOa9vMhjhj6bmrXLmx43YDVOWLnyKRXzzyz8nu3lgksSAUUaHnmV38Q4SQkXGGGpHSgHSs3RUJRzEhW7ieSxAiP0ZA4mnIUEumm+UKZdaqVgRVEQh+urFyd70hRKOUk9HVliNRILnpT8T/PSVRw5aaUx4kiHM8eChJmqciaRmQNqCBYsYkmCAuqd7XwCAmElQ6yrEOAi19eJp16Ddo1eHdeaTaKOErgGJyAMwDBJWiCW9ACbYDBI3gGr+DNeDJejHfjY1a6YhQ9R+APjM8fyAeatw== AAACFXicdVDLSsNAFL2pr1pfUZduBovgooREfC0LblxWsA9IQ5hMJ+3QyYOZiVBCf8KNv+LGhSJuBXf+jdO0ARU9MHA459w7lxOknEll259GZWl5ZXWtul7b2Nza3jF39zoyyQShbZLwRPQCLClnMW0rpjjtpYLiKOC0G4yvZn73jgrJkvhWTVLqRXgYs5ARrLTkm428XyxxxTDwcseyCzRs66wkc2UqfWfqm/UygcoEKhOotOqwQMs3P/qDhGQRjRXhWErXsVPl5VgoRjid1vqZpCkmYzykrqYxjqj08uKgKTrSygCFidAvVqhQv0/kOJJyEgU6GWE1kr+9mfiX52YqvPRyFqeZojGZfxRmHKkEzSpCAyYoUXyiCSaC6VsRGWGBidJF1r6X8D/pnFiObTk3p/Xm+aKOKhzAIRyDAxfQhGtoQRsI3MMjPMOL8WA8Ga/G2zxaMRYz+/ADxvsXkpqalg==
C C
AAACFXicbVBNS8MwGE79nPOr6tFLcQgexmiGTr0NvHic4D6gKyXN0i0sTUuSCqP0T3jxr3jxoIhXwZv/xqyrMDcfCDx5nvd98+bxY0alsu1vY2V1bX1js7RV3t7Z3ds3Dw47MkoEJm0csUj0fCQJo5y0FVWM9GJBUOgz0vXHN1O/+0CEpBG/V5OYuCEachpQjJSWPLOa9vMhjhj6bmrXLmx43YDVOWLnyKRXzzyz8nu3lgksSAUUaHnmV38Q4SQkXGGGpHSgHSs3RUJRzEhW7ieSxAiP0ZA4mnIUEumm+UKZdaqVgRVEQh+urFyd70hRKOUk9HVliNRILnpT8T/PSVRw5aaUx4kiHM8eChJmqciaRmQNqCBYsYkmCAuqd7XwCAmElQ6yrEOAi19eJp16Ddo1eHdeaTaKOErgGJyAMwDBJWiCW9ACbYDBI3gGr+DNeDJejHfjY1a6YhQ9R+APjM8fyAeatw== AAACFXicdVDLSsNAFL2pr1pfUZduBovgooREfC0LblxWsA9IQ5hMJ+3QyYOZiVBCf8KNv+LGhSJuBXf+jdO0ARU9MHA459w7lxOknEll259GZWl5ZXWtul7b2Nza3jF39zoyyQShbZLwRPQCLClnMW0rpjjtpYLiKOC0G4yvZn73jgrJkvhWTVLqRXgYs5ARrLTkm428XyxxxTDwcseyCzRs66wkc2UqfWfqm/UygcoEKhOotOqwQMs3P/qDhGQRjRXhWErXsVPl5VgoRjid1vqZpCkmYzykrqYxjqj08uKgKTrSygCFidAvVqhQv0/kOJJyEgU6GWE1kr+9mfiX52YqvPRyFqeZojGZfxRmHKkEzSpCAyYoUXyiCSaC6VsRGWGBidJF1r6X8D/pnFiObTk3p/Xm+aKOKhzAIRyDAxfQhGtoQRsI3MMjPMOL8WA8Ga/G2zxaMRYz+/ADxvsXkpqalg==
H3 C O O H3 C O O
g1
g1
AAACFXicbVDLSsNAFL3xWesr6tJNsAguSklE1GXBjcsK9gFpCJPpJB06mYSZiVBCfsKNv+LGhSJuBXf+jdO0BW29MHDmnHvu3DlByqhUtv1trKyurW9sVraq2zu7e/vmwWFHJpnApI0TlohegCRhlJO2ooqRXioIigNGusHoZqJ3H4iQNOH3apwSL0YRpyHFSGnKN+t5vxziiijwcqdhl1W3F0ER+U7hm7X53VoGc3MNZtXyza/+IMFZTLjCDEnpOnaqvBwJRTEjRbWfSZIiPEIRcTXkKCbSy8uFCutUMwMrTIQ+XFkl+9uRo1jKcRzozhipoVzUJuR/mpup8NrLKU8zRTiePhRmzFKJNYnIGlBBsGJjDRAWVO9q4SESCCsdZFWH4Cx+eRl0zhuO3XDuLmrNy1kcFTiGEzgDB66gCbfQgjZgeIRneIU348l4Md6Nj2nrijHzHMGfMj5/AG+Kmn8=
g4 CH 3
functional groups g4 g2
AAACFXicbVDLSsNAFL3xWesr6tJNsAguSklE1GXBjcsK9gFpCJPpJB06mYSZiVBCfsKNv+LGhSJuBXf+jdO0BW29MHDmnHvu3DlByqhUtv1trKyurW9sVraq2zu7e/vmwWFHJpnApI0TlohegCRhlJO2ooqRXioIigNGusHoZqJ3H4iQNOH3apwSL0YRpyHFSGnKN+t5vxziiijwcqdhl1W3F0ER+U7hm7X53VoGc3MNZtXyza/+IMFZTLjCDEnpOnaqvBwJRTEjRbWfSZIiPEIRcTXkKCbSy8uFCutUMwMrTIQ+XFkl+9uRo1jKcRzozhipoVzUJuR/mpup8NrLKU8zRTiePhRmzFKJNYnIGlBBsGJjDRAWVO9q4SESCCsdZFWH4Cx+eRl0zhuO3XDuLmrNy1kcFTiGEzgDB66gCbfQgjZgeIRneIU348l4Md6Nj2nrijHzHMGfMj5/AG+Kmn8=
AAACFXicbVDNS8MwHE39nPOr6tFLcAgexmhlqMeBF48T3Ad0paRZ2oWlaUlSYZT+E178V7x4UMSr4M3/xqzrRDcfBB7vvV/yy/MTRqWyrC9jZXVtfWOzslXd3tnd2zcPDrsyTgUmHRyzWPR9JAmjnHQUVYz0E0FQ5DPS88fXU793T4SkMb9Tk4S4EQo5DShGSkueWc8GxSWOCH03sxpWgfoPsUuSh14z98za3IDLZB6tgRJtz/wcDGOcRoQrzJCUjm0lys2QUBQzklcHqSQJwmMUEkdTjiIi3axYKIenWhnCIBb6cAUL9fdEhiIpJ5GvkxFSI7noTcX/PCdVwZWbUZ6kinA8eyhIGVQxnFYEh1QQrNhEE4QF1btCPEICYaWLrOoS7MUvL5PuecO2GvZts9a6KOuogGNwAs6ADS5BC9yANugADB7AE3gBr8aj8Wy8Ge+z6IpRzhyBPzA+vgF0B5qC
H2
AAACFXicbVDNS8MwHE39nPOr6tFLcAgexmhlqMeBF48T3Ad0paRZ2oWlaUlSYZT+E178V7x4UMSr4M3/xqzrRDcfBB7vvV/yy/MTRqWyrC9jZXVtfWOzslXd3tnd2zcPDrsyTgUmHRyzWPR9JAmjnHQUVYz0E0FQ5DPS88fXU793T4SkMb9Tk4S4EQo5DShGSkueWc8GxSWOCH03sxpWgfoPsUuSh14z98za3IDLZB6tgRJtz/wcDGOcRoQrzJCUjm0lys2QUBQzklcHqSQJwmMUEkdTjiIi3axYKIenWhnCIBb6cAUL9fdEhiIpJ5GvkxFSI7noTcX/PCdVwZWbUZ6kinA8eyhIGVQxnFYEh1QQrNhEE4QF1btCPEICYaWLrOoS7MUvL5PuecO2GvZts9a6KOuogGNwAs6ADS5BC9yANugADB7AE3gBr8aj8Wy8Ge+z6IpRzhyBPzA+vgF0B5qC AAACFXicbVBNS8MwGE79nPOr6tFLcAgeRmmHTI8DLx4nuA/oSkmztAtLP0hSYZT+CS/+FS8eFPEqePPfmHYd6OYDIU+e532TN4+XMCqkaX5ra+sbm1vbtZ367t7+waF+dNwXccox6eGYxXzoIUEYjUhPUsnIMOEEhR4jA296U/iDB8IFjaN7OUuIE6Igoj7FSCrJ1ZvZqLzE5oHnZKbRMgs0TaO9IOVu5oHbyl29sTjDVWJVpAEqdF39azSOcRqSSGKGhLAtM5FOhrikmJG8PkoFSRCeooDYikYoJMLJyoFyeK6UMfRjrlYkYan+7shQKMQs9FRliORELHuF+J9np9K/djIaJakkEZ4/5KcMyhgWEcEx5QRLNlMEYU7VrBBPEEdYqiDrKgRr+curpN8yLNOw7i4bnXYVRw2cgjNwASxwBTrgFnRBD2DwCJ7BK3jTnrQX7V37mJeuaVXPCfgD7fMHfDeahw==
C C
g3
AAACFXicbZDLSsQwFIZTr+N4q7p0ExwEF0NJvS8H3LgcwblAp5Q0k3bCpBeSVBhKX8KNr+LGhSJuBXe+jelcQGc8EPj4/3OSk99POZMKoW9jaXlldW29slHd3Nre2TX39tsyyQShLZLwRHR9LClnMW0ppjjtpoLiyOe04w9vSr/zQIVkSXyvRil1IxzGLGAEKy15Zj3vjS9xROi7ObIuUFl1ZKEZTJQi9M4Kz6zNDLgI9hRqYFpNz/zq9ROSRTRWhGMpHRulys2xUIxwWlR7maQpJkMcUkdjjCMq3Xy8UAGPtdKHQSL0iRUcq78nchxJOYp83RlhNZDzXin+5zmZCq7dnMVppmhMJg8FGYcqgWVEsM8EJYqPNGAimN4VkgEWmCgdZFWHYM9/eRHap5aNLPvuvNa4nMZRAYfgCJwAG1yBBrgFTdACBDyCZ/AK3own48V4Nz4mrUvGdOYA/Cnj8weA2JqK
H3 C
g3
O
g2
O
AAACFXicbVBNS8MwGE79nPOr6tFLcAgeRmmHTI8DLx4nuA/oSkmztAtLP0hSYZT+CS/+FS8eFPEqePPfmHYd6OYDIU+e532TN4+XMCqkaX5ra+sbm1vbtZ367t7+waF+dNwXccox6eGYxXzoIUEYjUhPUsnIMOEEhR4jA296U/iDB8IFjaN7OUuIE6Igoj7FSCrJ1ZvZqLzE5oHnZKbRMgs0TaO9IOVu5oHbyl29sTjDVWJVpAEqdF39azSOcRqSSGKGhLAtM5FOhrikmJG8PkoFSRCeooDYikYoJMLJyoFyeK6UMfRjrlYkYan+7shQKMQs9FRliORELHuF+J9np9K/djIaJakkEZ4/5KcMyhgWEcEx5QRLNlMEYU7VrBBPEEdYqiDrKgRr+curpN8yLNOw7i4bnXYVRw2cgjNwASxwBTrgFnRBD2DwCJ7BK3jTnrQX7V37mJeuaVXPCfgD7fMHfDeahw==
AAACFXicbZDLSsQwFIZTr+N4q7p0ExwEF0NJvS8H3LgcwblAp5Q0k3bCpBeSVBhKX8KNr+LGhSJuBXe+jelcQGc8EPj4/3OSk99POZMKoW9jaXlldW29slHd3Nre2TX39tsyyQShLZLwRHR9LClnMW0ppjjtpoLiyOe04w9vSr/zQIVkSXyvRil1IxzGLGAEKy15Zj3vjS9xROi7ObIuUFl1ZKEZTJQi9M4Kz6zNDLgI9hRqYFpNz/zq9ROSRTRWhGMpHRulys2xUIxwWlR7maQpJkMcUkdjjCMq3Xy8UAGPtdKHQSL0iRUcq78nchxJOYp83RlhNZDzXin+5zmZCq7dnMVppmhMJg8FGYcqgWVEsM8EJYqPNGAimN4VkgEWmCgdZFWHYM9/eRHap5aNLPvuvNa4nMZRAYfgCJwAG1yBBrgFTdACBDyCZ/AK3own48V4Nz4mrUvGdOYA/Cnj8weA2JqK
Legend connectedness incidence mereology
Figure 7: The breakdown of ethyl acetate. g1, g2, g3, g4 embody the primitive functional groups. s1,
s2, s3, s4 embody the skeletons. The bolded black lines represent connectivity among functional groups.
The dotted blue lines represent parthood relations. The dash-dotted green lines represent incident
relations.
allowing a skeleton to be part of another skeleton. In other words, functional groups are one
dimension lower than skeletons. Due to the nature of chemistry, not all decomposition of
molecules is viable. There is a fixed set of criteria to outline how functional groups can be
associated to form a skeleton. As outlined by Figure 7, even though g3 is connected to g2, they
do not have a sum. If we were to explain MoSt in the language of multimereology, we would
identify an equi-dimensional parthood relation among skeletons, and partitioning criteria across
dimensions to accommodate for chemical bonding.
Another application can be seen from the motivating scenario, in which we want to represent
hollow object like a gap in the wall. In 𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 , we define a surface to be two-dimensional.
Holes and surfaces share the same subgraph in CISCO since a hole forms a cycle, exhibiting the
properties of a two-dimensional surface. Nevertheless, intuitively, we know that a solid surface
should be a dimension higher than a hollow surface. Since multimereology has the freedom
to arbitrarily partition partial orderings according to dimensions, we can modify 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 to
characterize the fact that not every cycle constitutes a plane. In Figure4(a), imagine 123 is a
hole, while 234 is a surface. The representation of this simplicial complex would still be true
as shown in Figure4(b). The incidence structure will differ, however, since 123 is no longer
classified as a two-dimensional plane, but is instead a 1-simplex.
3.4. Summary
We began this section with questions about which mereology should be used in a multimereology
and which criteria should be used to partition it. Starting from the combinatorial structures
used to represent manifolds in algebraic topology, we claim that 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 is the right mereology,
since it extends the notion of CW complex. Furthermore, the expressiveness of 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 allows
us to specify different partitionings, depending on the classes of subgraphs (e.g. paths, cycles,
blocks).
4. Hahmann’s Multidimensional Mereotopology
A further application of multimereology is that it can be used as the basis for evaluating
other multidimensional mereotopologies. In particular, we can use multimereology to provide
a verification of CODI [9]. Recall that the major benefit of CODI, compared to Gotts’ and
Galton’s approaches, is its expressive power insofar as it generalizes relations between spatial
entities up to finite dimensions within a single model. 𝑇𝑐𝑜𝑑𝑖_𝑑𝑜𝑤𝑛 introduces three primitive
relations for specifying a multidimensional mereotopology. The first two relations (<𝑑𝑖𝑚 (𝑥, 𝑦),
𝐸𝑞𝐷𝑖𝑚(𝑥, 𝑦)) specify the relative dimension of two elements; in particular, <𝑑𝑖𝑚 (𝑥, 𝑦) is a
linear ordering over dimensions. The third relation (𝐶𝑜𝑛𝑡(𝑥, 𝑦)) is a parthood relation that
applies to all elements regardless of their dimension such that the dimension of an element has
a dimension greater than or equal to its parts.
To compare our axiomatization in 𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 with 𝑇𝑐𝑜𝑑𝑖_𝑑𝑜𝑤𝑛 , we want to find a common
ground where the set of minimal and maximal dimensions allowed is the same for both sides.
Since 𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 can only capture entities up to two-dimensional planes, we must restrict
𝑇𝑐𝑜𝑑𝑖_𝑑𝑜𝑤𝑛 to two dimensions as well:
Definition 10. 𝑇2𝑑_𝑐𝑜𝑑𝑖_𝑑𝑜𝑤𝑛 is the following set of axioms:
∀𝑥, 𝑦 (𝐶𝑜𝑛𝑡(𝑥, 𝑦) ⊃ (<𝑑𝑖𝑚 (𝑥, 𝑦) ∨ 𝐸𝑞𝐷𝑖𝑚(𝑥, 𝑦))) (16)
∃𝑥 (𝑀 𝑖𝑛𝐷𝑖𝑚(𝑥)) (17)
∀𝑥 (¬ <𝑑𝑖𝑚 𝑥, 𝑥) (18)
∀𝑥, 𝑦 (<𝑑𝑖𝑚 𝑥, 𝑦) ⊃ (¬ <𝑑𝑖𝑚 (𝑦, 𝑥)) (19)
∀𝑥, 𝑦, 𝑧 (<𝑑𝑖𝑚 (𝑥, 𝑦) ∧ (<𝑑𝑖𝑚 (𝑦, 𝑧) ∨ 𝐸𝑞𝐷𝑖𝑚(𝑦, 𝑧)) ⊃ (<𝑑𝑖𝑚 (𝑥, 𝑧))) (20)
∀𝑥, 𝑦 ((𝐶𝑜𝑛𝑡(𝑥, 𝑦) ∧ 𝐶𝑜𝑛𝑡(𝑦, 𝑥)) ⊃ (𝑥 = 𝑦)) (21)
∀𝑥, 𝑦, 𝑧 ((𝐶𝑜𝑛𝑡(𝑥, 𝑦) ∧ 𝐶𝑜𝑛𝑡(𝑦, 𝑧)) ⊃ (𝐶𝑜𝑛𝑡(𝑥, 𝑧))) (22)
∀𝑥 (𝑀 𝑖𝑛𝐷𝑖𝑚(𝑥) ∨ 𝐶𝑢𝑟𝑣𝑒(𝑥) ∨ 𝐴𝑟𝑒𝑎𝑙𝑅𝑒𝑔𝑖𝑜𝑛(𝑥)) (23)
Theorem 4. 𝑇2𝑑_𝑐𝑜𝑑𝑖_𝑑𝑜𝑤𝑛 is logically synonymous with 𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 .
By this Theorem, we know that there is a one-to-one correspondence between models of
𝑇𝑚𝑢𝑙𝑡𝑖𝑚𝑒𝑟𝑒𝑜 and models of 𝑇2𝑑_𝑐𝑜𝑑𝑖_𝑑𝑜𝑤𝑛 . Furthermore, because of Theorem 1, we know that all
models of 𝑇2𝑑_𝑐𝑜𝑑𝑖_𝑑𝑜𝑤𝑛 can be constructed by partitioning a partial ordering into intervals.
5. Unifying boundary definitions
We want the weakest possible mereotopology in which the notions of boundary can be rep-
resented and unified. In this section, we demonstrate the capabilities of multimereologies in
terms of representing different notions of boundary within the same theory.
5.1. Nondimensional Approach to Boundary
We exploit the dual nature of the models of 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 – on the one hand, models of 𝑇𝑐𝑖𝑠𝑐𝑜 are
nondimensional mereologies, but because the elements of these mereologies correspond to
connected subgraphs of a graph, they can also represent properties of graphs. The obvious place
to start is to note that within graph theory there is a notion of the boundary of a subgraph [14]:
Definition 11. Let 𝐺 = (𝑉, 𝐸) be a simple graph. The boundary of v ∈ 𝑉 is the set of all
vertices of 𝐺 which are adjacent to V:
𝐵(v) = {u ∈ 𝑉 : (u, v) ∈ 𝐸}.
The edge boundary of H is the set of edges in 𝐺 that contain one vertex in 𝐻 and one vertex not in
𝐻.
Based on this definition of graph boundary, we can define the corresponding notion for
elements within the mereology in a model of 𝑇𝑐𝑖𝑠𝑐𝑜_𝑚𝑡 :
Definition 12.
∀𝑥 (𝑒𝑑𝑔𝑒(𝑥) ≡ ∃𝑦, 𝑧 (𝑎𝑡𝑜𝑚(𝑦) ∧ 𝑎𝑡𝑜𝑚(𝑦) ∧ 𝑠𝑢𝑚(𝑦, 𝑧) ∧ ¬(𝑦 = 𝑧))) (24)
∀𝑥, 𝑦 (𝑔𝑟𝑎𝑝ℎ_𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦(𝑥, 𝑦) ≡ 𝑒𝑑𝑔𝑒(𝑥) ∧ 𝑃 𝑂(𝑥, 𝑦)) (25)
The boundary of the cycle 1234 in Figure 6 is {45}, while the boundary of the edge 12 in
the same Figure is {23, 14}.
5.2. Multidimensional Approach to Boundary
Inspired by algebraic topology [15], our interpretation of multidimensional notion of boundary
exhibits the following two characteristics:
1. The boundary is codimension 1 to the space that contains it.
2. 𝑥 is said to be the boundary of 𝑦 iff 𝑥 is a pendant element 7 of y.
which can be axiomatized as
Definition 13.
∀𝑥, 𝑦 (𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦(𝑥, 𝑦) ≡ 𝑖𝑛(𝑥, 𝑦) ∧ 𝑚_𝑐𝑜𝑣𝑒𝑟𝑠(𝑦, 𝑥) ∧ 𝑝𝑒𝑛𝑑𝑎𝑛𝑡(𝑥, 𝑦)) (26)
∀𝑥, 𝑦 (𝑝𝑒𝑛𝑑𝑎𝑛𝑡(𝑥, 𝑦) ≡ ∃𝑧, 𝑢 (𝐶(𝑥, 𝑧) ∧ 𝐶(𝑥, 𝑢) ∧ 𝑖𝑛(𝑢, 𝑦) ∧ 𝑖𝑛(𝑧, 𝑦) ∧ ¬(𝑧 = 𝑢))) (27)
∀𝑥, 𝑦 (𝑚_𝑐𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑦) ≡
(𝑝𝑎𝑟𝑡(𝑦, 𝑥) ∧ 𝑥 ̸= 𝑦 ∧ ¬∃𝑧 (𝑝𝑎𝑟𝑡(𝑦, 𝑧) ∧ 𝑝𝑎𝑟𝑡(𝑧, 𝑥) ∧ 𝑖𝑛(𝑧, 𝑦)) ∧ 𝑦 ̸= 𝑧 ∧ 𝑥 ̸= 𝑧)) (28)
According to this multidimensional characterization of boundary, in the same Figure6(b),
element 2 is not in the boundary of 123 because it is incident to more than one incomparable
equi-dimensional parts. Contrarily, 1 and 3 are the boundary of 123 because it is incident to
exactly one incomparable equi-dimensional part. In the same example in Figure5(a), the set
of boundary that was generated based on graph-theory definition is different from that of a
multidimensional based one. Note that even though multimereology successfully captures both
non- and multi-dimensional approaches to boundary, the exact relationship that holds between
these two approaches remains an open question.
7
In graph theory, a pendant element is connected to exactly one neighboring element.
6. Summary
This work was motivated by the need to integrate different characterizations of boundary.
From the Winograd Schema Challenge to Kandinsky’s painting, we saw that boundary is only
of a special form to the dimension discrepancy founded by different mathematical theories,
and borrowed by various mereotopologies. Instead of favoring one theory over the other, we
established a common ground for non- and multi-dimensional approaches.
We started the discussion of unification by specifying ontological commitments that a multi-
dimensional mereology must satisfy. We introduced the notion of multimereology, an amalga-
mation of a partial ordering and incidence structure that formalizes these commitments. A key
result is that each partitioning of a partial ordering corresponds to a unique multimereology,
and each multimereology specifies a partitioning of a partial ordering. This forms the basis of
the representation theorem for the class of multimereologies with respect to the set of poset
homomorphisms between partial orderings and finite linear orderings. Finally, we axiomatized
the class of multimereologies up to isomorphism.
Coming up with root theories for multimereology left us with the problem of amalgamating
an appropriate partial ordering with partitioning criteria. We therefore turned to the field of
algebraic topology to find the “right" manifold that can characterize our notion of partitioning
a partial ordering. We examined abstract simplicial complexes and regular CW complexes
as two possibilities. Due to the sets of construction rules they impose on topological spaces,
both CW and simplicial complexes fall short in representing non-trivial mereology within a
single dimension. We proceeded to showcase the expressiveness of our multimereology by
interpreting and representing other non- and multi-dimensional approaches.
So far, this work provided a middle ground to compare different notions of boundary. In
the future, we plan to explore and extend what multimereology is capable of. This can be
achieved in two directions. First, it would be useful to compose an exact theorem that captures
the relationship between non- and multi-dimensional boundary. Next, if we could use multi-
mereology to validate Smith’s and Hahmann’s mereotopologies, we would be one step closer to
harmonizing this dimension disagreement.
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