=Paper=
{{Paper
|id=Vol-3637/paper8
|storemode=property
|title=Transforming Geospatial Ontologies by Homomorphisms
|pdfUrl=https://ceur-ws.org/Vol-3637/paper8.pdf
|volume=Vol-3637
|authors=Xiuzhan Guo,Wei Huang,Min Luo,Priya Rangarajan
|dblpUrl=https://dblp.org/rec/conf/jowo/GuoHLR23
}}
==Transforming Geospatial Ontologies by Homomorphisms==
https://ceur-ws.org/Vol-3637/paper8.pdf
Transforming Geospatial Ontologies by Homomorphisms
Xiuzhan Guo1,* , Wei Huang1 , Min Luo1 and Priya Rangarajan1
1
Chief Data Office, Royal Bank of Canada, 181 Bay St., Toronto, ON M5J 2V1, Canada
Abstract
In this paper, we study the geospatial ontologies that we are interested in together as a geospatial ontology
system, consisting of a set of the geospatial ontologies and a set of geospatial ontology operations, without any
internal details of the geospatial ontologies and their operations being needed, algebraically. A homomorphism
between two geospatial ontology systems is a function between two sets of geospatial ontologies in the systems,
which preserves the geospatial ontology operations. We view clustering a set of the ontologies as partitioning the
set or defining an equivalence relation on the set or forming a quotient set of the set or obtaining the surjective
image of the set. Each geospatial ontology system homomorphism can be factored as a surjective clustering to
a quotient space, followed by an embedding. Geospatial ontology merging systems, natural partial orders on
the systems, and geospatial ontology merging closures in the systems are then transformed under geospatial
ontology system homomorphisms that are given by quotients and embeddings.
Keywords
Equivalence relation, quotient, surjection, injection, clustering, embedding, geospatial ontology, geospatial
ontology merging system, homomorphism, natural partial order, merging closure
1. Introduction
An ontology was considered as an explicit specification of a conceptualization that provides the ways of
thinking about a domain [14]. Ontologies are the silver bullet for many applications, such as, database
integration, peer to peer systems, e-commerce, etc. [13]. A geospatial ontology is an ontology that
implements a set of geospatial entities in a hierarchical structure [7, 10, 27, 28].
In the age of artificial intelligence, geospatial data, from multiple platforms with many different
types, not only is big, heterogeneous, connected, but also keeps changing continuously, which results
in tremendous potential for dynamic relationships. Geospatial data, ontologies, and models must be
robust enough to the dynamic changes.
After mathematical operations, e.g., +, −, ×, and ÷, being introduced, natural numbers can be
used not only to count but also to solve real life problems. The set of natural numbers, along with
the operations, forms an algebraic system that can be studied by its properties without any internal
details of the numbers and operation. These operations establish the relations among natural numbers,
which make more sense than isolated natural numbers. Geospatial ontologies are not isolated but
connected by their relations. For example, an ontology of Ontario climate data entities can be viewed
as a directed subgraph of Canada digital twin knowledge graph, the data management ontology of
Canada digital twin data is a super ontology of Ontario farm data ontology, etc. Geospatial ontologies
can be aligned, matched, mapped, merged, and transformed and so they are linked by these operations.
Relations between the ontologies, given by the operations, may make more sense than the single isolated
ontologies. In this paper, we shall assume that the geospatial ontologies that we are interested in, can
be viewed as a set of entities and their relations that carry certain algebraic structures and make more
Workshop on Geospatial Ontologies 2023: 9th Joint Ontology Workshops (JOWO 2023), co-located with FOIS 2023, 19-20 July,
2023, Sherbrooke, Québec, Canada
*
Corresponding author: xiuzhan@gmail.com
© 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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sense. We shall collect the ontologies together as a set G, along with a set 𝑃 of their operations that
give rise to their relations, called a geospatial ontology system (G, 𝑃 ).
Recall that a directed graph, the mathematical concept to model entities and their pairwise relations,
consists of a set of nodes (or vertices) and a set of edges (or arrows), given by an ordered pair of nodes. It
has been shown that relations can be queried, updated, computed, analyzed, and visualized efficiently
and provide the robustness to the models in a graph setting.
A geospatial ontology, viewed as a set of geospatial ontologies and their relations, can be represented
as a knowledge graph so that it, along with knowledge graph computing capabilities, provides an
efficient setting to align, integrate, transform, update, query, compute, analyze, and visualize the
geospatial ontologies. However, due to its complexity and size, the geospatial data is unlikely to be
entirely modeled by one single ontology or knowledge graph. To tackle such a big dynamic data or
ontology, we group or summarize it at multiple layers or dimensions.
In Sets, grouping objects (elements) amounts to clustering or partitioning them, which turns out
to be equivalent to an equivalence relation that produces a quotient set, a surjective function, and an
injective function, where injection (sub object) and surjection (quotient object) are the dual concepts.
Each function can factor through a quotient set, followed by an injection (embedding). In this paper,
we shall introduce equivalence relation, quotient, embedding to geospatial ontology systems, study
how geospatial ontologies are transformed under geospatial ontology system homomorphisms, each of
which can be viewed as a quotient surjection, followed by an embedding.
Ontologies and ontology operations, e.g., aligning and merging, are studied and implemented exten-
sively in different settings, such as, categorical operations [1, 4, 8, 9, 17, 18, 23, 31], relation algebras
[12], typed graph grammars [21]. In this paper, we shall group the geospatial ontologies and their
operations without any internal details of the ontologies and the operations being needed in any specific
setting but we shall utilize the generic algebraic properties they share, to study the geospatial ontologies
algebraically.
The paper proceeds as follows: First, in Section 2, we recall the basic notions and notations of a
binary relation in Sets, such as, an equivalence relation, a partition, a quotient set, a projection, a
kernel, an embedding, etc.
In Section 3, we consider the geospatial ontologies that we are interested in, collectively as a set and
cluster or partition them as a quotient set, which will also produce a surjective homomorphism.
In Section 4, we model the set of geospatial ontologies and their operations as a geospatial ontology
system. A homomorphism between geospatial ontology systems, a function between the systems
preserving the operations, is factored through the quotient geospatial ontology system, followed by an
embedding.
In [15], Guo et al. introduced ontology merging systems, the natural partial order on the systems, and
the merging closure of an ontology repository and studied the properties shared algebraically without
any internal details. In Sections 5, 6, and 7, we transform the geospatial ontology merging systems,
the natural partial order on the systems, and the merging closure of a geospatial ontology repository
using geospatial ontology merging system homomorphisms that amount to quotients and embeddings,
respectively. Finally, we complete the paper with our concluding remarks in Section 8.
2. Preliminaries
In this section, we recall the basic notations, concepts, and results of binary relations, equivalence
relations, partitions, and quotients on a nonempty set or a directed graph.
Given a nonempty set 𝑆, a binary relation on 𝑆 is a subset 𝜌 ⊆ 𝑆×𝑆, where 𝑆×𝑆 = {(𝑠1 , 𝑠2 ) | 𝑠1 , 𝑠2 ∈
2
𝑆} is the Cartesian product of 𝑆 and 𝑆. The inverse relation of 𝜌 is the relation
def
𝜌−1 = {(𝑠2 , 𝑠1 ) | (𝑠1 , 𝑠2 ) ∈ 𝜌} ⊆ 𝑆 × 𝑆.
If 𝜌 and 𝜎 are two binary relations on 𝑆,
def
𝜌𝜎 = {(𝑠1 , 𝑠3 ) | (𝑠1 , 𝑠2 ) ∈ 𝜌, (𝑠2 , 𝑠3 ) ∈ 𝜎} ⊆ 𝑆 × 𝑆.
A binary relation 𝜌 on 𝑆 is called reflexive if (𝑠, 𝑠) ∈ 𝜌 for all 𝑠 ∈ 𝑆, symmetric if 𝜌−1 = 𝜌, and transitive
if 𝜌𝜌 = 𝜌. An equivalence relation on 𝑆 is a reflexive, symmetric, and transitive binary relation on 𝑆.
Clearly, ∆𝑆 = {(𝑠, 𝑠) | 𝑠 ∈ 𝑆} and 𝑆 × 𝑆 are equivalence relations on 𝑆.
For a binary relation 𝜑 on 𝑆, the transitive closure 𝜑𝑡 of 𝜑 is the smallest binary relation on 𝑆, which
contains 𝜑 and is transitive. Since 𝑆 × 𝑆 is transitive and contains 𝜑, 𝜑𝑡 always exists and 𝜑𝑡 = ∪+∞ 𝑖=1 𝜑 ,
𝑖
which can be computed efficiently when |𝑆| < +∞.
A function 𝑓 : 𝑆 → 𝑇 is an injection or a monomorphism if for all set 𝑋 and functions 𝑔1 , 𝑔2 : 𝑋 → 𝑆,
𝑓 𝑔1 = 𝑓 𝑔2 implies 𝑔1 = 𝑔2 . The dual concept of an injection (a monomorphism) is a surjection (an
epimorphism).
Let 𝑓 : 𝑆 → 𝑇 be a function and let 𝜅𝑓 ⊆ 𝑆 × 𝑆 be such that
(𝑠1 , 𝑠2 ) ∈ 𝜅𝑓 if and only if 𝑓 (𝑠1 ) = 𝑓 (𝑠2 ).
Then 𝜅𝑓 is an equivalence relation on 𝑆, called the kernel of 𝑓 .
If 𝜌 and 𝜎 are equivalence relations on 𝑆 and 𝑇 , respectively, then the image of 𝜌 under 𝑓 :
def
𝑓 𝜌 = {(𝑓 (𝑠1 ), 𝑓 (𝑠2 )) | (𝑠1 , 𝑠2 ) ∈ 𝜌}
and the inverse image of 𝜎 under 𝑓 :
def
𝑓 −1 𝜎 = {(𝑠1 , 𝑠2 ) ∈ 𝑆 × 𝑆 | (𝑓 (𝑠1 ), 𝑓 (𝑠2 )) ∈ 𝜎}
are equivalence relations on 𝑓 (𝑆) and 𝑆, respectively. Obviously, 𝜅𝑓 = 𝑓 −1 (∆𝑇 ).
A partition of 𝑆 is a set 𝒫𝑆 of subsets 𝑆𝑖 ⊆ 𝑆 such that
each 𝑆𝑖 ̸= ∅, 𝑆𝑖 ∩ 𝑆𝑗 = ∅ for all distinct 𝑆𝑖 , 𝑆𝑗 ∈ 𝒫𝑆 , and 𝑆 = ∪𝑆𝑖 ∈𝒫𝑆 𝑆𝑖 .
Given an equivalence relation 𝜌 on 𝑆 and 𝑠 ∈ 𝑆, the subset [𝑠]𝜌 = {𝑎 | 𝑎 ∈ 𝑆, (𝑠, 𝑎) ∈ 𝜌} is called
the equivalence class of 𝑠 with respect to 𝜌. Each equivalence relation 𝜌 on 𝑆 partitions 𝑆 into the set
of all equivalence classes with respect to 𝜌, called the quotient set or quotient of 𝑆 with respect to 𝜌,
denoted by 𝑆/𝜌.
Conversely, each partition 𝒫𝑆 of 𝑆 gives rise to an equivalence relation 𝜌𝒫𝑆 , whose quotient set is
𝒫𝑆 , where (𝑠1 , 𝑠2 ) ∈ 𝜌𝒫𝑆 if and only if there is 𝑆𝑖 ∈ 𝒫𝑆 such that (𝑠1 , 𝑠2 ) ∈ 𝑆𝑖 × 𝑆𝑖 .
There is a canonical projection 𝜋𝜌 : 𝑆 → 𝑆/𝜌, sending 𝑠 to its equivalence class [𝑠]𝜌 , which is
surjective. Obviously, 𝑆/∆𝑆 = 𝑆 and 𝑆/(𝑆 × 𝑆) = {𝑆}. Equivalence relations, partitions, quotients,
and surjective images are equivalent in Sets and so they are interpreting the same thing. Therefore,
the results of the operations on equivalence relations (e.g., in [2]) can be mapped to clusters, partitions,
quotients, and surjective images.
Proposition 2.1. Given a nonempty set 𝑆, the set E𝑆 of all equivalence relations on 𝑆, the set P𝑆 of all
partitions of 𝑆, the set Q𝑆 of all quotients of 𝑆, and the set I𝑆 of all surjective images of 𝑆 are isomorphic
in Sets, namely, there exist the bijections between them.
3
All equivalence relations (partitions or quotients) on 𝑆 form a complete lattice.
Proposition 2.2. Let 𝑆 be a nonempty set.
1. The set E𝑆 of all equivalence relations on 𝑆 forms a complete lattice with ∧𝑖∈𝐼 𝜌𝑖 = ∩𝑖∈𝐼 𝜌𝑖 , ∨𝑖∈𝐼 𝜌𝑖 =
(∪𝑖∈𝐼 𝜌𝑖 )𝑡 , the greatest element 𝑆 × 𝑆, and the least element ∆𝑆 , where 𝜌𝑖 ∈ E𝑆 , 𝑖 ∈ 𝐼 and (𝑋)𝑡 is the
transitive closure of the subset 𝑋 ⊆ 𝑆;
2. Given 𝜌, 𝜎 ∈ E𝑆 , if 𝜌 ⊆ 𝜎, then there is a unique surjective function (𝜌 ≤ 𝜎)* : 𝑆/𝜌 → 𝑆/𝜎, sending
[𝑠]𝜌 to [𝑠]𝜎 , such that
𝑆
𝜋𝜌 𝜋𝜎
↙ (𝜌≤𝜎)* ↘
𝑆/𝜌 → 𝑆/𝜎
commutes.
Each quotient set (object) 𝑆/𝜌 gives rise to a surjection 𝜋𝜌 : 𝑆 → 𝑆/𝜌 and conversely, each surjection
𝑓 : 𝑆 → 𝑇 generates a quotient set (object) 𝑆/𝜅𝑓 ( ∼ = 𝑓 (𝑆) = 𝑇 ). A quotient object can be characterized
by a surjection while a sub object is characterized by an injection. Hence a quotient object and a sub
object (an embedding) are the dual concepts as a surjection and an injection are dual in Sets.
Each function 𝑓 : 𝑆 → 𝑇 factors through the quotient set 𝑆/𝜅𝑓 , followed by an injection 𝑓̃︀ : 𝑆/𝜅𝑓 →
𝑇 , sending [𝑠]𝜅𝑓 to 𝑓 (𝑠). Hence, combining with Proposition 2.2.2, one has:
Proposition 2.3. Given a nonempty set 𝑆, 𝜌 ∈ E𝑆 , and a function 𝑓 : 𝑆 → 𝑇 , if 𝜌 ⊆ 𝜅𝑓 , then there are
a unique injection 𝑓̃︀ : 𝑆/𝜅𝑓 → 𝑇 and a unique surjection (𝜌 ≤ 𝜅𝑓 )* : 𝑆/𝜌 → 𝑆/𝜅𝑓 such that
𝑓
𝑆 →𝑇
↗
𝜋𝜌 𝜋 𝜅𝑓
↙ (𝜌≤𝜅𝑓 )* ↘ 𝑓̃︀
𝑆/𝜌 → 𝑆/𝜅𝑓
commutes.
Each function 𝑓 : 𝑆 → 𝑇 is lifted to 𝑓̃︀ : 𝑆/𝜌 → 𝑇 /𝜎 when 𝑓 𝜌 can be embedded to 𝜎.
Proposition 2.4. Given a nonempty set 𝑆, a function 𝑓 : 𝑆 → 𝑇 , 𝜌 ∈ E𝑆 , and 𝜎 ∈ E𝑇 , if 𝑓 𝜌 ⊆ 𝜎, then
there is a unique function 𝑓̃︀ : 𝑆/𝜌 → 𝑇 /𝜎, sending [𝑠]𝜌 to [𝑓 (𝑠)]𝜎 , such that
𝑓
𝑆 →𝑇
𝜋𝜌 𝜋𝜎
↓ 𝑓̃︀ ↓
𝑆/𝜌 → 𝑇 /𝜎
commutes. If 𝑓 is surjective and so is 𝑓̃︀.
Since 𝑓 𝑓 −1 𝜎 ⊆ 𝜎, by Proposition 2.4 one has:
Corollary 2.5. Let 𝑓 : 𝑆 → 𝑇 be a function, 𝜌 ∈ E𝑆 , and 𝜎 ∈ E𝑇 . Then there are a unique surjection
𝑓̃︀ : 𝑆/𝜌 → 𝑓 (𝑆)/𝑓 𝜌 and a unique function 𝑓 * : 𝑆/𝑓 −1 𝜎 → 𝑇 /𝜎 such that
𝑓
𝑆 → 𝑓 (𝑆)
𝜋𝜌 𝜋𝑓 (𝜌)
↓ 𝑓̃︀ ↓
𝑆/𝜌 → 𝑓 (𝑆)/𝑓 𝜌
4
and
𝑓
𝑆 →𝑇
𝜋𝑓 −1 𝜎 𝜋𝜎
↓ 𝑓* ↓
𝑆/𝑓 −1 𝜎 → 𝑇 /𝜎
commute.
Let 𝜌 be an equivalence relation and ∼ a binary relation on 𝑆. ∼ is compatible with 𝜌 (or ∼ is invariant
under 𝜌) if and only if 𝑠1 ∼ 𝑠2 implies [𝑠1 ]𝜌 ∼𝜌 [𝑠2 ]𝜌 . That is, ∼𝜌 is a well-defined binary relation on
𝑆/𝜌, where ∼𝜌 is the relation on 𝑆/𝜌 by mapping ∼ from 𝑆 to 𝑆/𝜌: [𝑠1 ]𝜌 ∼𝜌 [𝑠2 ]𝜌 in 𝑆/𝜌 if and only if
𝑠1 ∼ 𝑠2 in 𝑆.
Given a binary operation ∘ on 𝑆, ∘ is compatible with 𝜌 if and only if 𝜌 is a congruence equivalence
relation on 𝑆 with respect to ∘, namely, [𝑠1 ]𝜌 ∘𝜌 [𝑠2 ]𝜌 = [𝑠1 ∘ 𝑠2 ]𝜌 is well defined.
If ∘ is not compatible with 𝜌, then the congruence (compatible) closure 𝜌𝑐 of 𝜌 for ∘ is the smallest
equivalence relation 𝜚 such that 𝜌 ⊆ 𝜚 and ∘ is compatible with 𝜚. 𝜌𝑐 exists and is unique since ∘ is
always compatible with 𝑆 × 𝑆.
Recall that a directed graph is an ordered pair 𝐺 = (𝑉𝐺 , 𝐸𝐺 ), where 𝑉𝐺 is a set of vertices (or nodes),
and 𝐸𝐺 ⊆ {(𝑥, 𝑦) | (𝑥, 𝑦) ∈ 𝑉𝐺 × 𝑉𝐺 and 𝑥 ̸= 𝑦} is a set of edges (or arrows or arcs).
A quotient graph 𝐺/𝑅 of 𝐺 is a directed graph whose vertices are blocks of a partition of the vertices
𝑉𝐺 , where there is an edge of 𝐺/𝑅 from block 𝐵 to block 𝐶 if there is an edge from some vertex in 𝐵
to some vertex in 𝐶 from 𝐸𝐺 . That is, if 𝑅 is the equivalence relation induced by the partition of 𝑉𝐺 ,
then the quotient graph 𝐺/𝑅 has vertex set 𝑉𝐺 /𝑅 and edge set {([𝑢]𝑅 , [𝑣]𝑅 ) | (𝑢, 𝑣) ∈ 𝐸𝐺 }.
Proposition 2.6. 1. Given a directed graph 𝐺, the set of all equivalence relations of 𝐺𝑉 , the set of all
partitions of 𝑉𝐺 , the set of all quotient graphs of 𝐺, and the set of all graph homomorphic images of 𝐺,
are isomorphic.
2. Every directed graph homomorphism ℎ : 𝐺 → 𝐻 can be factored as ℎ = 𝑖𝜋, where 𝜋 : 𝐺 → 𝐺/𝜅ℎ
is a surjective directed graph homomorphism and 𝑖 : 𝐺/𝜅ℎ → 𝐻 is an injective directed graph
homomorphism:
𝐺
ℎ →𝐻
↗
𝜋
↘ 𝑖
𝐺/𝜅ℎ
3. Geospatial Ontologies, Clustering, and Quotients
In this section, we group geospatial ontologies together and discuss clustering and quotienting operations
in geospatial ontology setting.
Recall that a geospatial ontology is an ontology that has a set of geospatial entities in a hierarchical
structure [7, 10, 27, 28]. Geospatial ontologies are not isolated but connected by their relations.
Numbers are linked by their operations (e.g., +, −, ×, ÷) so that they are used to solve real life
problems. Geospatial ontologies can be aligned, matched, mapped, merged, and transformed and they
are linked by these operations. Relations between numbers (geospatial ontologies), given by operations,
make more sense than single numbers (geospatial ontologies). Hence we study the ontologies we are
interested in together as a set collectively, e.g., the geospatial data ontologies in [27], the sub set of the
objects of the category Ont+ of the ontologies defined in [31], or the ontology structures considered in
[4].
Here are some examples of sets of the connected geospatial ontologies.
5
Example 3.1. 1. In [27], Sun et al. defined
GeoDataOnt = {(𝐸, 𝑅(𝐸𝑖 ,𝐸𝑗 ) ) | 𝐸𝑖 , 𝐸𝑗 ∈ 𝐸, 0 ≤ 𝑖, 𝑗 ≤ |𝐸|},
where 𝐸 is the set of geographic entities concerned and 𝑅 the set of relations between the entities
from 𝐸. Clearly, GeoDataOnt can be represented as a directed graph with geospatial entities as
nodes and their relations as edges. Since these geospatial ontologies (directed graphs) are connected
and share certain geospatial properties, we collect them together as a set Gd.
2. Assume that there is a climate data repository, which collects the climate data from a number of
data silos and covers a variety of climate domain application areas, e.g., location, weather condition,
climate hazard, wildfire, air quality, events, etc., each of which is managed by a geospatial ontology.
We group these geospatial ontologies as a set, denoted by Cd. Here is a directed subgraph, showing
the relations between some objects in Cd, e.g., temperature ontology and location ontology, from
both English and France systems at a time point.
Quebec City Ville de Québec
-34.06 -36.7 ∘ C
22/01/22:07:35 Jan 22, 2022
Entity resolution tools match Quebec City with Ville de Québec and merge their records together,
with the existing relations (their neighborhoods in the knowledge graphs) being preserved, to obtain
the standardized record with the maximal information.
Generally, clustering aims to group a set of the objects in such a way that objects in the same cluster
(group) are more similar to each other. There exist a number of the approaches to clustering. The
interested reader may consult [30] for a comprehensive survey of clustering approaches.
Geospatial ontology clustering can facilitate a better understanding and improve the reusability of
the ontologies at the different summarization granularities [19, 25]. If a similarity approach is applied
to the climate ontologies from Cd in Example 3.1.2 above, then the clusters are:
{Quebec City, Ville de Québec}, {−36.7∘ 𝐶, −34.06}, {Jan 22, 2022, 22/01/22:07:35}.
If a geospatial ontology 𝐺 is represented as a set of entities and their relations, which is a directed
graph and 𝜌 is an equivalence relation on the set of entities, then we have the quotient geospatial
ontology 𝐺/𝜌 by quotienting the directed graph and so the results of Proposition 2.6 can be mapped to
the quotient geospatial ontology 𝐺/𝜌.
For a set O of the ontologies, a clustering algorithm may produce a partition of O, which is equivalent
to a quotient set or a surjective image of O. Hence, by Propositions 2.1 and 2.2, we have:
Proposition 3.2. Given a nonempty set O of geospatial ontologies, the set EO of all equivalence relations
of O, the set PO of all partitions of O, the set QO of all quotients of O, and the set IO of all surjective
images of O are isomorphic and form a complete lattice.
6
Hence clustering a set O of the ontologies can be interpreted as partitioning O or defining an
equivalence relation on O or forming a quotient of O or finding a surjective image of O. The results
of clustering O at the different summarization granularities are linked by the complete lattice in
Proposition 2.2.
Using entity resolution tools to group Cd in Example 3.1.2 above, amounts to:
• clustering Cd by their similarities, e.g., {-36.7 ∘ C, -34.06}, {Jan 22, 2022, 22/01/22:07:35},
• forming a quotient by identifying the similar objects from Cd in each cluster, e.g., Quebec City =
Ville de Québec, and -36.7 ∘ C = -34.06 ∘ F,
• taking the surjective image of Cd by mapping the similar ontologies in each cluster to the merged
ontology, e.g., {Jan 22, 2022, 22/01/22:07:35} to Jan 22, 2022, 07:35 am,
• defining the equivalence relation 𝜌 on Cd by the clusters, e.g.,
(Quebec City, Ville de Québec), (−36.7 ∘ C, −34.06 ∘ F) ∈ 𝜌.
4. Geospatial Ontology System Homomorphisms and Embeddings
The word homomorphism, from Greek homoios morphe, means "similar for". In an algebra, e.g., groups,
semigroups, rings, a homomorphism is a map that preserves the algebra operation(s).
In [4] Cafezeiro and Haeusler defined an ontology homomorphism between ontology structures
introduced in [20], as a pair of functions (𝑓, 𝑔), where 𝑓 is a function between the concepts and 𝑔 a
function between relations, which preserve the ontology structures. Geospatial ontologies carry some
structures and can be viewed as a set of entities and their relations, as assumed. Given two geospatial
ontologies 𝑂1 and 𝑂2 , a geospatial ontology homomorphism 𝑓 : 𝑂1 → 𝑂2 is a function that preserves
the ontology structures. For example, given a geospatial ontology 𝐺 and 𝜌 is an equivalence relation on
the set of the entities in 𝐺, we have a canonical geospatial ontology homomorphism 𝜋 : 𝐺 → 𝐺/𝜌.
In this section, we move to the second layer: geospatial ontology systems and homomorphisms
between them.
After collecting the geospatial ontologies into a set, we need to introduce their relations by a set of
operations and form a greospatial ontology system.
Definition 4.1. A geospatial ontology system (O, 𝑃 ) consists of a set O of geospatial ontologies and a
(finite) set 𝑃 of geospatial ontology operations.
A geospatial ontology system homomorphism ℎ : (O, 𝑃 ) → (P, 𝑄) is a function ℎ : O → P that
preserves all operations in 𝑃 to 𝑄.
Guo et al. [15] studied the ontologies and their operations (aligning and merging) together within the
partial groupoid or semigroup using the properties the operations share without any ontology internal
details being needed. They defined an ontology merging system as follows.
Let O be the non-empty set of the ontologies concerned, ∼ a binary relation on O that models a
generic ontology alignment relation, and ! a partial binary operation on O that models a merging
operation defined on alignment pairs: For all 𝑂1 , 𝑂2 ∈ O, 𝑂1 ! 𝑂2 exists if 𝑂1 ∼ 𝑂2 and 𝑂1 ! 𝑂2
is undefined, denoted by 𝑂1 ! 𝑂2 = ↑, otherwise. (O, ∼, !) forms an ontology merging system
[15]. Similarly, we define a geospatial merging system (G, ∼, !) to be a geospatial system (G, 𝑃 ) with
𝑃 = {∼, !}.
Let (O, ∼, !) and (P, ≈, ≬) be two geospatial ontology merging systems. A geospatial ontology
7
merging system homomorphism 𝑓 : (O, ∼, !) → (P, ≈, ≬) is a function 𝑓 : O → P such that
𝑓 ×𝑓
𝑑! →𝑑
≬
! ≬
↓ 𝑓 ↓
O →P
commutes, where 𝑑! (𝑑≬ ) is the domain of ! (≬), specified by ∼ (≈). That is, for all 𝑂1 , 𝑂2 ∈ O if
𝑂1 ! 𝑂2 is defined then 𝑓 (𝑂1 ) ≬ 𝑓 (𝑂2 ) is defined and 𝑓 (𝑂1 ! 𝑂2 ) = 𝑓 (𝑂1 ) ≬ 𝑓 (𝑂2 ).
In mathematics, an embedding in a mathematical structure (e.g., semigroup, group, ring) is a sub-
mathematical structure (e.g., sub-semigroup, sub-group, sub-ring). An object 𝐸 is embedded in another
object 𝑂 if there is an injective structure-preserving map 𝑒 : 𝐸 → 𝑂 and 𝑒 is an embedding of 𝑂.
Embeddings and surjections that preserve the structures are dual.
Ontology embeddings aim to map ontologies from a high dimension space to a much lower dimension
space with certain ontology structures being preserved. Ontology embeddings were studied extensively,
e.g., [5, 6, 16, 10]. Word embeddings and graph embeddings were employed in the approaches widely
[5, 6, 16, 29].
A word feature vector or word embedding is a function that converts words into points in a vec-
tor space. Word embeddings are usually injective functions (i.e. two words do not share the same
word embedding), and highlight not-so-evident features of words. Hence, one usually says that word
embeddings are an alternative representation of words [3, 26].
Word2vec is a popular model that generates vector expressions for words. Since it was proposed in
2013 [24], embedding technology has been extended from natural language processing to other fields,
such as, graph embedding, ontology embedding [5, 6, 10, 11, 16], etc.
However, geospatial ontology systems may carry many structures and can be very complex. These
embeddings may fail to capture a lot of important properties, e.g., hierarchy, closedness, completeness,
insights in a logic sentence etc. [10, 22]. The embeddings may not be injective. But in this case, the
injective one can be obtained by factoring the original one through its quotient using the kernel. On
the other hand, injective transformers, e.g., shaving one’s beard with a mirror, can change working or
computing environments but cannot reduce the difficulty of the problem one tries to solve in general.
5. Transforming Geospatial Ontology Merging Systems
Given a geospatial ontology merging system (O, ∼, !) and an equivalence relation 𝜌 on O, we have
a quotient set O/𝜌. If both ∼ and ! are compatible with 𝜌, then we have (O/𝜌, ∼𝜌 , !𝜌 ), called a
quotient ontology merging system, where ∼𝜌 is the equivalence relation on O/𝜌, given by [𝑠1 ]𝜌 ∼ [𝑠2 ]𝜌
if and only if 𝑠1 ∼ 𝑠2 , and [𝑠1 ]𝜌 !𝜌 [𝑠2 ]𝜌 = [𝑠1 ! 𝑠2 ]𝜌 . It is routine to verify that both ∼𝜌 and !𝜌 are
well-defined. In this section, we study how geospatial ontology merging systems are transformed by
quotienting.
As in Propositions 2.3 and 2.4, and Corollary 2.5, we have the following Propositions 5.1, 5.2, and 5.3,
and Corollary 5.4, on quotient ontology merging systems.
Proposition 5.1. Given a geospatial ontology merging system (O, ∼, !) and 𝜌 ∈ EO , if ∼ and ! are
compatible with both 𝜌, then (O/𝜌, ∼𝜌 , !𝜌 ) is a geospatial ontology system and
𝜋𝜌 : (O, ∼, !) → (O/𝜌, ∼𝜌 , !𝜌 ),
sending 𝑂 to [𝑂]𝜌 , is a geospatial ontology merging system homomorphism.
8
Each geospatial ontology merging system homomorphism is factored through the quotient geospatial
ontology merging system.
Proposition 5.2. Let ℎ : (O, ∼, !) → (P, ≈, ≬) be a geospatial ontology merging system homomorphism
and 𝜌 ∈ EO . If 𝜌 ⊆ 𝜅ℎ , then there are a unique injective homomorphism (embedding)
ℎ : (O/𝜅ℎ , ∼𝜅ℎ , !𝜅ℎ )/ → (P, ≈, ≬)
̃︀
and an unique surjection (𝜌 ≤ 𝜅ℎ )* : O/𝜌 → O/𝜅ℎ such that
(O, ∼, !)
ℎ → (P, ≈, ≬)
→
𝜋𝜌 𝜋 𝜅ℎ
← (𝜌≤𝜅ℎ )* → ℎ
̃︀
(O/𝜌, ∼𝜌 , !𝜌 ) → (O/𝜅ℎ , ∼𝜅 , !𝜅 )
ℎ ℎ
commutes.
Each geospatial ontology merging system homomorphism can be lifted to the quotient geospatial
ontology merging systems.
Proposition 5.3. Let ℎ : (O, ∼, !) → (P, ≈, ≬) be a geospatial ontology merging system homomor-
phism, 𝜌 ∈ EO , and 𝜎 ∈ EP . If ℎ(𝜌) ⊆ 𝜎, then there is a unique geospatial ontology merging system
homomorphism
ℎ : (O/𝜌, ∼𝜌 , !𝜌 ) → (P/𝜎, ≈𝜎 , ≬𝜎 ),
̃︀
sending [𝑠]𝜌 to [ℎ(𝑠)]𝜎 , such that
(O, ∼, !)
ℎ → (P, ≈, ≬)
𝜋𝜌 𝜋𝜎
↓ ↓
(O/𝜌, ∼𝜌 , !𝜌 )
ℎ → (P/𝜎, ≈𝜎 , ≬𝜎 )
̃︀
commutes. If ℎ is a surjection and so is ̃︀
ℎ.
There are also the image and inverse image cases of an equivalence relation on a geospatial ontology
merging system.
Corollary 5.4. Let ℎ : (O, ∼, !) → (P, ≈, ≬) be a geospatial ontology merging system homomorphism,
𝜌 ∈ EO , and 𝜎 ∈ EP . Then there are unique geospatial ontology merging system homomorphisms
ℎ : (O/𝜌, ∼𝜌 , !𝜌 ) → (ℎ(O), ∼ℎ𝜌 , !ℎ𝜌 )
̃︀
and
ℎ* : (O/ℎ−1 𝜎, ∼ℎ−1 𝜎 , !ℎ−1 𝜎 ) → (P/𝜎, ≈𝜎 , ≬𝜎 )
such that
(O, ∼, !)
ℎ → (ℎ(O), ≈, ≬)
𝜋𝜌 𝜋ℎ𝜌
↓ ↓
(O/𝜌, ∼𝜌 , !𝜌 )
ℎ → (ℎ(O)/ℎ𝜌, ≈ℎ𝜌 , ≬ℎ𝜌 )
̃︀
9
and
(O, ∼, !)
ℎ → (P, ≈, ≬)
𝜋ℎ−1 𝜎 𝜋𝜎
↓ ↓
ℎ* → (P/𝜎, ≈𝜎 , ≬𝜎 )
(O/ℎ−1 𝜎, ∼ℎ−1 𝜎 , !ℎ−1 𝜎 )
commute.
Hence geospatial ontology aligning and merging operations behave like binary relations in Sets.
6. Transforming Natural Partial Orders
Given a geospatial ontology merging system (O, ∼, !), ! aims to obtain more information by combin-
ing the aligned geospatial ontologies together. In [15], the natural ontology partial order 𝑂1 ≤! 𝑂2
was defined if merging 𝑂1 to 𝑂2 does not yield the more information than 𝑂2 . In this section, we
introduce the natural partial order to a geospatial ontology merging system (O, ∼, !) and show that
the natural partial order can be mapped to the quotient of (O, ∼, !).
Definition 6.1. For all 𝑂1 , 𝑂2 ∈ O, 𝑂1 ≤! 𝑂2 if and only if 𝑂1 ∼ 𝑂2 , 𝑂2 ∼ 𝑂1 , and 𝑂1 ! 𝑂2 =
𝑂2 ! 𝑂1 = 𝑂2 .
In [15], it was shown that (O, ≤! ) is a partially ordered set (poset), namely, ≤! is a reflexive,
antisymmetric, and transitive binary relation on O, if (I) and (CA), defined in Proposition 6.2 below, are
satisfied.
Proposition 6.2. If geospatial ontology merging system (O, ∼, !) satisfies
• for all 𝑂 ∈ O,
𝑂 ∼ 𝑂 and 𝑂 ! 𝑂 = 𝑂 (I)
• for all 𝑂1 , 𝑂2 , 𝑂3 ∈ O such that 𝑂1 ! 𝑂2 and 𝑂2 ! 𝑂3 exist,
(𝑂1 ! 𝑂2 ) ! 𝑂3 = 𝑂1 ! (𝑂2 ! 𝑂3 ) ̸= ↑, (CA)
then ≤! is a partial order on O and so (O, ≤! ) is a poset.
Proof. It is routine to verify by the same proof process of Proposition 3.2 [15]. □
The natural partial order ≤! is mapped to the quotient space shown in Proposition 6.3 below.
Proposition 6.3. Let (O, ∼, !) be a geospatial ontology merging system and 𝜌 ∈ EO such that both ∼
and ! are compatible with 𝜌. If (O, ∼, !) satisfies (I) and (CA), so does (O/𝜌, ∼𝜌 , !𝜌 ) and (O/𝜌, ≤!𝜌 )
is a poset.
Proof. By Proposition 5.1, (O/𝜌, ∼𝜌 , !𝜌 ) is a geospatial ontology merging system. Since 𝜋𝜌 : (O, ∼
, !) → (O/𝜌, ∼𝜌 , !𝜌 ) is a geospatial ontology merging system homomorphism and (I) and (CA) are
preserved under geospatial ontology merging system homomorphisms, (O/𝜌, ∼𝜌 , !𝜌 ) satisfies (I) and
(CA). Hence (O/𝜌, ≤!𝜌 ) is a poset. □
Since ≤! is natural, namely, it is defined by !, each geospatial ontology merging system homomor-
phism gives rise to a poset homomorphism:
10
Proposition 6.4. Given a geospatial ontology merging system homomorphism ℎ : (O, ∼, !) → (P, ≈, ≬),
𝜌 ∈ EO , and 𝜎 ∈ EO , if ℎ(𝜌) ⊆ 𝜎, then there is a unique poset homomorphism
ℎ : (O/𝜌, ≤!𝜌 ) → (P/𝜎, ≤≬𝜎 ),
̃︀
sending [𝑂]𝜌 to [ℎ(𝑂)]𝜎 , such that
O
ℎ →P
𝜋𝜌 𝜋𝜎
↓ ↓
(O/𝜌, ≤!𝜌 )
ℎ → (P/𝜎, ≤ )
̃︀
≬𝜎
commutes. If ℎ is a surjection and so is ̃︀
ℎ.
A partial order on a geospatial ontology merging system (O, ∼, !), where ∼ is reflexive and commu-
tative, must be the natural partial order ≤! if merges give the least upper bounds and ∼ is compatible
with !, shown in [15] (See Theorem 3.3 in [15] for the detail).
7. Transforming Geospatial Ontology Merging Closures
A geospatial ontology repository or instance in a geospatial ontology merging system (O, ∼, !) is a
finite set O ⊆ O. In [15], Guo et al. introduced the merging closure of O and showed that the merging
closure of a repository is a finite poset if some reasonable conditions are satisfied (Theorem 4.3 [15]). In
this section, we introduce geospatial ontology merging closure and show the interactions between the
closure operator and quotienting.
Definition 7.1. Given a geospatial repository O ⊆ O, the merging closure of O, denoted by O,
̂︀ is the
smallest set P ⊆ O such that
1. O ⊆ P,
2. P is closed with respect to merging: for all 𝑂1 , 𝑂2 ∈ P such that 𝑂1 ∼ 𝑂2 , 𝑂1 ! 𝑂2 ∈ P.
By the same process of Theorem 4.2 [15], O
̂︀ exists and is unique.
Proposition 7.2. Given a geospatial repository O ⊆ O, the merging closure O
̂︀ exists and it is unique.
The merging closure operation (̂︁) can be transformed to the quotient space and is commutative with
the quotient operation /.
Proposition 7.3. Given a geospatial ontology merging system (O, ∼, !) and an equivalence relation 𝜌,
if ∼ and ! are compatible with 𝜌, then [O]
̂︂𝜌 = [O]
̂︀ 𝜌 .
Proof. Since O ⊆ O,
̂︀ clearly [O]𝜌 ⊆ [O]
̂︀ 𝜌 . For all [𝑂1 ]𝜌 , [𝑂2 ]𝜌 ∈ [O],
̂︀ where 𝑂1 , 𝑂2 ∈ O,
̂︀
[𝑂1 ]𝜌 !𝜌 [𝑂2 ]𝜌 = [𝑂1 ! 𝑂2 ]𝜌 ∈ [O]
̂︀ 𝜌 .
Hence [O]
̂︀ 𝜌 is closed with respect to !𝜌 .
For each P ⊇ [O]𝜌 such that P is closed with respect to !𝜌 ,
̂︀ 𝜌 ⊆ [̂︂
[O] ̂︀ 𝜌 ⊆ P
O] ̂︀ = P.
Then O ̂︀ 𝜌 as O
̂︀ = [𝑂] ̂︀ is the smallest set, containing [O]𝜌 and closed with respect to !𝜌 . □
Combining Proposition 7.3 with the finiteness result (Theorem 4.3) in [15], we have:
11
Corollary 7.4. Given a geospatial ontology merging system (O, ∼, !), 𝜌 ∈ EO , and a repository O ⊆ O
if ∼ and ! are compatible with 𝜌 and each cluster (equivalence class) produced by 𝜌 is finite, then O
̂︀ is
finite if and only if [O]
̂︂𝜌 is finite.
8. Conclusions
Relations between geospatial ontologies make more sense than isolated geospatial ontologies. Geospatial
ontology operations provide the relations between these ontologies. We studied the geospatial ontologies
that we are interested in, together as a geospatial ontology system algebraically, which consists of a
set G of the ontologies and a set 𝑃 of geospatial ontology operations, without any internal details of
the ontologies and the operations being needed. A homomorphism between two geospatial ontology
systems is a function between two sets of geospatial ontologies, which preserves the geospatial ontology
operations. Clustering a set of the ontologies was interpreted as partitioning the set or defining an
equivalence relation on the set or forming the quotient of the set or obtaining the surjective image of the
set. Clustering (Quotienting) and embedding can be utilized at multiple layers, e.g., geospatial ontology
layer and geospatial ontology system layer. The results at the different layers behave like a complete
lattice. Each geospatial ontology system homomorphism was factored as a surjective clustering to a
quotient space, followed by an embedding. Clustering and embedding are the dual concepts in general.
Geospatial ontology (merging) systems, natural partial orders on the systems, and geospatial ontology
merging closures in the systems were transformed by geospatial ontology system homomorphisms.
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