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.
An ontology was considered as an explicit specification of a conceptualization that provides the ways of
thinking about a domain [
In the age of artificial intelligence, geospatial data, from multiple platforms with many diferent 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 sense. We shall collect the ontologies together as a set G, along with a set of their operations that give rise to their relations, called a geospatial ontology system (G, ).
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 eficiently 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 eficient 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
extensively in diferent settings, such as, categorical operations [
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 [
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 , a binary relation on is a subset ⊆ × , where × = {(1, 2) | 1, 2 ∈ } is the Cartesian product of and . The inverse relation of is the relation
− 1 =def {(2, 1) | (1, 2) ∈ } ⊆ × .
=def {(1, 3) | (1, 2) ∈ , (2, 3) ∈ } ⊆ × . which can be computed eficiently when
|| < +∞. if A binary relation on is called reflexive if (, ) ∈ for all ∈ , symmetric if − 1 = , and transitive = . An equivalence relation on is a reflexive, symmetric, and transitive binary relation on . Clearly, ∆ = {(, ) | ∈ } and × are equivalence relations on .
For a binary relation on , the transitive closure of is the smallest binary relation on , which contains and is transitive. Since × is transitive and contains , always exists and = ∪+=∞1 ,
A function : → is an injection or a monomorphism if for all set and functions 1, 2 : → , 1 = 2 implies 1 = 2. The dual concept of an injection (a monomorphism) is a surjection (an epimorphism).
Let : → be a function and let ⊆ × be such that
(1, 2) ∈ if and only if (1) = (2).
Then is an equivalence relation on , called the kernel of .
If and are equivalence relations on and , respectively, then the image of under : =def {( (1), (2)) | (1, 2) ∈ } and the inverse image of under :
− 1 =def {(1, 2) ∈ × | ( (1), (2)) ∈ } are equivalence relations on () and , respectively. Obviously, = − 1(∆ ).
A partition of is a set of subsets ⊆ such that
each ̸= ∅, ∩ = ∅ for all distinct , ∈ , and = ∪∈ .
Given an equivalence relation on and ∈ , the subset [] = { | ∈ , (, ) ∈ } is called the equivalence class of with respect to . Each equivalence relation on partitions into the set of all equivalence classes with respect to , called the quotient set or quotient of with respect to , denoted by / .
Conversely, each partition of gives rise to an equivalence relation , whose quotient set is , where (1, 2) ∈ if and only if there is ∈ such that (1, 2) ∈ × .
There is a canonical projection : → / , sending to its equivalence class [] , which is
surjective. Obviously, /∆ = and /( × ) = {}. 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 [
partitions of , the set Q of all quotients of , and the set I of all surjective images of are isomorphic in Sets, namely, there exist the bijections between them.
All equivalence relations (partitions or quotients) on form a complete lattice.
Proposition 2.2. Let be a nonempty set. 1. The set E of all equivalence relations on forms a complete lattice with ∧∈ = ∩∈ , ∨∈ = (∪∈ ), the greatest element × , and the least element ∆ , where ∈ E, ∈ and () is the transitive closure of the subset ⊆ ; 2. Given , ∈ E, if ⊆ , then there is a unique surjective function ( ≤ )* : / → / , sending [] to [] , such that / ↙
( ≤ )* ↘ → / commutes.
Each quotient set (object) / gives rise to a surjection : → / and conversely, each surjection : → generates a quotient set (object) / ( ∼= () = ). 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 : → factors through the quotient set / , followed by an injection ̃︀ : / → , sending [] to (). Hence, combining with Proposition 2.2.2, one has: Proposition 2.3. Given a nonempty set , ∈ E, and a function : → , if ⊆ , then there are a unique injection ̃︀ : / → and a unique surjection ( ≤ )* : / → / such that / ↙
( ≤ )*
↘ → / ̃︀ →↗ commutes.
Each function : → is lifted to ̃︀ : / → / when can be embedded to . Proposition 2.4. Given a nonempty set , a function : → , ∈ E, and ∈ E , if ⊆ , then there is a unique function ̃︀ : / → / , sending [] to [ ()] , such that commutes. If is surjective and so is ̃︀.
Since − 1 ⊆ , by Proposition 2.4 one has: Corollary 2.5. Let : → be a function, ∈ E, and ∈ E . Then there are a unique surjection ̃︀ : / → ()/ and a unique function * : / − 1 → / such that
↓ / ̃︀ → /↓ →
↓ / → ()
↓ ̃︀ → ()/
( ) and commute. 1 ∼ 2 in .
− 1 ↓ / − 1 * → ↓ → / are isomorphic. homomorphism:
Let be an equivalence relation and ∼ a binary relation on . ∼ is compatible with (or ∼ is invariant under ) if and only if 1 ∼ 2 implies [1] ∼ [2] . That is, ∼ is a well-defined binary relation on / , where ∼ is the relation on / by mapping ∼ from to / : [1] ∼ [2] in / if and only if always compatible with × .
Given a binary operation ∘ on , ∘ is compatible with if and only if is a congruence equivalence relation on with respect to ∘ , namely, [1] ∘ [2] = [1 ∘ 2] is well defined.
If ∘ is not compatible with , then the congruence (compatible) closure of for ∘ is the smallest equivalence relation such that ⊆ and ∘ is compatible with . exists and is unique since ∘ is
Recall that a directed graph is an ordered pair = (, ), where is a set of vertices (or nodes), and ⊆ { (, ) | (, ) ∈ × and ̸= } is a set of edges (or arrows or arcs).
A quotient graph / of is a directed graph whose vertices are blocks of a partition of the vertices , where there is an edge of / from block to block if there is an edge from some vertex in to some vertex in from . That is, if is the equivalence relation induced by the partition of , then the quotient graph / has vertex set / and edge set {([], []) | (, ) ∈ }. Proposition 2.6. 1. Given a directed graph , the set of all equivalence relations of , the set of all partitions of , the set of all quotient graphs of , and the set of all graph homomorphic images of , 2. Every directed graph homomorphism ℎ : → can be factored as ℎ = , where : → / ℎ is a surjective directed graph homomorphism and : / ℎ → is an injective directed graph
ℎ ↘ / ℎ → ↗
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 [
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 [
Here are some examples of sets of the connected geospatial ontologies.
GeoDataOnt = {(, (,)) | , ∈ , 0 ≤ , ≤ | |}, where is the set of geographic entities concerned and the set of relations between the entities from . 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.
-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 [
Geospatial ontology clustering can facilitate a better understanding and improve the reusability of
the ontologies at the diferent summarization granularities [
{Quebec City, Ville de Québec}, {− 36.7∘ , − 34.06}, {Jan 22, 2022, 22/01/22:07:35}.
If a geospatial ontology is represented as a set of entities and their relations, which is a directed graph and is an equivalence relation on the set of entities, then we have the quotient geospatial ontology / by quotienting the directed graph and so the results of Proposition 2.6 can be mapped to the quotient geospatial ontology / .
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:
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.
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 diferent 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 on Cd by the clusters, e.g.,
(Quebec City, Ville de Québec), (− 36.7 ∘ C, − 34.06 ∘ F) ∈ .
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 [
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.
(finite) set of geospatial ontology operations.
A geospatial ontology system homomorphism ℎ : (O, ) → (P, ) is a function ℎ : O → P that preserves all operations in to .
Guo et al. [
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 1, 2 ∈ O, 1 ! 2 exists if 1 ∼ 2 and 1 ! 2
is undefined, denoted by 1 ! 2 = ↑, otherwise. (O, ∼ , !) forms an ontology merging system
[
Let (O, ∼ , ) and (P, ≈ , ≬) be two geospatial ontology merging systems. A geospatial ontology ! merging system homomorphism : (O, ∼ , ) → (P, ≈ , ≬) is a function : O → P such that !
! ! ↓ O × → ≬
↓ → P ≬ co1m!mut2esis, wdehfineerde th!en(≬) (ist1h)e≬do(ma2i)nisofde!fine(d≬)a,nsdpecifie(db1y! ∼2()≈ =). T(ha1t)is≬, fo(ra2l)l.1, 2 ∈ O if
In mathematics, an embedding in a mathematical structure (e.g., semigroup, group, ring) is a submathematical structure (e.g., sub-semigroup, sub-group, sub-ring). An object is embedded in another object if there is an injective structure-preserving map : → and is an embedding of . 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., [
A word feature vector or word embedding is a function that converts words into points in a
vector 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 [
Word2vec is a popular model that generates vector expressions for words. Since it was proposed in
2013 [
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. [
Given a geospatial ontology merging system (O, ∼ , !) and an equivalence relation on O, we have a quotient set O/ . If both ∼ and ! are compatible with , then we have (O/, ∼ , ! ), called a quotient ontology merging system, where ∼ is the equivalence relation on O/ , given by [1] ∼ [2] if and only if 1 ∼ 2, and [1] ! [2] = [1 ! 2] . It is routine to verify that both ∼ and ! 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 ∈ EO, if ∼ and ! are compatible with both , then (O/, ∼ , ! ) is a geospatial ontology system and : (O, ∼ , ) → (O/, ∼ , ! ),
! sending to [] , is a geospatial ontology merging system homomorphism.
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 ∈ EO. If ⊆ ℎ, then there are a unique injective homomorphism (embedding) ̃ℎ︀ : (O/ ℎ, ∼ ℎ , ! ℎ )/ → (P, ≈ , ≬) and an unique surjection ( ≤ ℎ)* : O/ → O/ ℎ such that (O, ∼ , )
! ( ≤ ℎ)*
ℎ ℎ
→ → (O/ ℎ, ∼ ℎ , ! ℎ ) ℎ ̃︀ → (P, ≈ , ≬) →
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 homomorphism, ∈ EO, and ∈ EP. If ℎ( ) ⊆ , then there is a unique geospatial ontology merging system homomorphism sending [] to [ℎ()] , such that ̃ℎ︀ : (O/, ∼ , ! ) → (P/, ≈ , ≬ ),
↓ ↓ (O/, ∼ , ! ) ̃ℎ︀ → (P/, ≈ , ≬ ) commutes. If ℎ is a surjection and so is ̃ℎ︀.
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, ! ∈ EO, and ∈ EP. Then there are unique geospatial ontology merging system homomorphisms and such that
̃ℎ︀ : (O/, ∼ , ! ) → (ℎ(O), ∼ ℎ , !ℎ ) ℎ* : (O/ℎ− 1, ∼ ℎ− 1 , !ℎ− 1 ) → (P/, ≈ , ≬ ) and
Hence geospatial ontology aligning and merging operations behave like binary relations in Sets.
Given a geospatial ontology merging system (O, ∼ , !), ! aims to obtain more information by combin2ing the aligned geospatial1otnotolo2gdieosetsongoetthyeire.lIdnt[h1e5]m, tohree ninaftourrmalaotinotnoltohgaynpar2t.iaIlnotrhdiesrsec1t i≤on!, we was defined if merging 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 1, 2 ∈ O, 1 ≤ ! 2 if and only if 1 ∼ 2, 2 ∼ 1, and 1 ! 2 = 2 ! 1 = 2.
In [
Proposition 6.2. If geospatial ontology merging system (O, ∼ , ) satisfies ! • for all ∈ O,
∼ and ! = • for all 1, 2, 3 ∈ O such that 1 ! 2 and 2 ! 3 exist,
(1 ! 2) ! 3 = 1 ! (2 ! 3) ̸= ↑, 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 [
(I) (CA) □
The natural partial order ≤ ! is mapped to the quotient space shown in Proposition 6.3 below. aPnrdop!osairteioconm6p.3a.tibLleetw(Oith, ∼ .,I!f()Ob,e∼ a,g!eo)spsaattiisafielson(tIo)laongyd (mCeArg),insog dsyoesste(mOa/n,d∼ ∈, !EO) sauncdh(tOha/t,b o≤th!∼ ) is a poset.
Proof. By Proposition 5.1, (O/, ∼ , ! ) is a geospatial ontology merging system. Since : (O, ∼ , !) → (O/, ∼ , ! ) is a geospatial ontology merging system homomorphism and (I) and (CA) are preserved under geospatial ontology merging system homomorphisms, (O/, ∼ , ! ) satisfies (I) and (CA). Hence (O/, ≤ ! ) is a poset. □
Since ≤ ! is natural, namely, it is defined by ! phism gives rise to a poset homomorphism: , each geospatial ontology merging system homomorProposition 6.4. Given a geospatial ontology merging system homomorphism ℎ : (O, ∼ , ) → (P, ≈ , ≬), ! ∈ EO, and ∈ EO, if ℎ( ) ⊆ , then there is a unique poset homomorphism sending [] to [ℎ()] , such that ̃ℎ︀ : (O/, ≤ ! ) → (P/, ≤ ≬ ),
O ↓ (O/, ≤ ! ) ℎ ℎ ̃︀ → P ↓ → (P/, ≤ ≬ ) commutes. If ℎ is a surjection and so is ̃ℎ︀.
A partial order on a geospatial ontology merging system (O, ∼ , !), where ∼ is reflexive and
commutative!,m,suhsot wbentihne[n1a5t]u(rSaelepTarhteiaolroemrde3r. 3≤ in! [
if merges give the least upper bounds and ∼ is compatible with
A geospatial ontology repository or instance in a geospatial ontology merging system (O, ∼ , !) is a
ifnite set O ⊆ O. In [
Definition 7.1. smallest set P ⊆ 1. O ⊆ P,
Given a geospatial repository O ⊆
O, the merging closure of O, denoted by Ô︀ , is the 2. P is closed with respect to merging: for all 1, 2 ∈ P such that 1 ∼ 2, 1 ! 2 ∈ P.
By the same process of Theorem 4.2 [
Proposition 7.2. Given a geospatial repository O ⊆
O, the merging closure Ô︀ exists and it is unique.
The merging closure operation (̂︁) can be transformed to the quotient space and is commutative with the quotient operation /. if ∼ and ! are compatible with , then [̂O︂] = [ Ô︀ ] .
Proposition 7.3. Given a geospatial ontology merging system (O, ∼ , ) and an equivalence relation , ! Proof. Since O ⊆ Ô︀ , clearly [O] ⊆ [ Ô︀ ] . For all [1] , [2] ∈ [ Ô︀ ], where 1, 2 ∈ Ô︀ , [1] ! [2] = [1 ! 2] ∈ [ Ô︀ ] .
Hence [ Ô︀ ] is closed with respect to ! .
For each P ⊇ [O] such that P is closed with respect to ! ,
Then Ô︀ = [̂︀] as Ô︀ is the smallest set, containing [O] and closed with respect to ! .
□
Combining Proposition 7.3 with the finiteness result (Theorem 4.3) in [
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 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 diferent 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.