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 [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., +, β, Γ, and Γ·, 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 π of their operations that give rise to their relations, called a geospatial ontology system (G, π ).
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 extensively 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.
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 π, a binary relation on π is a subset π β πΓπ, where
π} is the Cartesian product of π and π. The inverse relation of π is the relation
If π and π are two binary relations on π,
A
There is a canonical projection π π : π β π/π, sending π to its equivalence class [π ] π , which is surjective. Obviously, π/β π = π and π/(π Γ π) = {π}. 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 π, the set E π of all equivalence relations on π, the set P π of all partitions of π, the set Q π of all quotients of π, and the set I π of all surjective images of π are isomorphic in Sets, namely, there exist the bijections between them.
All equivalence relations (partitions or quotients) on π form a complete lattice. Proposition 2.2. Let π be a nonempty set.
The set E π of all equivalence relations on π forms a complete lattice with β§ πβπΌ π π = β© πβπΌ π π , β¨ πβπΌ π π = (βͺ πβπΌ π π ) π‘ , the greatest element π Γ π, and the least element β π , where π π β E π , π β πΌ and (π) π‘ is the transitive closure of the subset π β π;
Each quotient set (object) π/π gives rise to a surjection π π : π β π/π and conversely, each surjection π : π β π generates a quotient set (object) π/π π ( βΌ = π (π) = π ). 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
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., +, β, Γ, Γ·) 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.
where πΈ is the set of geographic entities concerned and π the set of relations between the entities from πΈ. 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.
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. Entity resolution tools match Quebec City with Ville de QuΓ©bec 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]
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 (π, π), where π is a function between the concepts and π 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 π 1 and π 2 , a geospatial ontology homomorphism π : π 1 β π 2 is a function that preserves the ontology structures. For example, given a geospatial ontology πΊ and π is an equivalence relation on the set of the entities in πΊ, we have a canonical geospatial ontology homomorphism π : πΊ β πΊ/π.
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.
A geospatial ontology system homomorphism β : (O, π ) β (P, π) is a function β : O β P that preserves all operations in π to π.
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, βΌ 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
) forms an ontology merging system [15]. Similarly, we define a geospatial merging system (G, βΌ, ) to be a geospatial system (G, π ) with π = {βΌ, }.
Let (O, βΌ, ) and (P, β, β¬) be two geospatial ontology merging systems. A geospatial ontology merging system homomorphism π : (O, βΌ, ) β (P, β, β¬) is a function π : O β P such that
commutes, where π (π β¬ ) is the domain of (β¬), specified by βΌ (β). That is, for all
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 πΈ is embedded in another object π if there is an injective structure-preserving map π : πΈ β π and π is an embedding of π. 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 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 [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.
Given a geospatial ontology merging system (O, βΌ, ) and an equivalence relation π on O, we have a quotient set O/π. If both βΌ and are compatible with π, then we have (O/π, βΌ π , π ), called a quotient ontology merging system, where βΌ π is the equivalence relation on O/π, given by
It is routine to verify that both βΌ π and π 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, βΌ, ) and π β E O , if βΌ and are compatible with both π, then (O/π, βΌ π , π ) is a geospatial ontology system and
sending π to [π] π , 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 β : (O, βΌ, ) β (P, β, β¬) be a geospatial ontology merging system homomorphism and π β E O . If π β π β , then there are a unique injective homomorphism (embedding)
and an unique surjection
Each geospatial ontology merging system homomorphism can be lifted to the quotient geospatial ontology merging systems. Proposition 5.3. Let β : (O, βΌ, ) β (P, β, β¬) be a geospatial ontology merging system homomorphism, π β E O , and π β E P . If β(π) β π, then there is a unique geospatial ontology merging system homomorphism
commutes. If β is a surjection and so is ΜοΈ β.
There are also the image and inverse image cases of an equivalence relation on a geospatial ontology merging system. Corollary 5.4. Let β : (O, βΌ, ) β (P, β, β¬) be a geospatial ontology merging system homomorphism, π β E O , and π β E P . Then there are unique geospatial ontology merging system homomorphisms
Hence geospatial ontology aligning and merging operations behave like binary relations in Sets.
Given a geospatial ontology merging system (O, βΌ, ), aims to obtain more information by combining the aligned geospatial ontologies together. In [15], the natural ontology partial order π 1 β€ π 2 was defined if merging π 1 to π 2 does not yield the more information than π 2 . In this section, we introduce the natural partial order to a geospatial ontology merging system (O, βΌ, ) and show that the natural partial order can be mapped to the quotient of (O, βΌ, ).
commutes. If β is a surjection and so is ΜοΈ β.
A partial order on a geospatial ontology merging system (O, βΌ, ), where βΌ is reflexive and commutative, must be the natural partial order β€ if merges give the least upper bounds and βΌ is compatible with , shown in [15] (See Theorem 3.3 in [15] for the detail).
A geospatial ontology repository or instance in a geospatial ontology merging system (O, βΌ, ) is a finite set O β 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.
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 π 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.