Transforming Geospatial Ontologies by Homomorphisms

In this paper, we study the (geospatial) ontologies we are interested in together as an ontology (a geospatial ontology) system, consisting of a set of the (geospatial) ontologies and a set of ontology operations. A homomorphism between two ontology systems is a function between two sets of ontologies, which preserves these ontology operations. We view clustering a set of the ontologies we are interested in 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 ontology system homomorphism can be factored as a surjective clustering to a quotient space, followed by an embedding. Ontology (merging) systems, natural partial orders on the systems, and ontology merging closures in the systems are then transformed under ontology system homomorphisms, given by quotients and embeddings.

Another Generic Setting for Entity Resolution: Basic Theory

Benjelloun et al. \cite{BGSWW} considered the Entity Resolution (ER) problem as the generic process of matching and merging entity records judged to represent the same real world object. They treated the functions for matching and merging entity records as black-boxes and introduced four important properties that enable efficient generic ER algorithms. In this paper, we shall study the properties which match and merge functions share, model matching and merging black-boxes for ER in a partial groupoid, based on the properties that match and merge functions satisfy, and show that a partial groupoid provides another generic setting for ER. The natural partial order on a partial groupoid is defined when the partial groupoid satisfies Idempotence and Catenary associativity. Given a partial order on a partial groupoid, the least upper bound and compatibility ($LU_{pg}$ and $CP_{pg}$) properties are equivalent to Idempotence, Commutativity, Associativity, and Representativity and the partial order must be the natural one we defined when the domain of the partial operation is reflexive. The partiality of a partial groupoid can be reduced using connected components and clique covers of its domain graph, and a noncommutative partial groupoid can be mapped to a commutative one homomorphically if it has the partial idempotent semigroup like structures. In a finitely generated partial groupoid $(P,D,\circ)$ without any conditions required, the ER we concern is the full elements in $P$. If $(P,D,\circ)$ satisfies Idempotence and Catenary associativity, then the ER is the maximal elements in $P$, which are full elements and form the ER defined in \cite{BGSWW}. Furthermore, in the case, since there is a transitive binary order, we consider ER as ``sorting, selecting, and querying the elements in a finitely generated partial groupoid."

Merging Ontologies Algebraically

Ontology operations, e.g., aligning and merging, were studied and implemented extensively in different settings, such as, categorical operations, relation algebras, typed graph grammars, with different concerns. However, aligning and merging operations in the settings share some generic properties, e.g., idempotence, commutativity, associativity, and representativity, labeled by (I), (C), (A), and (R), respectively, which are defined on an ontology merging system $(\mathfrak{O},\sim,\merge)$, where $\mathfrak{O}$ is a set of the ontologies concerned, $\sim$ is a binary relation on $\mathfrak{O}$ modeling ontology aligning and $\merge$ is a partial binary operation on $\mathfrak{O}$ modeling ontology merging. Given an ontology repository, a finite set $\mathbb{O}\subseteq \mathfrak{O}$, its merging closure $\widehat{\mathbb{O}}$ is the smallest set of ontologies, which contains the repository and is closed with respect to merging. If (I), (C), (A), and (R) are satisfied, then both $\mathfrak{O}$ and $\widehat{\mathbb{O}}$ are partially ordered naturally by merging, $\widehat{\mathbb{O}}$ is finite and can be computed efficiently, including sorting, selecting, and querying some specific elements, e.g., maximal ontologies and minimal ontologies. We also show that the ontology merging system, given by ontology $V$-alignment pairs and pushouts, satisfies the properties: (I), (C), (A), and (R) so that the merging system is partially ordered and the merging closure of a given repository with respect to pushouts can be computed efficiently.