In this paper, we study the machine learning elements which we are interested in together as a machine learning system, consisting of a collection of machine learning elements and a collection of relations between the elements. The relations we concern are algebraic operations, binary relations, and binary relations with composition that can be reasoned categorically. A machine learning system transformation between two systems is a map between the systems, which preserves the relations we concern. The system transformations given by quotient or clustering, representable functor, and Yoneda embedding are highlighted and discussed by machine learning examples. An adjunction between machine learning systems, a special machine learning system transformation loop, provides the optimal way of solving problems. Machine learning system transformations are linked and compared by their maps at 2-cell, natural transformations. New insights and structures can be obtained from universal properties and algebraic structures given by monads, which are generated from adjunctions.

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.

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."