In this setting, a type is a function from the places of a relation, that is, from the index set of its components, to a collection of sets that are called the domains of the relation."
We have also tied in the definition of a system as:
"A system is a relationship associated with a collection of objects."
"A system is a relation mapped over a set of objects."
The connection between and among the concepts of function, relation, type and system are very powerful and interesting.
We further expand the application of system by making a distinction between a natural language "relationship" and a mathematical "relation."
The mapping of a natural language relationship to a mathematical relation has many useful and valuable attributes in the assessment of generic design processes and generic ontology structures.
I will review our work in this area when I get time and post a more detailed statement.
Bottom line, this is a key, fundamental area of integration.
Take care, be good to yourself and have fun,
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Words represent a category of concepts whose structures, therefore meanings, differ in different contexts. The combinatorics of concepts and contexts boggle the mind (at the level of lint). The problem may be assuming that the relations are isomorphies – and particularly binary isomorphisms. At best, they represent homomorphisms, but even that is not assured. By focusing on one isomorphism, which may not hold true, you often miss other, perhaps more correct, morphisms whose workings provide interesting information as to their development of meaning.
Furthermore, there are several dimensions in the location of the “system of interest” within the contexts. As this locus changes within those metric spaces, the dynamics involved are useful and informative. In other words, the structure one is modeling is not static, but dynamic.
“In times of change, learners inherit the earth, while the learned find themselves beautifully equipped to deal with a world that no longer exists.” – Eric Hoffer.
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