Piaget’s “structure” would be what I would call APPLIED DOMAIN with its “laws” what I call the PRINCIPLES OF THE DOMAIN. APPLIED DOMAIN and TOOL _may_ be one and the same _or_ perhaps APPLIED DOMAIN is where the TOOL interfaces with the whole DOMAIN.

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Mathematical Structure:

What is structure? Jean Piaget apparently gives a great answer:

First of all, when we compare the use of the term “structure” in the various natural and human sciences, we find the following characteristics.

Structure is, in the first place, a system of transformations having its laws, as a system, these therefore being distinct from the properties of the elements.

In the second place, these transformations have a self-regulating device in the sense that no new element engendered by their operation breaks the boundaries of the system (the addition of two numbers still gives a number, etc.) and that the transformations of the system do not involve elements outside it.

In the third place, the system may have sub-systems by differentiation from the total system (for example, by a limitation of the transformations making it possible to leave this or that characteristic constant, etc.) and there may be some transformations from one sub-system to another

https://en.wikipedia.org/wiki/Mathematical_structure

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More from Piaget (I had no idea):

First of all, a structure is a totality; this is, it is a system governed by laws that apply to the system as such, and not only to one or another element in the system. The system of whole numbers is an example of a structure, since there are laws that apply to the series [of whole numbers] as such. Many different mathematical structures can be discovered in the series of whole numbers. . . A second characteristic of these laws is that they are laws of transformation; they are not static characteristics. In the case of addition of whole numbers, we can transform one number into another by adding something to it. The third characteristic of a structure is that a structure is self-regulating; that is, in order to carry out these laws of transformation, we need not go outside the system to find some external element. Similarly, once a law of transformation has been applied, the result does not end up outside the system. Referring to the additive group again, when we add one whole number to another, we do not have to go outside the series of whole numbers in search of any element that is not in the series. And once we have added the two whole numbers together, our result still remains within the series

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Ok. I’m starting to think mathematical structures stay within their domain and these may be subdomains interacting, which can also be called domains. What will change is the wilderness: So now a stacked side map is needed, with Wilderness at the bottom, and each layer indicated as the ‘new wilderness’ (background) — but each wilderness comes with its own principles.

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Ok: Algebraic structures conform to either APPLIED or TOOL

https://en.wikipedia.org/wiki/Algebraic_structure

Algebraic structures

Group-like

Ring-like

Lattice-like

Module-like

Algebra-like

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TOPOLOGICAL STRUCTURE: Class and Operation. So that’s: PRINCIPLES OF A DOMAIN and APPLIED DOMAIN.

Topological structure package TopologicalStructure contains following stereotypes:

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TopologicalCycle—topological cycle represents directed functional cycle of system; it consists of elements and relationships between them. It can show the main functionality that has a vital importance in the functioning of system, i.e., by interrupting the main cycle the system can no longer function or its functioning is deformed.

•

TopologicalOperation—topological operation is a behavioral feature of classifier that specifies the name, type, parameters, and constraints for invoking an associated behavior, and related functional features and topological relationships for specifying cause-and-effect relations within system, thus allowing a cause-and-effect relations to be modeled within the system by means of behavioral features (e.g., in topological class diagram).

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UML – hm: Their generalizations may fit perfectly.

“A class is a classifier which describes a set of objects that share the same

features,

constraints,

semantics (meaning).

“

“Operation is a behavioral feature that may be owned by an interface, data type, or class. Operations may also be templated and used as template parameters. An operation may be directly invoked on instances of its featuring classifiers. The operation specifies the name, type, parameters, and constraints for such invocations.”

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Digress for a sec:

Ah! Piaget addresses the three mathematical structures:

The first is what the Bourbaki call the algebraic structure. The prototype of this structure is the mathematical notion of a group. There are all sorts of mathematical groups: the group of displacements, as found in geometry; the additive group that I have already referred to in the series of whole numbers; and any number of others. Algebraic structures are characterized by their form of reversibility, which is inversion in the sense I described above

The second type of structure is the order structure. This structure applies to relationships, whereas the algebraic structure applies essentially to classes and numbers. The prototype of an order structure is the lattice, and the form of reversibility characteristic of order structures is reciprocity. We can find this reciprocity of the order relationship if we look at the logic of propositions, for example. In one structure within the logic of propositions, P and Q is the lower limit of a transformation, and P or Q is the upper limit P and Q, the conjunction, precedes P or Q, the disjunction. But this whole relationship can be expressed in the reverse way. We can say that P or Q follows P and Q just as easily as we can say that P and Q precedes P or Q. This is the form of reversibility that I have called reciprocity; it is not at all the same thing as inversion or negation. There is nothing negated here.

The third type of structure is the topological structure based on notions such as neighborhood, borders, and approaching limits. This applies not only to geometry but also to many other areas of mathematics. Now these three types of structure appear to be highly abstract. Nonetheless, in the thinking of children as young as 6 or 7 years of age, we find structures resembling each of these three type

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ok back to algebraic structure:

” a collection of operations on A of finite arity (typically binary operations), and a finite set of identities, known as axioms, that these operations must satisfy.”

So: Operations are APPLIED and identities/axioms are PRINCIPLES.

and each is called an Algebraic Structure.

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