Sociological Space

DEFINITIONS
DEFINITION 1 (SOCIAL SYSTEM): A social system is system that governs a society. Under a social system, there specific rules and laws; and there are social norms and customs that govern the behavior of members of the society.
DEFINITION 2 (CORE VALUES): The core values of a social system are the values or attributes that uniquely identify the system. These values drive the systems's rules and laws, and its social norms.
DEFINITION 3 (TRANSACTION): A social transaction (or simply, a transaction ) is any form of action carried out in a society.
DEFINITION 4 (CLOSURE): A social system is closed if the result of any transaction is a part of the same social system, or if any transaction leads to another transaction within the same social system.
DEFINITION 5 (SOCIOLOGICAL SPACE): A sociological space is a closed social system.
DEFINITION 6 (SUBSET): A subset of a society is any element (or member) or group within the society.
DEFINITION 7 (SUBSPACE): A sociological subspace (or simply, a subspace) is a subset of a society that is closed under the transactions defined on the larger sociological space.
EXAMPLES OF SOCIOLOGICAL SPACES AND SUBSPACES
1. Every independent nation is a sociological space.
2. Examples of Sociological Subspaces—Public Schools, Public Hospitals, Prisons, Jails, etc.
IMPORTANT NOTE SOCIOLOGICAL SUBSETS
A sociological subset need not be a sociological subspace.
THE SMALLEST SOCIOLOGICAL SUBSPACE AND THE LARGEST SOCIOLOGICAL SUBSPACE
The smallest sociological subspace consists of just the core values and the transactions that are defined on the larger sociological space. We call this space the hollow space. The largest sociological subspace is the sociology itself.

OTHER IMPORTANT POINTS ABOUT MEMBERS OR VALUES, AND SUBSETS WITHIN A SOCIOLOGICAL SPACE
If a member or value A has the same representation as a member or value B within a sociological space, then A must be the same as B. In other words, every member or value can be clearly, singulary defined within a sociological space. The same holds true for subsets.
COMPATIBILITY BETWEEN DIFFERENT SOCIETIES
SOCIOLOGICAL COMPATIBILITY (OR SIMPLY, COMPATIBILITY)
While it is very difficult fully layout the full details behind every internation conflict, this theory of sociological spaces makes some simple suggestion.
DEFINITION 8 (ISOMORPHISM): As used in mathematics, an isomorphism is a one-to-one and onto mapping. In the case of sociological spaces, we have an isomorphism between any map between two social sets allows for the mapping of every element in one set onto its indivual partner in the other set.

DEFINITION 9 (STRONG COMPATIBILTY RULE): Two sociological spaces are strongly compatible if and only if there is an isomorphism between their core values.
DEFINITION 10 (WEAK COMPATIBILITY (ALSO CALLED LOCAL COMPATIBILITY) RULE): Two sociological spaces are weakly compatible whenever there is no isomorphism between them.
Two sociological spaces can be strongly compatible in two ways, as follows. 1: They may be globally compatible. In this case there is an isomorphism between the two sociological spaces themselves. Global compatibility is extremely difficult to achieve. 2: They may be core compatible. In this case, there is an isomorphism between their core values, but no isomorphism between the sociological spaces themselves.
It should be immediately clear that global compatibility implies core compatibility, but the converse is not true.
THEOREM: Wherever there is a core compatible relationship between sociological spaces, this relationship may be upgraded to global compatibility.
This can clearly be done by removing the values outside the core that prevents isomorphism between the sociological spaces, or values may also be added to effect the same.
THEOREM: A locally compatible relationship cannot be upgraded to a strong compatible relationship unless there is a core value change in at least one of the two interacting sociological spaces.
This follows from the definition of strong compatibility.
Note: While it may be theoretically possible to upgrade sociologies, the condition required upgrade from a local to a strong compatibilty makes that upgrade extremely difficult. Upgrading from core to global compatibility is also difficult, but it is not as difficult as the local to strong upgrade.
APPLICATION OF SOCIOLOGICAL SPACE THEORY
Although this idea of sociological space is so simple, and may be even a little purile, it may inform us on some of the reality of international relations. For, we can clearly see that some independent nations share values that are so disparate that social interaction between them need to be thought out clearly before proceeding. Any other way may be purely presumptuous.

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