A scope is a marked zone where an abstract/non-abstract discussable entity can be examined.

A scope is a marked zone where an abstract/non-abstract discussable entity can be examined.

An atom is a non-composed scope.

Examples: {} (= an empty scope), . (= a point), ._. (=a segment),
__ or .__ or __. (= an infinitely long entity).

An empty scope is a marked zone without any content.

An example: {}

A point is a non-composed and non-empty scope that has no directions where a direction is < , > or < > .

An example: .

A segment is a non-composed and non-empty scope that has directions which are closed upon themselves, or has at least two reachable edges.

An example: O , .__.

Each segment can have a unique name, which is based on its ratio to some arbitrary segment, which its name is 0_1.

An infinitely long entity is a non-composed non-empty scope which is not closed on itself and has no more than one reachable edge.

An example: __ , .__ , __.

Non-atom (or notom) is a scope that includes at leat one scope as its content.

An example: {{}} ,{__} ,{ {},{...,{{}},{},{}}}, {... ,{{}} , . , ._.} etc.
A sub-scope is a scope that exists within another scope.
An Open notom (or Onotom) is a collection of sub-scopes that has no first sub-scope and not a last sub-scope, or a one and only one infinitely long entity with no edges.

An example: {... ,{},{},{}, ...} ,{__} ,{... ,{{}},{},{}, ...} etc.

A Half-Closed notom (or Hnotom) is a scope that includes a first sub-scope but not a last sub-scope, or a last sub-scope and not a first sub-scope.

Also a Hnotom can be based on a one infinitely long entity that has at least one reachable edge.

An example: {{},{},{},…}, {.__}, {__.} etc.

A Closed notom (or Cnotom) is a scope that includes a first sub-scope and a last sub-scope, and it does not include Hnotom or Onotom.

An example: {{},{},{}}, {{}}, {{},{{},{{}}},._.} etc.

A Nested-Level is a common environment for a finite or non-finite collection of sub-scopes.

If a notom includes identical sub-scopes ( __ , .__ or __. are excluded), then it is called a First-Order Collection (or FOC).

An example:

,{... ,{},{},{}} ,{... ._. , ._. , ._.} ,{... ,{},{}, ...} ,{... , ._. , ._. , ...}
{{},{}} ,{{{}},{{}},{{}}} ,{... , {.},{.},{.}} ,{{._.},{._.}} etc.

The name of an atom or a notom within some FOC is determined by its internal property and/or its place in the collection. From this definition it is understood that each atom or notom within a FOC, has more than one name.

Non-FOC (or NFOC) is a nested-level that does not include identical sub-scopes.

An example:

,{... , {} , . , {}} ,{... , {._.} , ._. , ._.} ,{... ,{.},{}, ...} ,{... , ._. ,{._.} , ...}
{{},{.}} ,{{{}} ,{} ,{{}}} ,{... ,{},{.},{.}} ,{{},{._.}} etc.

Any atom ( __ is excluded) or notom has a unique name only if it can be distinguished from the other atoms or notoms that share with it the same nested level.

Let redundancy be more than one copy of the same entity can be found.

Let uncertainty be more than a one unique name is related to an entity.

An edge and a point:

A point is a non-composed and non-empty scope that has no directions where a direction is < , > or < > .

An example: .

An edge is an inseparable part of an atom that has a direction.

An example: ._. , .__ , __.

Let p be a point.

Let s be a segment.

The axiom of independency:
p and s cannot be defined by each other.

Huh, some guys are lucky.