Generators

Definition

Generative element to the typed structure S is such element that if contained in that structure, would become a primer to hierarchical type ordering. In other words, element can be considered a generative to typed structure of
S[Tn]S[T_{n}]
only if there can exist structure of higher order, that would take exactly, one rewrite rule to generate identity equation, for all the generated objects. Generative element element is always of type
Tn+1T_{n+1}
xXg(x)=ø\forall_{x \in X} |g(x)=ø

Identity

An element, can be considered a generative to a structure S, only when all objects, belonging to that structure, can be said to be have uniform type T and only when there can exist a hypothetical structure of higher order, than S.
In other words, we can speak about generator, only when all generated objects, are distinguishable to with the generator, by exactly the same property. Formally:
xSD(x,g)=T(x)T(g)=1\forall{x \in S} | D(x,g) = T(x)-T(g) = -1
Where, g is generative element of the structure and denotes a deffer as a functor of arity 2.

Generator of natural numbers

Generator of
N+N_{+}
is
()(-)
, because
n(n)=ø\forall_{n} -(n) = ø
, this is to be interpreted, simply as an algebraic expression
nN+nn=0\forall_{n \in N_{+}}| n-n=0
.

Equivalent definition

Generator of structure S is such object, which if contained in its' structure would be a primer to the symmetrical object of the one that generates. Therefore every structure, to which we can assign a generative element, is itself a primer to a symmetrical object to itself, that is constructed, through the composition of itself and its' generator.

Duality and symmetry

Given the upper, we can conclude there exists duality between hierarchical ordering and existence of symmetrical structures.
This is more, or less what I, by "All is dual [...] and up to it's anti-auto-morphism".
Up to the anti-morphism:
Can be understood as lacking of equivalence relation, and absence of morphism between objects higher of different consecutive orders. This is requirement in order, for the existence of generative element, because if morphism Y -> X existed, then g couldn't be considered a primer to the generator. Primer requires unambiguous implication.
Formally:
Let Y, be of higher type than x. Then we can speak about a generative element g=y that belongs to the structure Y <==> Pr(X,X+y) => (Pr(y) -> G(y, X).
Pr(X, Y) is a function of arity 2, that returns a primer from Y and the object, to which it is a primer(in this case a generator, of structure X and generative element g=y)

Auto-anti-morphism (without "up")

Anti + Auto can be thought as the functor created through the composition of self-reference operator and the anti operator.

TODO

  • Primer | introduce the property set and why it's not necessary to specify the feature set when speaking about generator