Introduction to NEPT and NOPT structures

[MikeSmith]

Can you explain what you mean by OrderType?

Also, what does NOPT stand for?

ok maybe i didn't explicitly say that
NOPT =df Nested Operational Power Towers
NOPT structures are defined for any "suitable" operation.

OrderType...
First consider how to
write 3^^3, 3^^^3, 3^^^^3, 3^^^^^3 and so on
in terms of exponentiation power towers
*not* tetration powers etc
the higher the hyperoperation the more layers of nesting are required.
With 3^^3 , only a single exponential power tower is created
tetration is hyperoperation with n=4 of course
but we say (or define) the first nontrivial NOPT structure
to have OrderType=4 to make a suitable correspondence
with tetration (where tetration is expressed in NEPT form)

You could say that the "trivial" NOPT structure is just "n".
Here n is a small number expressed with a linear sequence of digits with standard interpretation, using SPN (Standard Positional Notation)
And SPN implicitly contains the first 3 operations (+, *, exp)

The first "nontrivial" NOPT structure is just "theta ) n"
"theta" by itself is a "formal power tower", not a number
as there is no height information.
"theta ) n" obviously is a power tower, as the height of the power tower is given by n.