Extensible-E System

The Extensible-E System (ExE) is a work-in-progress array notation by Sbiis Saibian. It is a blanket term for Hyper-E Notation, Extended Hyper-E Notation, Cascading-E Notation, Extended Cascading-E Notation, and any future extensions to come.

All notations in Extensible-E have a number of standardized traits. All notations can be expressed generically in the form:

Ea&a&a& ... &a&a

where the a's are all positive integer arguments, and the &'s are delimiters chosen from the delimiter set defined for the particular notation. Each notation also defines fundamental sequences for all decomposition delimiters.

Hyper-E only allows use of the single hyperion ( # ) as a delimiter. Extended Hyper-E uses delimiters below #^#. Cascading-E Notation uses delimiters below #^^# Extended Cascading-E Notation uses delimiters below #{#}#.

All ExE type notations follow 5 fundamental laws. These laws have priority, where the earlier the rule, the higher priority it has. Only if the condition of that rule are not met does one progress to check for the next rule. The 5 law are:

1. Base Case If there is only a single argument: En = 10^n

2. Decomposition Case If the delimiter between the last two arguments is a decomposition delimiter: @m&n = @m&[n]m

3. Terminating Case If the last argument = 1:

@&1 = @

4. Expansion Case If the delimiter between the last two arguments is not the proto-hyperion: @m&*#n = @m&m&*#(n-1)

5. Recursive Case Otherwise: @m#n = @(@m#(n-1))

For the lowest level notations, some of the rules may never apply. For example in xE#, there are no decomposition delimiters, so Rule 2 never applies. In E#, there is only the hyperion as a delimiter, so neither Rule 2 or Rule 4 ever applies.

ExE could also be called "SAN" (Saibian's array notation), similar to BAN (Bird's array notation) or HAN (Hyperfactorial array notation or Hollom's array notation).