User blog:Rgetar/Change of my array notation

Recently I came up with a "rule": creating a notation system, write entities in order of "strength" - first write the "strongest" entities, then more "weaker".

But then I noticed that my array notation does not fit this "rule": first I write an element of array, then its coordinates. But coordinates are "stronger" than elements.

For example, in array of φ(array), for φ(1,0), increasing the element ("1"), we get φ(2,0), φ(3,0), ..., and increasing its coordinates, we get φ(1,0,0), φ(1,0,0,0), ...

So, my array notation was something like this:

α7<β7>α6<β6>α5<β5>α4<β4>α3<β3>α2<β2>α1<β1>α0<β0>

where αn is an element, and <βn> is its coordinates. (βn also may be an array).

Now I will write first coordinates, and then an element at this coordinates, and I will add a separator between coordinates-element pairs like this:

<β7>α7, <β6>α6, <β5>α5, <β4>α4, <β3>α3, <β2>α2, <β1>α1, <β0>α0

And, some coordinates may be omitted: <0> and adjacent values of coordinates left to known coordinates.

So, if coordinates of last element of an array is omitted, it is <0>. And if coordinates of not last element is omitted, it is coordinates of the element right to it increased by one (if coordinates is an array itself, its zero-coordinates element is increased by one).

For example:

<ω + 1>α7, <ω>α6, <8>α5, <7>α4, <3>α3, <2>α2, <1>α1, <0>α0 =

α7, <ω>α6, α5, <7>α4, α3, α2, α1, α0

Example with array coordinates:

<<1>1<0>1>α2, <<1>1>α1, <5>α0 =

<<1>1<0>1>α2, <<1>1<0>0>α1, <5>α0 =

<1,1>α2, <1,0>α1, <5>α0 =

α2, <1,0>α1, <5>α0

All zero elements may be omitted. For example,

1, 0, 0, 0, 0, 1 =

<5>1, <4>0, <3>0, <2>0, <1>0, <0>1 =

<5>1, <0>1 =

<5>1, 1

With one exception: if all elements are zeros, at least one element should not be omitted. For example:

<0>0 =

0

Also, "standard form" of an array: its elements should be written in order of decreasing coordinates (if coordinates are arrays, should be compared first elements with largest coordinates, then, if they equal, second largest, etc.)