User blog comment:Rgetar/Variations of ordinal array function/@comment-32697988-20171215224922/@comment-32213734-20171215235405

Also, other designation used here.

Coordinates are counted from the right beginning from 0. An array can be written as row of ⟨coordinates⟩element pairs separated with commas. So,

X = 5, 0, 0, 1, 1, 0, 0, 8, 7, 0, 0, 0

can be written as

X = ⟨11⟩5, ⟨10⟩0, ⟨9⟩0, ⟨8⟩1, ⟨7⟩1, ⟨6⟩0, ⟨5⟩0, ⟨4⟩8, ⟨3⟩7, ⟨2⟩0, ⟨1⟩0, ⟨0⟩0

X0 is array X with its last element (i. e. element with coordinates 0) set to 0. Element of X with coordinates 0 in my example is 0 (⟨0⟩0). So,

X0 = 5, 0, 0, 1, 1, 0, 0, 8, 7, 0, 0, 0 = X

X' is coordinates of last base element of X. In my example last base element with coordinates: ⟨3⟩7. So,

X' = 3

X" is X' applied twice. So,

X' = 3 = ⟨0⟩3

X" = 0

(So, X" can be non-zero only for multi-dimensional X, when its coordinates themselves are arrays).

(X; a; b) is a, if X' = 0, and b, if X' ≠ 0

(that is (X; a; b) is a, if last element of X is not zero, and b, if it is zero)

sup is supremum

Also (if short forms are used): [-1]a = a, and elements with negative coordinates should be ignored.