In my last blog post, I formally defined Hybrid Hypermathematics, a stronger variant of hypermathematics where standard hyperoperations are applied before concatenation. In the comments on that post, P-bot noted the extensionality of the system, specifying that it would be very easy for someone to apply my work into their own definitions. However, what wasn't mentioned was that I deliberately designed the formal rules to allow for extensions using arrays. In this blog post, I will be extending Hybrid Hypermathematics even further by creating an array notation for it. But before that, we must first extend the standard hyperoperator sequence into an array notation as well. It's go time:
Hyperoperator Array Notation[]
Domain: positive integers; Order of operations: right-to-left
Rule 1: a[1]b=a+b
Rule 2: a[#,0]b=a[#]b, where # is the rest of the array
Rule 3: a[c+1,#]b=a[c,#]a[c,#]...length b...a[c,#]a; length measures the number of a's
Rule 4: a[#0,0,c+1,#]b=a[#0,b,c,#]b, where #0 denotes a string of 0's
Example: 3[0,0,3,0]3=3[0,0,3]3=3[0,3,2]3=3[3,2,2]3=3[2,2,2]3[2,2,2]3=3[2,2,2]3[1,2,2]3[1,2,2]3
Hybrid Hypermathematics Array Notation[]
Domain: same as before; Order of operations: same as before
Rule 1: a[1]b=ccc...length c...ccc, where c is the sum of a and b
Rule 2: a[#,0]b=a[#]b, where # is the rest of the array
Rule 3: a[c+1,#]b=d[c,#]d[c,#]...length d...d[c,#]d, where d is the answer to a[c+1,#]b in Hyperoperator Array Notation; length measures the number of d's
Rule 4: a[#0,0,c+1,#]b=d[#0,d,c,#]d, where #0 is a string of 0's, and d is the answer to a[#0,0,c+1,#]b in Hyperoperator Array Notation
Examples: 3[0,0,3,0]3=3[0,0,3]3=a[0,a,2]a; 3[2,2]3=a[1,2]a[1,2]...length a...a[1,2]a (in both examples, a is the answer to that expression in Hyperoperator Array Notation)