User blog:Nayuta Ito/Extended extended chain notation

I will start with extended chain notation. For more extension, I will write numbers below the chain.

$$a\rightarrow_b c=a\xrightarrow[b]{} c$$

This looks like chemical equation, so I call {what is below the chain} {catalyst}. The catalyst makes number really bigger.

Catalyst must not be one. Some reactions require more than one catalysts.

$$a\xrightarrow[1,0]{} b = a\xrightarrow[a\xrightarrow[a\xrightarrow[\cdots a\xrightarrow[]{} a]{} a]{} a]{} a $$ (b stories)

Next, using the same rule, we can define up to $$a\xrightarrow[1,x]{} b $$.

Then,

$$a\xrightarrow[2,0]{} b = a\xrightarrow[1,a\xrightarrow[1,a\xrightarrow[\cdots 1,a\xrightarrow[]{} a]{} a]{} a]{} a $$ (b stories)

Keep going on until $$a\xrightarrow[x,y]{} b$$. At this stage, I think this is as big as $$f_{\omega ^4} (x)$$.

Next, three catalysts come.

$$a\xrightarrow[1,0,0]{} b = a\xrightarrow[a\xrightarrow[a\xrightarrow[\cdots a\xrightarrow[]{} a,0]{} a,0]{} a,0]{} a $$ (b stories)

See the pattern? As you know, definition can be up to n-entry.

This notation makes up to $$f_{\omega^\omega}(x)$$.

If you have a question, ask in a comment.