Supremus Array Notation is denoted with [] brackets instead of {} brackets because it is easier to understand [] brackets than it is {} brackets. [n]a single letter inside brackets just represents a number. [n,b] this is just exponentiation. [n,b,c] this uses up arrows for example [5,6,7] means 5↑(6)7 where there are 6 arrows between the 5 and 7. The 4th level uses Conway chained arrows as this is the Natural next step to go after Knuth up arrows. [5,6,7,8,9]=5→6→7→8→9. The 5th level uses repeated or iterated Conway chained arrows for example [5,6,7,8,9(5,6,7,8,9)] means 5→6→7→8→9...5 →6→7→8→9. This means that there are 5→6→7→8→9 number of arrows in the chain and also 5→6→7→8→9 things in the chain. Or in other words the number of arrows is equivalent to the number of things in the chain. The 6th level uses the basic levels of the Fast growing Hierarchy up to omega level. For example [f(n),5]=[f(5),5)=fw(5)= f5(5). The 7th level uses the fast growing hiearchy from omega+n level to omega^omega level. For example [f(w+p),6]=[f(w+4),6]= fw+4(6) and [f(w^w),5]=fw^w(5)=fw^w(5). The following levels will follow shortly. Before I publish I have a few questions is this a good notation? Second how could I improve it?