User:Inthar/Endoparticular extensions

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Goal: Formalize Leri's notion of temp extension naturalness or a some notion that is stronger than Leri!naturalness.

Special case 1

Suppose the base temp tempers out (Sk)^m S(k+1)^n for nonzero integers m and n. An extension by one basis element is natural if it tempers out Sk and S(k+1).

Examples

Porcupine

2.3.5 Porc -> 2.3.5.11 Porc is natural:

250/243
= (10/9)^3/(4/3)
= (10/9)^2(11/10)S(10)/(4/3)
= (10/9)(11/10)(12/11)/(4/3)*S(10)^2*S(11)
= S(10)^2*S(11)

And 2.3.5.11 Porc indeed tempers out S10 and S11.

Kleismic

2.3.5 Kleismic -> 2.3.5.13 Kleismic is natural because 15625/15552 = S(25)^2*S(26).