A NOTE ON PIFFLES

by A. B. Smith

Introduction.

A.C.Jones in his paper "A Note on the Theory of Boffles", Proceedings of the National Society, 13, first defined a Biffle to be a non-definite Boffle and asked if every Biffle was reducible.

C.D. Brown in "On a paper by A.C.Jones", Biffle, 24, answered in part this question by defining a Wuffle to be a reducible Biffle and he was then able to show that all Wuffles were reducible.

H. Green, P. Smith, and D. Jones in their review of Brown's paper, Wuffle Review, 48, suggested the name Woffle for any Wuffle other than the nontrivial Wuffle and conjectured that the total number of Woffles would be at least as great as the number so far known to exist. They asked if this conjecture was the strongest possible.

T. Brown in "A collection of 250 papers on Woffle Theory dedicated to the honor of R.S.Green on his 23rd birthday" defined a Piffle to be an infinite multi-variable sub-polynormal Woffle which does not satisfy the lower regular Q-property. He states, but was unable to prove, that there was at least a finite number of Piffles.

T.Smith, L.Jones, R.Brown, and A.Green in their collected works "A short introduction to the classical theory of the Piffle", Piffle Press, $20, showed that all bi-universal Piffles were strictly descending and conjectured that to prove a stronger result would be harder.

It is this conjecture which motivated the present paper.

Dept. of Pure Mathematics, The University. Sheffield, England, 1966.