The triviality of English Dan Asimov Adam P. Gaucher Michael Kleber (Posed problem 25 sept 2015) Andy Latto Wouter Meeussen Warren D Smith Allan Wechsler Regard English as a left-cancellative and right-cancellative multiplicative semigroup with identity, i.e. obeying the relations XY=ZY or YZ=YX implies X=Z, and having an element "1" such that 1X=X1=X. If any two different-meaning words which sound the same are "equivalent" we shall show each letter of the alphabet (as well as space and apostrophe) generates the trivial group, i.e. all alphabet letters (and space and apostrophe) equal the identity element. A=1: AISLE=ISLE B=E=1: PLUM=PLUMB, CLIME=CLIMB C=S=L=1: SCENT=CENT=SENT, CELL=SELL=CEL=SEL, MILL=MIL D=A=1: AD=ADD, REDD=READ=RED E=1: AID=AIDE, AR=ARE G=1: GNU=NU, GNOME=NOME, SIGN=SINE H=W=1: HOUR=OUR, O=OH=OWE I=U=1: BAIT=BATE, MAID=MADE, AID=ADE, FIR=FUR K=W=1: KNIGHT=NIGHT, SCULL=SKULL, KNOT=NOT, KNOW=NO M=1: MUMM=MUM, DAMNED=DAMMED N=1: DAMN=DAM, IN=INN, SUNN=SUN, CONN=CON O=1: TO=TOO P=1: PSI=SIGH, COUP=COO, WOO=WHEW R=1: TOR=TORR, CURR=CUR T=1: BUT=BUTT, BAT=BATT U=1: BY=BUY, GUILT=GILT W=1: WRAP=RAP, WRUNG=RUNG, WRITE=RITE=RIGHT, TWO=TO, SO=SOW, WHO=HOO X=1: LOX=LOCKS, TAX=TACKS, FAUX=FOE A=1: ROAD=ROWED E=L=U=1: WHOLLY=HOLEY=HOLY, SEW=SOW, NU=NEW, PER=PURR F=1: TOUGH=TUFF, STAPH=STAFF, PHI=FI, PHAT=FAT J=1: GIN=JINN, JEANS=GENES L=1: WOOD=WOULD O=1: SUE=SOU Q=1: QI=KI, CUE=QUEUE V=1: LUVVY=LOVEY=LUVVIE, BEVY=BEVVY Y=N=1: HYMN=HIM, BITE=BYTE, LIE=LYE Z=1: DAYS=DAZE, PHIZ=FIZZ, FAT=PHAT, FAZE=PHASE miscellaneous other non-obvious relations: REIGN=RAIN, HEIR=AIR, WHETHER=WEATHER, COLONEL=KERNEL space=1: CATWALK=CAT_WALK, BE_CALM=BECALM aprostrophe=1: CANT=CAN'T A weaker result had been shown by Jean-Francois Mestre, Rene Schoof, Lawrence Washington, Don Zagier: http://projecteuclid.org/download/pdf_1/euclid.em/1062620828 Quotients Homophones des Groupes Libres, Experimental Mathematics 2,3 (1993) 153-155. They also claimed there was an unpublished proof by Herbert Gangl & Norbert Schappacher of the triviality of German; speculated Japanese katakana was relation-free; and gave as proof of the triviality of French SOIE=SOI, SERRE=SERT=CERF, CE=SE, BALLE=BAL, LAID=LAIT, HAUT=AU, PARLENT=PARLE, ALLEZ=ALLER, SEPT=CET, CHAMPS=CHANT, FARD=PHARE, AN=EN, MAIS=METS, BAYER=BAILLER, SANG=CENT, JET=GEAI, ABBESSE=ABAISSE, KHAN=QUAND, LACQUE=LAC, COQ=COKE, POT=PEAU, WATT=OUATE, VAGUONS=WAGON. (They also showed various French accent marks are trivial.) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)