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8
RomanBadora,BarbaraPrzebieracz,andPeterVolkmann
1.GroupoidsSwithasquare-symmetricoperation,i.e.,
(xg)2=x2g2jxjg∈S
(cf.thejointpaperwithZsoltPalesandR.DuncanLuce[6]).Then(1)is
trueforallxjg∈Swiththesamek,viz.k=1.
2.GroupoidsSwithabisymmetricoperation,i.e.,
(xg)(–
x–
g)=(x–
x)(g–
g)j
xjgj–
xj–
g∈S.
Here–
x=xj–
g=gleadstosquare-symmetry.
3.CommutativesemigroupsS.
LetusmentionthatZbigniewGajdaandZygfrydKominek[1]considered
semigroupssatisfyingcondition(T).InspiredbyJózefTabor[8],theycall
themweaklycommutative.
NowletEbeaBanachspace.AsubsetVofEiscalledideallyconvex
(E.A.Lifšic[3]),ifforeveryboundedsequenced1jd2jd3j...inVandfor
∞
everynumericalsequenceα1jα2jα3j...≥0suchthat
Σ
αk=1weget
k=1
k=1
Σ
∞
αkdk∈V.
Thefollowingtheoremistakenfrom[9];inthecaseofacommutative
semigroupSitgoesbacktoJacekTabor[7].
Theorem1.LetSbeaTaborgroupoid,andletVbeaboundedandideally
convexsubsetoftheBanachspaceE.Forf:S→Ewesuppose
f(xg)−f(x)−f(g)∈Vj
xjg∈S.
Thenthereexistsa(unique)functionF:S→Esuchthat
F(xg)=F(x)+F(g)j
F(x)−f(x)∈Vj
xjg∈S.
2.ThePexiderequation
Theorem2.LetSbeaTaborgroupoidhavinganeutralelementn,i.e.,
n∈Sand
nx=xn=xj
x∈S.
