“main” — 2005/6/23 — 17:58 — page 187 — #1
The gene al se in he MCIM Iso opic Model
R. M. Falcón and J. Núñez
Abs ac . To ob ain a bigge numbe o ma hema ical and physical applica ions o
he San illi’s iso heo y, he la es s udies ha e shown he necessi y o analyzing iso opic
models which use non associa i e laws. The main goal o his pape is o gi e a gene -
aliza ion o he iso opic cons uc ion model based on he mul iplica ion (MCIM), which
is use ul o ob ain non associa i e ma hema ical isos uc u es.
Keywo ds: San illi’s Iso heo y, iso opic model, isos uc u e.
Ma hema ical subjec classi ica ion: 03H05, 08A05, 03C65.
In oduc ion
In 1978, R.M. San illip oposed agene aliza ion o he con en ional Lie’s heo y
by using iso opies. The iso opies o Lie’s heo y we e cons uc ed o li he
heo y om i s cu en sole applicabili y o linea sys ems o nonlinea sys ems,
li ing achie ed ia he econs uc ion o linea i y on isospaces o e iso ields.
I was he i s s age o wha is ac ually known as San illi’s Iso heo y [1]. He
conside ed ha he basic uni Io each ma hema ical s uc u e can depend on
se e al ac o s ex e nal o he sys em in which we a e placed, like coo dina es,
speed, ime, densi y, empe a u e, and so on. I in ol es an isouni o he ype
b
I=b
I(x, , , μ, τ, ...). By using his p inciple, San illi ca ied ou a s ep by
s ep cons uc ion which gene alizes he mos common ma hema ical s uc u es,
o igina ing hose denomina ed ma hema ical isos uc u es [2], [3]. I allowed
him o p og ess in he de elopmen o some physical applica ions, mainly in
Quan um Mechanics and Dynamics o Pa icles [4].
In2001, heiso opiccons uc ionmodelbasedon hemul iplica ion( omnow
on,i willbedeno edbyMCIM)wasin oducedbyou sel es(see[5])in hesame
way as he one p oposed by San illi, al hough by pu ing a special emphasis in
he use o ∗-laws. La e , i was imp o ed in [6] and [7]. Ne e heless, o enla ge
he numbe o ma hema ical isos uc u esand o ge newp ac ical applica ions, i
is necessa y o weaken he associa i i y hypo hesis, ob aining in his way mo e
Recei ed 7 July 2004.
“main” — 2005/6/23 — 17:58 — page 188 — #2
188 R. M. FALCÓN and J. NÚÑEZ
gene al cases. To do i , all he elemen s which o m he gene al se o e e y
iso opy mus be se led. This will be hen he main objec i e o his pape .
1 P elimina y Concep s
F om now on an iso opic li ing o iso opy will be any co espondence be ween
a ma hema ical s uc u e and ano he one o he same ype, ha is o say, in such
a way ha bo h e i y he same p ope ies. No e ha acco ding o his de ini ion
an iso opy could no be a map. The image is hen called iso opic s uc u e o
isos uc u e [1].
San illi’s iso opic model o 1978 is based on he gene aliza ion o he ini ial
uni : I→b
I=b
I(x, , , μ, τ, ...). So, ixed any ma hema ical s uc u e E,
endowed wi h an inne law ×, his model conside s a se V⊇E, endowed wi h
an associa i e law ∗and I,b
I,T∈V, whe e I∈Eis he uni o ∗in Vand
T=b
I−I.V,Tand b
Ia e espec i ely called gene al se , iso opic elemen and
isouni y o he iso opy. So, i is de ined he isos uc u e b
E,endowed wi h he
lawb
×wi h uni b
Ias: b
E={bx=x∗b
I:x∈E},
b
ab
×b
b=ba∗T∗b
b=(a∗b)∗b
I, o all ba,b
b∈b
E.
The MCIM iso opic model gene alizes he San illi’s one, by using as many
∗-laws as he ini ial ones in E, in such a way ha i Eis endowed wi h an inne
law ◦, i will ha e associa ed a ∗-law ?, such ha :
b
ab◦b
b=(a?b)∗b
I, o all ba,b
b∈b
E.
Le us obse e ha i ?≡ ∗, henb◦ ≡ b
×is he San illi’s p e ious law.
So, any iso opy is gi en in he ollowing way:
Con en ional le el −−−−−−−−−−−−−−−−−−−→
Gene al le el
(V, ?, ∗, ...)
∪
(E,+,×, ...) (E, ?, ∗, ...)
a a
↓]∼
=↓I
P ojec ion le el π
←−−−−−−−−−−−−−−−−−−−− Iso opic le el
(b
E,b
+,b
×, ...) (b
E,b
+,b
×, ...)
π◦I(a)=ba=a∗b
II(a)=ba
b b
Ibab
+b
b=d
a?b
b b
Ibab
×b
b=[
a∗b
abb+b = (a ? b) ∗
abb×b = (a ∗ b) ∗
“main” — 2005/6/23 — 17:58 — page 189 — #3
THE GENERAL SET IN THE MCIM ISOTOPIC MODEL 189
2A Gene aliza ion o he MCIM
2.1 A i s gene aliza ion o he MCIM
To ge a gene aliza ion o he MCIM, we a e going o weaken some o he
condi ions which a e necessa y when using his iso opic model. To do i , le us
conside he ame o hypo hesis:
(1) A se E, endowed wi h an inne law ◦, a se V⊇E, endowed wi h a inne
law ∗, wi h uni I∈V, and an a bi a y elemen b
Iin V.
So, le us de ine:
b
E={bx=x∗b
I},bab◦b
b=(a∗b)∗b
I.
We will say ha b
Eis injec i e i ba=b
b∈b
Eimplies a=b∈E.
Le us obse e ha i ∗is associa i e and he e exis s T=b
I−I∈V, hen we
ha e he MCIM iso opic model. In his case, b
Ewill be injec i e.
In he gene al case, we ha e he ollowing esul :
P oposi ion 2.1. I ∗is associa i e in V, henb◦is associa i e in b
V, p o ided i
is well-de ined, ha is, i o all a,b,c∈Esuch ha ba=b
b, i is e i ied ha
b
ab◦bc=b
bb◦bcand bcb◦ba=bcb◦b
b.Mo eo e , i b
Eis injec i e, he con e se is also
e i ied.
P oo . Le ∗be associa i e. Asb◦is well-de ined, i means ha such a law has
a pe ec sense o e b
E. Sob◦is associa i e:
b
ab◦b
bb
◦bc=((a∗b)∗c)∗b
I=(a∗(b∗c))∗b
I=bab◦b
bb◦bc.
Le us now suppose ha b◦is associa i e and b
Eis injec i e. So:
((a∗b)∗c)∗b
I=b
ab◦b
bb
◦bc=bab◦b
bb◦bc=(a∗(b∗c))∗b
I.
Now, as b
Eis injec i e, we ha e ha (a∗b)∗c=a∗(b∗c)and hus, ∗is
injec i e. 
Le us obse e ha i Eb is injec i e, hen b◦ is well-de ined. Howe e , as we
will see nex , he con e se is no ue in gene al. Indeed, le us suppose ha Eb
is no injec i e. So, hey should exis , a leas , wo di e en elemen s x, y ∈ E
“main” — 2005/6/23 — 17:58 — page 190 — #4
190 R. M. FALCÓN and J. NÚÑEZ
such ha x∗b
I=y∗b
I. Besides, as b◦should be well-de ined, i mus be
(x∗a)∗b
I=(y∗a)∗b
I, o all a∈E. I we also conside ha Ehas he in e se
elemen p ope y wi h espec o ∗, we can ake a=x−I∈E,and so:
b
I=(y∗x−I)∗b
I.
Then, le us obse e ha i ∗is associa i e in V, i will imply he non exis ence
o T=b
I−I,because in he o he case, by mul iplyingon he igh by Tin helas
equali y, i is deduced ha I=y∗x−I; his is, x=y, which is a con adic ion.
On he o he hand, i ∗is non associa i e in V, he co esponding iso opic
elemen could exis o no , al hough i i exis s, hen he ollowing equali y has
o be sa is ied: I=y∗x−I∗b
I∗T.
o equi alen ly:
T=y∗x−I∗b
I−I.
So, we ha e hen p o ed he ollowing esul :
P oposi ion 2.2. By adding in (1) ha Ehas he in e se elemen p ope y wi h
espec o ∗and ha b
Eis no injec i e, hen ei he T=b
I−Idoes no exis o i
i exis s, hen ∗is non associa i e in V. Besides, in his case, o all x,y∈E
such ha bx=by,i is e i ied ha T=y∗x−I∗b
I−I.
2.2 A second gene aliza ion o he MCIM
Le us now conside he ollowing se o hypo hesis:
(2) A se E, endowed wi h wo inne laws ◦and •, a se V⊇E, endowed
wi h wo inne laws, ∗, wi h uni I∈V, and ?, and a bi a y elemen s b
I,T
in V.
Le us de ine:
b
E={bx=x∗b
I},
b
ab◦b
b=[(ba∗T)∗(b
b∗T)]∗b
I,
b
ab•b
b= [(ba∗T) ? (b
b∗T)] ∗ b
I.
b
Le us obse e ha i ∗ is associa i e and he e exis s T = I
−I ∈ V , hen we
ha e he MCIM iso opic model.
“main” — 2005/6/23 — 17:58 — page 191 — #5
THE GENERAL SET IN THE MCIM ISOTOPIC MODEL 191
This gene aliza ion is use ul o s udy some o he p ope ies ha he gene al
se Vshould e i y. To see i , we will show i s an explici example. We a e
going o ge he complex numbe s s uc u e (C,+,∙)as an iso opic p ojec ion o
he eal numbe s s uc u e (R,+,×). To do i , we will employ he laws ?|R≡ +
and ∗|R≡ × and he isouni b
I=b
I( ), depending on a ac o “ ime” ∈R, such
ha x∗b
I( )=x+ i ∈C, o all x, ∈R. In his way, b
R=C.
The ollowing s ep is o ge +and ∙as laws in he p ojec ion le el. To achie e
his objec i e, i will be necessa y o use again he ime ac o , al hough now in
an in e se sense wi h espec o he p ocedu e used o ob ain he iso opic se in
ques ion. Fo his eason, a good way o deal wi h his aspec would be o use
he iso opic elemen T=T( )=b
I( )−1. So:
(a+bi)b
+(c+di)=([(a+bi)∗T]?[(c+di)∗T])∗b
I,
(a+bi)b∙(c+di)=([(a+bi)∗T]∗[(c+di)∗T])∗b
I.
Then, we can ake ad an age ha he iso opic elemen depends on he ime o
conse e he ollowing use ul in o ma ion: ixed (a+bi)∈C, his elemen was
jus ob ained when he ime is =b. To ge i , we can de ine he ope a ion ∗in
he gene al se Vas ollows:
(a+bi)∗T=ab
whe e ab(which could be iden i ied wi h he pai (a,b)in R2) would belong o
V. So, i is use ul o conside he ollowing se :
CT={ab=(a+bi)∗T:a,b∈R}.
Rema k. Inasimila way,i wea enowconside inganiso opyo anys uc u e
E, i will be use ul o conside he se :
ET=nba∗T:ba∈b
Eo.
The e o e, he co esponding gene al se Vcan be de ined as:
V=E∪b
E∪ET∪b
I,T.
Now, a his poin , i is impo an o no e ha in he case o he MCIM iso opic
model,se s Eand ETcoincide. I issobecauseunde hismodel heassocia i i y
o he ope a ion ∗in he gene al se Vand he exis ence o he iso opic elemen
T=b
I−Ia e sa is ied by hypo hesis. So, o all a∈Ei is e i ied:
a∗b
I∗T=a∗b
I∗T=a.

“main” — 2005/6/23 — 17:58 — page 192 — #6
192 R. M. FALCÓN and J. NÚÑEZ
In pa icula , his p o es he necessi y o gene alizing such a model o he
ollowing h ee cases: i s , when he associa i i y o ∗in Vis no e i ied.
Second, when he iso opic elemen does no exis and hi d, when none o hese
wo condi ions is sa is ied. These h ee cases will be deal in his pape .
On he o he hand, i would be necessa y o de ine he co esponding ∗-laws
be weenelemen so ET.By example, in he conc e ecase whichwe a e dealing,
i would be con enien o de ine he laws ?and ∗in RT={ab:a,b∈R}as
ollows:
ab?cd=(a+c)b+d,ab∗cd=((a×c)−(b×d))b×c+a×d
I would be also necessa y o de ine he ope a ion ∗be ween elemen s o RTand
he isouni b
I:ab∗b
I=a+bi
In his way, we would ob ain in he p ojec ion le el:
(a+bi)b
+(c+di)=([(a+bi)∗T]?[(c+di)∗T])∗b
I=
=(ab?cd)∗b
I=((a+c)b+d)∗b
I=(a+c)+(b+d)i.
(a+bi)b
×(c+di)=([(a+bi)∗T]∗[(c+di)∗T])∗b
I=
=(ab∗cd)∗b
I=((a×c)−(b×d))b×c+a×d∗b
I=
=(ac −bd)+(bc +ad)i.
No e ha , in pa icula , we ha e ob ained ha :
b
R,b
+,b
×=(C,+,∙)
So, i is p o ed ha an iso ield, which con en ionally canno be do ed o a o al
o de , can be ob ained s a ing om a o ally o de ed ield, like eal numbe s, by
usinganiso opicli ing. In hisway, i isalsop o ed henecessi yo conside ing
he isoo de which was de inedin [6]. Mo eo e , he s udy o he isoo de should
be deepe in his conc e e case, o sol e some p oblems which appea . We will
deal wi h his s udy in u u e wo ks.
Finally, we can summa ize he in o ma ion ela i e o ou gene al se as ol-
lows: V=R∪C∪RT∪b
I,T
being, o all a,b,c,d∈R:
a?b=a+b,a∗b=a×b,a∗b
I( )=a+ i
“main” — 2005/6/23 — 17:58 — page 193 — #7
THE GENERAL SET IN THE MCIM ISOTOPIC MODEL 193
(a+bi)∗T=ab,ab∗b
I=a+bi,ab?cd=(a+c)b+d
ab∗cd=(ac−bd)ad+bc,b
I∗T=T∗b
I=I=1≡10
An aspec o ema k is ha by using his cons uc ion we ha e shown an
example in which he ope a ion ∗is non associa i e in V, since ixed a∈Rand
a ce ain ins an o ime 0,one has:
(a∗b
I)∗T=(a+ 0i)∗T=a 06= a=a∗I=a∗(b
I∗T)
Besides:
(a∗T)∗b
I=a0∗b
I=a=a∗(T∗b
I)
Mo eo e , i we demand he cons uc ion o be cohe en , ano he condi ion o
be sa is ied would be he ollowing:
a∗b
I∗T∗b
I=a∗b
I
As T=b
I−I,i mus be b
I=I∗b
I=(b
I∗T)∗b
Iand hus, we ge he Mou ang’s
Iden i y:a∗b
I∗T∗b
I=a∗(b
I∗T)∗b
I
2.3 The MCGIM iso opic model
We can y o gene alize he MCIM iso opic model by conside ing ha Tis no
he in e se o he isouni b
Iin V. In his case, we will call gene alized iso opic
elemen o T, which is a di e en concep om iso opic elemen :
De ini ion 2.3. I will be said ha an iso opy ollows he gene alized MCIM
( om now on, MCGIM) when, unde usual no a ions, he e exis s an elemen
T∈Vsuch ha , o all a∈E, i is e i ied:
ba∗T∗b
I=ba=a∗b
I
So, he se ETacqui es a g ea impo ance when we mus decide he iso opy o
cons uc , because such a se will be he one ha has o e i y p ope ies sa is ied
by he isos uc u e o ge . In his way, i is con enien o cen e us in he na u e
o he se ET.
P oposi ion 2.4. Le us conside an iso opic li ing by using MCGIM. Then,
E=ETi and only i o all a∈E,i is e i ied:
a∗b
I∗T=a.
“main” — 2005/6/23 — 17:58 — page 194 — #8
194 R. M. FALCÓN and J. NÚÑEZ
P oo . The su icien condi ion is e iden . So, le us suppose a∈E. As
E=ET, he e exis s b∈Esuch ha a=b∗b
I∗Tand so, a∗b
I=
b∗b
I∗T∗b
I=b∗b
I. Thus, a=a∗b
I∗T.
This esul allowsus oha econ idencein hegene aliza ionachie ed,because
all he basic aspec s o iso opies a e p ese ed. In ac , we a e imposing he
associa i i y o he ope a ion ∗a no ime, which is pa icula ly a o able o he
s udy abou he iso opical ela ionship be ween associa i e and non associa i e
ma hema ical s uc u es.
In he o he way, no e ha in he pa icula case in which ∗is associa i e in V
and T=b
I−Iis he iso opic elemen , we go hen he MCIM iso opic model as
a pa icula case o he MCGIM one. Mo eo e , he ollowing esul is e i ied:
P oposi ion 2.5. By using he MCGIM iso opic model, i ∗is an associa i e law
in V, wi h uni I, hen hey a e e i ied:
a) I he e exis s a∈Esuch ha a∗b
Ihas le in e se in Vwi h espec o ∗,
hen T∗b
I=I.
b) I ET=Eand he e exis s a∈Eadmi ing le in e se in Vwi h espec
o ∗, hen b
I∗T=I.
In pa icula , i (a) and (b) a e sa is ied, hen Tis he iso opic elemen o such
an iso opy.
P oo . No e ha he inal asse o he p oposi ion is e iden , because he
condi ion T=b
I−Iis di ec ly deduced om (a) and (b). So, i is su icien o
p o e bo h i ems. Wi hou los o gene ali y, we can suppose in bo h o hem
ha ∗is associa i e:
a) Le us suppose ha he e exis s a le in e se o a∗b
Iin V, o a gi en
a∈E. Then:
a∗b
I=ba∗T∗b
I=(a∗b
I)∗(T∗b
I)⇒I=T∗b
I.
b) Le us now suppose ha ET=Eand ha he e exis s a le in e se o
a∈Ein V, wi h espec o ∗. Then:
a=(a∗b
I)∗T=a∗(b
I∗T)⇒I=b
I∗T.
“main” — 2005/6/23 — 17:58 — page 195 — #9
THE GENERAL SET IN THE MCIM ISOTOPIC MODEL 195
So, when wo king wi h associa i e ∗-laws, i is no mo e in e es ing he injec-
i i y o he used iso opy bu i E=ETo no . Ne e heless, wha is he ma e
wi h iso opies in which E6= ET?They a e possible, because we ha e al eady
conside ed he case in which b
R=Cwas ob ained, being RT6= R.Such an ex-
ample is use ul o show ha in e e y iso opy bo h p ope ies and laws demanded
o b
Eha e o be sa is ied in ET,because such p ope ies will be inhe i ed by he
isos uc u e o ob ain. So, gi en an ini ial ma hema ical s uc u e and ano he o
he same ype, e e yiso opic li ingconsis s on indinga supe s uc u e Vwhich
con ains bo h s uc u es plus an isouni and an gene alized iso opic elemen . In
his way, le us conside inally he ollowing:
Example2.6. Le ussuppose henilpo en g oup(Z/Z2,+), wi h he usualsum.
We a e going o ca y ou an iso opic li ing o i by s a ing om elemen s o
iso opy b
I=0(associa ed wi h he iso opic elemen T) and ∗de ined bo h in
such a way ha , ixed a,b∈Z/Z2, hey a e e i ied:
a∗b=a+b;a∗b
I=ba=a+0=a
In his way, we ge [
Z/Z2=Z/Z2.We also de ine, o all a∈Z/Z2:
ba∗T=aT,aT∗b
I=a,0T∗0T=0T,
1T∗1T=1T=1T∗0T=0T∗1T.
So, ixed a,b∈Z/Z2,we ha e in he p ojec ion le el:
b
ab
+b
b=π◦I(ba∗T)∗(b
b∗T)=π◦I(aT∗bT).
Hence: 0b
+0=0,1b
+1=1=1b
+0=0b
+1,
and hus, [
Z/Z2,b
+=Z/Z2,b
+is a non nilpo en g oup.
So,in hisway,anonnilpo en ma hema icalisos uc u ehasbeenob ainedas
aniso opicp ojec iono anilpo en s uc u e. Howe e ,i hasbeennecessa y o
ge ingi o impose inou cons uc ion ha heco espondingpai (Z/Z2)T,∗
coincides wi h he isos uc u e sea ched.