

DOI: 10.2478/s12175-013-0181-7
Ma h. Slo aca 64 (2014), No. 1, 1–12
ON THE STABILITY OF SOME PROPERTIES
OF PARTIAL ALGEBRAS UNDER POWERS
N. Chaisansuk* — S. Lee a ana alee* — J. ˇ
Slapal**
(Communica ed by Jiˇ ´ıRach˚unek )
ABSTRACT. In he pape , we s udy eigh p ope ies o pa ial algeb as mos
o which a e ela ed o diagonali y. Fo each o he p ope ies, we gi e sufficien
condi ions unde which his p ope y is p ese ed by powe s o pa ial algeb as.
c
2014
Ma hema ical Ins i u e
Slo ak Academy o Sciences
1. In oduc ion
In his pionee ing pape [1], G. Bi khoff in oduced he ope a ion o a ca dinal
powe o pa ially o de ed se s and showed he alidi y o he fi s exponen ial
law o he ope a ion, i.e., he law (AB)C∼
=AB×C.In[6],J.
ˇ
Slapal ex ended
he concep o a powe om pa ially o de ed se s on o a bi a y n-a y ela ional
sys ems and s udied i s beha io . The same au ho in oduced and s udied pow-
e s o n-a y algeb as in [7] and also powe s o pa ial algeb as in [8]. He showed
he impo ance o idempo ency, diagonali y (in he sense o [5]), mediali y (in
he sense o [3]), and commu a i i y (in he sense o [4]) o algeb as and pa ial
algeb as when s udying hei powe s. Mo eo e , in [9], J. ˇ
Slapal defined and
s udied he ollowing eigh p ope ies o pa ial algeb as: idempo ency, diag-
onali y, commu a i i y, s ong diagonali y, weak diagonali y, local diagonali y,
local an idiagonali y, and weak local an idiagonali y. The aim o his pape
2010 M a h e m a i c s Subjec Classi ica ion: P ima y 08A05, 08A30, 08A55.
K ey w o d s: pa ial algeb a, eflexi e, diagonal and commu a i e pa ial algeb as, powe o
pa ial algeb as.
This esea ch was suppo ed by he Royal Golden Jubilee Ph.D. P og am o he Thailand
Resea ch Fund., he G adua e School and he Facul y o Science, Chiang Mai Uni e si y,
Thailand.
The hi d au ho acknowledges a suppo om he B no Uni e si y o Technology, p ojec
No. FSI-S-11-3.
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N. CHAISANSUK — S. LEERATANAVALEE — J. ˇ
SLAPAL
is o gi e sufficien condi ions o uni e sal pa ial algeb as unde which hese
p ope ies a e p ese ed by exponen ia ing he algeb as and unde which he
exponen ia ion sa isfies he fi s exponen ial law.
2. Pa ial algeb as
Fo he basic concep s used conce ning pa ial algeb as we e e o [2]. Le Ω
be a nonemp y se . A amily τ=(Kλ;λ∈Ω) o se s will be called a ype.By
apa ial algeb a o ype τwe unde s and a pai G=G, (pλ;λ∈Ω)whe e G
is a nonemp y se , he so-called unde lying se o G,andpλis a Kλ-a y pa ial
ope a ion on G, i.e., a map pλ:Dpλ→Gwhe e Dpλ⊆GKλ, o each λ∈Ω. The
se Dpλis called he domain o pλ.I Kλis fini e, hen pλis said o be ini a y.
Th oughou he pape , he ype τo any pa ial algeb a G, (pλ;λ∈Ω)is
unde s ood o be he amily deno ed by (Kλ;λ∈Ω). The unde lying se o
a pa ial algeb a Gwill be deno ed by |G|. O cou se, gi en a pa ial algeb a
G, (pλ;λ∈Ω), when w i ing pλ(xi;i∈Kλ)=x( o some x∈G), we
au oma ically mean ha pλ(xi;i∈Kλ) is defined, i.e., ha (xi;i∈Kλ)∈Dpλ.
Le H=H, (qλ;λ∈Ω),G=G, (pλ;λ∈Ω)be a pai o pa ial algeb as
o ype τ.ThenGis called a pa ial subalgeb a o Hp o ided ha G⊆Hand,
o each λ∈Ω, whene e (xi;i∈Kλ)∈GKλand x∈H,
pλ(xi;i∈Kλ)=x⇐⇒ qλ(xi;i∈Kλ)=x
(no e ha Dpλ=Dqλ∩GKλ). A map :H→Gis called a homomo phism o H
in o Gi , o each λ∈Ω, qλ(xi;i∈Kλ)=ximplies pλ( (xi); i∈Kλ)= (x).
The se o all homomo phisms o Hin o Gwill be deno ed by Hom(H,G).
The di ec p oduc o a amily Gi=Gi,(piλ;λ∈Ω),i ∈Io pa ial al-
geb as o ype τis he pa ial algeb a 
i∈I
Gi=
i∈I
Gi,(qλ;λ∈Ω)whe e

i∈I
Gideno es he ca esian p oduc o se s and, o any λ∈Ω, any ( k;
k∈Kλ)∈
i∈I
GiKλand any ∈
i∈I
Gi,qλ( k;k∈Kλ)= i and only
i piλ( k(i); k∈Kλ)= (i) o each i∈I.I Gi=G o e e y i∈I, henwe
w i e GIins ead o 
i∈I
Gi.
Gi en a pa ial algeb a G, (pλ;λ∈Ω)and λ∈Ω, an elemen x∈Gis said
o be a pλ-idempo en i pλ(xi;i∈Kλ)=xwhene e xi=x o each i∈Kλ.
Le G, K, L be se s. By a K×L-ma ix Mo e Gwe unde s and any map
M:K×L→G, i.e., M=(xij ;i∈K, j ∈L)whe exij ∈Gwhene e i∈K
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STABILITY OF SOME PROPERTIES OF PARTIAL ALGEBRAS UNDER POWERS
and j∈L. Wedeno ebyMT he ansposed ma ix o M, i.e., he ma ix
MT=(xji ;j∈L, i ∈K)o e G.
Le x∈Gand le M=(xij ;i∈K, j ∈L)beaK×L-ma ix Mo e G.Then
Mis said o be x-cons an p o ided ha xij =xwhene e i∈Kand j∈L,and
i is said o ha e x-cons an diagonal i K=Land xii =x o all i∈K.Le qbe
an L-a y pa ial ope a ion on G.ThenMis said o be q-ope a ional p o ided
ha (xij ;j∈L)∈Dq o each i∈K.Le pbe a K-a y pa ial ope a ion
on G.ThenMis said o be pq-ope a ional p o ided ha i is q-ope a ional
and (q(xij ;j∈L); i∈K)∈Dp; we pu pq(M)=p(q(xij ;j∈L); i∈K).
Finally, Mis said o be diagonally p-ope a ional p o ided ha K=Land
(xii ;i∈K)∈Dp; we pu ∆p(M)=p(xii ;i∈K).

2.1

([9]) A pa ial algeb a G, (pλ;λ∈Ω)o ype τis called
(1) idempo en i , o each λ∈Ω, e e y elemen x∈Gis a pλ-idempo en .
(2) commu a i e i , o any λ, µ ∈Ωandanypλpµ-ope a ional Kλ×Kµ-ma ix
Mo e Gsuch ha MTis pλ-ope a ional, MTis pµpλ-ope a ional wi h
pλpµ(M)=pµpλ(MT),
(3) diagonal i , o any λ∈Ω, e e y pλpλ-ope a ional Kλ×Kλma ix Mo e
Gis diagonally pλ-ope a ional wi h pλpλ(M)=∆
pλ(M),
(4) s ongly diagonal i , o any λ∈Ωandanyx∈G,anypλ-ope a ional
Kλ×Kλma ix Mo e Gis pλpλ-ope a ional wi h pλpλ(M)=xi and
only i i is diagonally pλ-ope a ional wi h ∆pλ(M)=x,
(5) weakly diagonal i , o any λ∈Ωandanypλpλ-ope a ional Kλ×Kλma ix
Mo e G, whene e MTis pλpλ-ope a ional wi h pλpλ(M)=pλpλ(MT),
Mis diagonally pλ-ope a ional wi h pλpλ(M)=∆
pλ(M),
(6) locally diagonal i , o any λ∈Ωandanyx∈G,e e ypλ-ope a ional
Kλ×Kλma ix Mo e Gwi h x-cons an diagonal is pλpλ-ope a ional
wi h pλpλ(M)=x,
(7) locally an idiagonal i , o any λ∈Ω, any x∈Gand any pλpλ-ope a ional
Kλ×Kλma ix Mo e Gwi h x-cons an diagonal, pλpλ(M)=ximplies
ha Mis x-cons an ,
(8) weakly locally an idiagonal i , o any λ∈Ω, any x∈Gand any pλpλ-ope -
a ional Kλ×Kλma ix Mo e Gwi h x-cons an diagonal such ha MT
is pλpλ-ope a ional oo, pλpλ(M)=pλpλ(MT)=ximplies ha Mis
x-cons an .
Clea ly, s ong diagonali y implies diagonali y, diagonali y implies weak di-
agonali y and local an idiagonali y implies weak local an idiagonali y. I Gis
idempo en , hen s ong diagonali y implies local diagonali y.
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N. CHAISANSUK — S. LEERATANAVALEE — J. ˇ
SLAPAL
Example 2.2.Le Gbe a se and ρbe a bina y ela ion on G.Le pbe he
bina y pa ial ope a ion on Ggi en by
Dp=ρand p(a, b)=awhene e (a, b)∈Dp.
Le qbe he bina y pa ial ope a ion on Gdual o p, i.e., gi en by
Dq=ρand q(c, d)=dwhene e (c, d)∈Dq.
Then G, p, qis a pa ial algeb a o ype (2,2) and we ha e:
(a) G, p, qis commu a i e,
(b) ρis eflexi e i and only i G, p, qis idempo en ,
(c) ρis ansi i e i and only i G, p, qis diagonal,
(d) G, p, qis locally diagonal i ρis symme ic,
(e) G, p, qis s ongly diagonal i ρis ansi i e and symme ic,
( ) ρis an isymme ic i and only i G, p, qis weakly locally an idiagonal.
Example 2.3.Le (G, p) be a pa ial ec angula band, i.e., a pa ial algeb a o
ype (2) such ha he e a e se s X, Y wi h G=X×Yand pis he bina y
pa ial ope a ion on Ggi en by
Dp=((a1,b
1),(a2,b
2)) ∈(X×Y)2:b1=a2
and
p((a1,b
1),(a2,b
2)) = (a1,b
2) whene e ((a1,b
1),(a2,b
2)) ∈Dp.
Then (G, p) is a weakly diagonal pa ial algeb a.
Example 2.4.Le Xbe a se and le pbe a bina y pa ial ope a ion on he
powe se P(X)o Xgi en by
(A, B)∈Dp⇐⇒ A=Bo (A=∅&B=∅&A∩B=∅)
and hen p(A, B)=A∩B.
Then (P(X),p) is a locally an idiagonal pa ial algeb a.
Le H=H, (qλ;λ∈Ω)and G=G, (pλ;λ∈Ω)be pa ial algeb as
o he same ype, le λ∈Ωandle ( i;i∈Kλ)∈(Hom(H,G))Kλ.Le
∈Hom(H,G) be a homomo phism such ha , o all (xi;i∈Kλ)∈HKλand
all x∈H,qλ(xi;i∈Kλ)=ximplies pλ( i(xi); i∈Kλ)= (x). Then need
no be a unique homomo phism o Hin o Gha ing his p ope y — see he
ollowing example.
Example 2.5.Le H=(H, q), G=(G, p) be pa ial algeb as o ype (2) whe e
H={1,2},G={a, b}and p, q a e gi en by he ables
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STABILITY OF SOME PROPERTIES OF PARTIAL ALGEBRAS UNDER POWERS
q12
122
222
and
pab
aaa
baa
Le :H→Gbe he map gi en by (1) = a, (2) = aand g:H→G
be he map gi en by g(1) = b,g(2) = a.Then ,g ∈Hom(H,G) and, o any
x, y, z ∈{1,2},q(x, y)=zimplies p( (x), (y)) = (z)andp( (x), (y)) = g(z).
Bu =g.

2.6

Le H=H, (qλ;λ∈Ω),G=G, (pλ;λ∈Ω)be pa ial
algeb as o he same ype, Hidempo en , and le λ∈Ωand ( i;i∈Kλ)∈
(Hom(H,G))Kλ.I ∈Hom(H,G)is a homomo phism such ha , o all
(xi;i∈Kλ)∈HKλand all x∈H,
qλ(xi;i∈Kλ)=x=⇒pλ( i(xi); i∈Kλ)= (x),
hen is uniquely de e mined.
P oo . Le ,g ∈Hom(H,G) be homomo phisms such ha , o all (xi;
i∈Kλ)∈HKλand all x∈H,qλ(xi;i∈Kλ)=ximplies bo h pλ( i(xi);
i∈Kλ)= (x)andpλ( i(xi); i∈Kλ)=g(x). Le x∈H.AsHis
idempo en , pu ing xi=x o each i∈Kλ,wege qλ(xi;i∈Kλ)=x,
pλ( i(x);i∈Kλ)= (x)andpλ( i(x);∈Kλ)=g(x). Hence, =g.
3. S abili y o p ope ies o pa ial algeb as
unde powe s

3.1

Le H=H, (qλ;λ∈Ω)and G=G, (pλ;λ∈Ω)be
pa ial algeb as o he same ype τ.Thepowe o Gand His he pa ial
algeb a GH=Hom(H,G),( λ;λ∈Ω)o ype τwhe e, o any λ∈Ω, any
( i;i∈Kλ)∈(Hom(H,G))Kλand any ∈Hom(H,G), λ( i;i∈Kλ)=
i and only i is a unique homomo phism o Hin o Gwi h he p ope y
ha , o all (xi;i∈Kλ)∈HKλand all x∈H,qλ(xi;i∈Kλ)=ximplies
pλ( i(xi); i∈Kλ)= (x).
Lemma 2.6 and Defini ion 3.1 immedia ely esul in

3.2

Le G,Hbe pa ial algeb as o he same ype. I His idem-
po en , hen he powe GHis idempo en , oo.
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N. CHAISANSUK — S. LEERATANAVALEE — J. ˇ
SLAPAL

3.3

Le G=G, (pλ;λ∈Ω),H=H, (qλ;λ∈Ω)be pa ial alge-
b as o he same ype, Hidempo en , and le GH=Hom(H,G),( λ;λ∈Ω).
Le λ, µ ∈Ωand le M=( ij ;i∈Kλ,j ∈Kµ)be a ma ix o e Hom(H,G).
I Mis µ-ope a ional wi h µ( ij ;j∈Kµ)= i o each i∈Kλ, λ µ-ope -
a ional wi h λ µ(M)= o diagonally λ-ope a ional wi h ∆ λ(M)= , e-
spec i ely, hen, whene e (yj;j∈Kµ)∈Dqµand qµ(yj;j∈Kµ)=y, hema-
ix M∗=( ij(yj);i∈Kλ,j ∈Kµ)o e Gis pµ-ope a ional wi h pµ( ij(yj);
j∈Kµ)= i(y) o each i∈Kλ,pλpµ-ope a ional wi h pλpµ(M∗)= (y)o
diagonally pλ-ope a ional wi h ∆pλ(M∗)= (y), espec i ely.
P oo .
(1) Le Mbe µ-ope a ional wi h µ( ij ;j∈Kµ)= i o each i∈Kλand
le (yj;j∈Kµ)∈Dqµ,qµ(yj;j∈Kµ)=y. Then i ollows immedia ely om
Defini ion 3.1 ha M∗is pµ-ope a ional wi h pµ( ij(yj); j∈Kµ)= i(y) o
each i∈Kλ.
(2) Le Mbe λ µ-ope a ional wi h λ µ(M)= and pu µ( ij ;j∈Kµ)= i
o each i∈Kλ.Le (yj;j∈Kµ)∈Dqµ,qµ(yj;j∈Kµ)=y.By(1),M∗is
pµ-ope a ional wi h pµ( ij(yj); j∈Kµ)= i(y) o each i∈Kλ.Thus,
pλpµ(M∗)=pλpµ( ij(yj)) ; j∈Kµ,i∈Kλ=pλ( i(y); i∈Kλ).
As His idempo en , we ge pλ( i(y); i∈Kλ)= (y). Consequen ly, M∗is
pλpµ-ope a ional wi h pλpµ(M∗)= (y).
(3) Le Mbe diagonally λ-ope a ional wi h ∆ λ(M)= , i.e., le λ( ii ;
i∈Kλ)= .Le (yj;j∈Kµ)∈Dqµ,qµ(yj;j∈Kµ)=y.Then
∆pλ(M∗)=pλ( ii(yi); i∈Kλ)= (y).
Hence, M∗is diagonally pλ-ope a ional wi h ∆pλ(M∗)= (y). 
Using Lemma 3.3, one can easily p o e he ollowing heo em.

3.4

Le G,Hbe pa ial algeb as o he same ype, Hidempo en . I
Gis commu a i e, diagonal, s ongly diagonal o weakly diagonal, espec i ely,
hen so is GH.

3.5

Le G=G, (pλ;λ∈Ω)be an idempo en , locally diagonal pa ial
algeb a and le λ∈Ω.I (xi;i∈Kλ),(yi;i∈Kλ)∈GKλ,pλ(yi;i∈Kλ)=y
and he e exis s i0∈Kλsuch ha xi0=yand xi=yi0 o each i∈Kλ−{i0},
hen pλ(xi;i∈Kλ)=yi0.
P oo . Le (xi;i∈Kλ),(yi;i∈Kλ)∈GKλ,pλ(yi;i∈Kλ)=yand le
i0∈Kλbe an elemen such ha xi0=yand xi=yi0 o each i∈Kλ−{i0}.
Le M=(aij ;i∈Kλ,j ∈Kλ)beaKλ×Kλ-ma ix o e Gsuch ha aij =yi0
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STABILITY OF SOME PROPERTIES OF PARTIAL ALGEBRAS UNDER POWERS
o each i∈Kλ−{i0}and ai0j=yj o each j∈Kλ.SinceGis idempo en , M
is pλ-ope a ional wi h yi0-cons an diagonal. As Gis locally diagonal, we ha e
pλpλ(M)=pλpλ(aij ;j∈Kλ); i∈Kλ=yi0.
Thus, pλ(xi;i∈Kλ)=yi0because xi=pλ(aij ;j∈Kλ) o e e y i∈Kλ.

3.6

Le G,Hbe pa ial algeb as o he same ype, Hidempo en and
locally diagonal. I Gis locally diagonal, hen so is GH.
P oo . Le H=H, (qλ;λ∈Ω),G=G, (pλ;λ∈Ω),GH=Hom(H,G),
( λ;λ∈Ω)and le Gbe locally diagonal. Le λ∈Ωand ∈Hom(H,G).
Le M=( ij ;i∈Kλ,j∈Kλ)bean λ-ope a ional ma ix o e Hom(H,G)
wi h -cons an diagonal. Pu λ( ij ;j∈Kλ)= i o each i∈Kλ.Le
(yi;i∈Kλ)∈Dqλ,qλ(yi;i∈Kλ)=yand, o e e y i∈Kλ, pu aii =yand
aij =yi o each j∈Kλ−{i}.AsHis idempo en and locally diagonal, we
ha e qλ(aij ;j∈Kλ)=yi o e e y i∈Kλby Lemma 3.5. Thus, pλ( ij(aij);
j∈Kλ)= i(yi) o e e y i∈Kλ.Pu M∗=( ij(aij); i∈Kλ,j∈Kλ). Then
M∗is a pλ-ope a ional Kλ×Kλ-ma ix o e Gwi h (y)-cons an diagonal.
Since Gis locally diagonal, we ha e pλpλ(M∗)= (y). The e o e,
pλ( i(yi);i∈Kλ)= (y).
Since His idempo en , is uniquely de e mined. Hence, λ( i;i∈Kλ)=
λ λ(M)= .

3.7

Le G,Hbe pa ial algeb as o he same ype. Hidempo en .
I Gis locally an idiagonal o weakly locally an idiagonal, espec i ely, hen so
is GH.
P oo . Le G=G, (pλ;λ∈Ω),H=H, (qλ;λ∈Ω)be pa ial algeb as
o he same ype, Hidempo en , and le GH=Hom(H,G),( λ;λ∈Ω).Le
λ∈Ωand ∈Hom(H,G). Le M=( ij ;i∈Kλ,j∈Kλ)bean λ λ-ope -
a ional ma ix o e Hom(H,G)wi h -cons an diagonal and le y∈H.Since
His idempo en , M∗=( ij(x); i∈Kλ,j∈Kλ)isapλpλ-ope a ional ma ix
o e Gby Lemma 3.3. I is also e iden ha M∗has (y)-cons an diagonal.
Using hese ac s, we easily ge he s a emen . 

3.8

Le G,H,Kbe pa ial algeb as o he same ype. I H,Ka e
idempo en , hen
(GH)K∼
=GH×K.
P oo . Le
H=H, (qλ;λ∈Ω),G=G, (pλ;λ∈Ω),
K=K, (sλ;λ∈Ω),GH=Hom(H,G),(Rλ;λ∈Ω),
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N. CHAISANSUK — S. LEERATANAVALEE — J. ˇ
SLAPAL
H×K=H×K, (Vλ;λ∈Ω),(GH)K=Hom(K,Hom(H,G)),(Tλ;λ∈Ω),
GH×K=Hom(H×K,G),(Uλ;λ∈Ω)
and le H,Kbe idempo en .
Le λ∈Ω and define a map ϕ:Hom(K,Hom(H,G)) →Hom(H×K,G)
by ϕ(g)(y,z)=g(z)(y) whene e g∈Hom(K,Hom(H,G)), z∈Kand
y∈H. We will show fi s ha he map ϕis well defined, i.e., ha ϕ(g)∈
Hom(H×K,G) whene e g∈Hom(K,Hom(H,G)). Fo his accoun , le
g∈(Hom(K,Hom(H,G)) and le ((yi,z
i);i∈Kλ)∈DVλ,Vλ((yi,z
i);i∈Kλ)
=(y,z). Then qλ(yi;i∈Kλ)=yand sλ(zi;i∈Kλ)=zand, since
g∈(Hom(K,Hom(H,G)), we ge Rλ(g(zi);i∈Kλ)=g(z). This implies
pλ(g(zi)(yi); i∈Kλ)=g(z)(y)
and, he e o e,
pλ(ϕ(g)(yi,z
i); i∈Kλ)=pλ(g(zi)(yi);i∈Kλ)=g(z)(y)=ϕ(g)(y,z).
Hence, ϕ(g)∈Hom(H×K,G).
Fu he , define a map α:Hom(H×K,G)→HomK,Hom(H,G)by
α(h)(z)(y)=h(y,z) whene e h∈Hom(H×K,G), z∈Kand y∈H. We will
show ha αis well defined, oo.
Le h∈Hom(H×K,G). To show ha α(h)(z)∈Hom(H,G) o each z∈K,
le z∈Kand (yi;i∈Kλ)∈Dqλ,qλ(yi;i∈Kλ)=y.SinceKis idempo en ,
we ha e Vλ((yi,z);i∈Kλ)=(y,z). F om h∈Hom(H×K,G) i ollows ha
pλ(h(yi,z);i∈Kλ)=h(y,z). Thus,
pλα(h)(z)(yi);i∈Kλ=pλh(yi,z); i∈Kλ=h(y, z)=α(h)(z)(y),
so ha α(h)(z)∈Hom(H,G).
To show ha α(h)∈HomK,Hom(H,G),le (zi;i∈Kλ)∈Dsλ,
sλ(zi;i∈Kλ)=zand le (yi;i∈Kλ)∈Dqλ,qλ(yi;i∈Kλ)=y.Then
Vλ((yi,z
i); i∈Kλ)=(y,z). Since h∈Hom(H×K,G), we ge pλ(h(yi,z
i);
i∈Kλ)=h(y,z). So,
pλ(α(h)(zi)(yi);i∈Kλ)=pλ(h(yi,z
i);i∈Kλ)=h(y,z)=α(h)(z)(y).
Since His idempo en , α(h)(z) is uniquely de e mined. Consequen ly,
Rλα(h)(zi); i∈Kλ=α(h)(z),
so ha α(h)∈Hom(K,Hom(H,G)).
Fu he , we ha e
ϕ(α(h))(y,z)=α(h)(z)(y)=h(y,z)andα(ϕ(g))(z)(y)=ϕ(g)(y, z)=g(z)(y)
o all h∈Hom(H×K,G), g∈Hom(K,Hom(H,G)), y∈Hand z∈K.
Consequen ly, ϕand αa e bijec ions in e se o each o he .
8
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STABILITY OF SOME PROPERTIES OF PARTIAL ALGEBRAS UNDER POWERS
To show ha ϕis a homomo phism, le (gi;i∈Kλ)∈DTλ,Tλ(gi;i∈Kλ)
=gand ((yi,z
i); i∈Kλ)∈DVλ,Vλ((yi,z
i); i∈Kλ)=(y, z). Then
qλ(yi;i∈Kλ)=yand sλ(zi;i∈Kλ)=z. Hence, Rλ(gi(zi); i∈Kλ)
=g(z) and, consequen ly, pλ(gi(zi)(yi); i∈Kλ)=g(z)(y)andg(z) is uniquely
de e mined because H×Kis idempo en . The e o e,
pλ(ϕ(gi)(yi,z
i); i∈Kλ)=pλ(gi(zi)(yi);i∈Kλ)=g(z)(y)=ϕ(g)(y, z)
and ϕ(g) is uniquely de e mined because H×Kis idempo en . We ha e
Uλ(ϕ(gi); i∈Kλ)=ϕ(g). Thus, ϕis a homomo phism.
To show ha αis a homomo phism, le (hi;i∈Kλ)∈DUλ,Uλ(hi;i∈Kλ)
=h,(zi;i∈Kλ)∈Dsλ,sλ(zi;i∈Kλ)=zand (yi;i∈Kλ)∈Dqλ,
qλ(yi;i∈Kλ)=y.Thenweha eVλ((yi,z
i);i∈Kλ)=(y,z) and, conse-
quen ly, pλ(hi(yi,z
i); i∈Kλ)=h(y,z). The e o e,
pλ(α(hi)(zi)(yi); i∈Kλ)=pλ(hi(yi,z
i);i∈Kλ)=h(y,z)=α(h)(z)(y)
and α(h)(z) is uniquely de e mined because His idempo en . Consequen ly,
Rλ(α(hi)(zi);i∈Kλ)=α(h)(z). The idempo ency o Kimplies he uniqueness
o α(h), hence Tλ(α(hi); i∈Kλ)=α(h). The e o e, αis a homomo phism.
We ha e shown ha ϕ:Hom(K,Hom(H,G)) →Hom(H×K,G)isaniso-
mo phism (wi h ϕ−1=α), i.e., (GH)K∼
=GH×K.

3.9

Le G,Hbe pa ial algeb as o he same ype. I Gis commu-
a i e, hen he e exis s a pa ial subalgeb a Ko he di ec p oduc G|H|such
ha |K|=Hom(H,G).
P oo . Le H=H, (qλ;λ∈Ω),G=G, (pλ;λ∈Ω)G|H|
=GH,(Rλ;λ∈Ω)and le Gbe commu a i e. Le λ∈Ω, le ( i;i∈Kλ)
∈DRλ∩(Hom(H,G))Kλ,Rλ( i;i∈Kλ)= .Le µ∈Ωandle (yj;
j∈Kµ)∈Dqµ,qµ(yj;j∈Kµ)=y. Conside he Kλ×Kµ-ma ix M=
( i(yj); i∈Kλ,j ∈Kµ)o e G. Then, o each i∈Kλ,pλ( i(yj);
j∈Kµ)= i(y) because i∈Hom(H,G) o each i∈Kλ.Fu he ,weha e
pλ( i(y);i∈Kλ)= (y)andpλ( i(yj); i∈Kλ)= (yj) o each j∈Kµ(by
he defini ion o he di ec p oduc o pa ial algeb as). Thus, Mis pλpµ-ope -
a ional wi h pλpµ(M)= (y)andMTis pλ-ope a ional. Since Gis commu a-
i e, we ha e pµpλ(MT)= (y). Thus, pλ( (yj); j∈Kµ)= (y), which yields
∈Hom(H,G). 

3.10

Le G,Hbe pa ial algeb as o he same ype. I Gis bo h
diagonal and commu a i e and His idempo en , hen he powe GHis a pa ial
subalgeb a o he di ec p oduc G|H|.
9
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