On the stability of some properties of partial algebras under powers

  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. Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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 2 Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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. 3 Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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 4 Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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. 5 Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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 6 Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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), 7 Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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 Unau hen ica ed Download Da e | 8/14/17 8:14 AM 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 Unau hen ica ed Download Da e | 8/14/17 8:14 AM