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7 Stabilizers of functional Menger systems 0 0 2 Wieslaw A. Dudek and Valentin S. Trokhimenko n a J Abstract 2 A functional Menger system isa set of n-placefunctionscontaining n 1 projectionsandclosedundertheso-calledMenger’scompositionofn-place functions. Wegivetheabstractcharacterizationforsubsetsofthesefunc- ] M tional systems which contain functions havingone common fixed point. G 1 Introduction . h t a Investigation of multiplace functions by algebraic methods plays a very impor- m tant role in modern mathematics were we consider various operations on sets [ of functions, which are naturally defined. The basic operation for functions is superposition (composition), but there are some other naturally defined op- 1 v erations, which are also worth of consideration. For example, the operation 4 of set-theoretic intersection and the operation of projections (see for example 6 [1, 2, 3, 6, 7]). The central role in these study play sets of functions with fixed 3 points. The study of such sets for functions of one variable was initiated by B. 1 M.Scheinin[4]and[5]. Next,forsetsoffunctionsofnvariables,wascontinued 0 7 by V.S. Trokhimenko (see [8, 9]). 0 In this paper, we consider the sets of n-place functions containing n- / projections and closed under the so-called Menger’s composition of n-place h t functions. For such functional systems we give the abstract characterization a for subsets of functions having one common fixed point. m : v 2 Preliminaries i X r Let An be the n-th Cartesian product of a set A. Any partial map from a An into A is called an n-place function on A. The set of all such maps is denoted by F(An,A). On F(An,A) we define one (n+1)-ary superposition O: (f,g ,...,g ) 7→ f[g ...g ], called the Menger’s composition, and n unary 1 n 1 n operations R : f 7→R f, i∈1,n={1,...,n} putting i i f[g ...g ](a ,...,a )=f(g (a ,...,a ),...,g (a ,...,a )), (1) 1 n 1 n 1 1 n n 1 n R f(a ,...,a )=a , where (a ,...,a )∈pr f (2) i 1 n i 1 n 1 forf,g ,...,g ∈F(An,A), (a ,...,a )∈An, wherepr f denotesthe domain 1 n 1 n 1 ofafunctionf. Itisassumedtheleftandrighthandsideofequality(1)arede- fined, or not defined, simultaneously. Algebras of the form (Φ,O,R ,...,R ), 1 n 1 where Φ ⊂ F(An,A), are called functional Menger systems of n-place functions. Algebras of the form (Φ,O,∩,R ,...,R ), where ∩ is a set-theoretic 1 n intersection, are called functional Menger ∩-algebras of n-place functions. In the literature such algebras are also called functional Menger P-algebras (see [1] and [9]). Let a be some fixed element (point) of A. The stabilizer of a is the set Ha Φ of such functions from Φ for which a is a fixed point, i.e., the set Ha ={f ∈Φ|f(a,...,a)=a}. Φ Let (G,o) be a nonempty set with one (n+1)-ary operation o: (x ,x ,...,x )7→x [x ...x ]. 0 1 n 0 1 n An abstractalgebra G =(G,o,R ,...,R ) of type (n+1,1,...,1) for all i,k ∈ 1 n 1,n satisfying the following axioms: A1: x[y1...yn][z1...zn]=x[y1[z1...zn]...yn[z1...zn]], A2: x[R1x...Rnx]=x, A3: x[u¯|iz][R1y...Rny]=x[u¯|iz[R1y...Rny]], A4: Rix[R1y...Rny]=(Rix)[R1y...Rny], A5: x[R1y...Rny][R1z...Rnz]=x[R1z...Rnz][R1y...Rny], A6: Rix[y1...yn]=Ri(Rkx)[y1...yn], A7: (Rix)[y1...yn]=yi[R1(x[y1...yn])...Rn(x[y1...yn])], where x[u¯|iz] means x[u1...ui−1zui+1...un], is called a functional Menger system of rank n. An algebra Gf = (G,o,f,R1,...,Rn) of type (n + 1,2,1,...,1), where (G,o,R ,...,R )is afunctionalMengersystemofranknand(G,f)isasemi- 1 n lattice, is called a functional Menger f-algebra of rank n if it satisfies the iden- tities: A8: xfy[R1z...Rnz]=(xfy)[R1z...Rnz], A9: xfy =x[R1(xfy)...Rn(xfy)], A10: (xfy)[z1...zn]=x[z1...zn]fy[z1...zn]. Any Menger algebra of rank n, i.e., an abstract groupoid (G,o) with an (n+1)-ary operation satisfying A1 is isomorphic to some set of n-place functions closed under Menger’s composition [6]. Functional Menger f-algebras and Menger systems of rank n are isomorphic, respectively, to some functional Menger ∩-algebras and Menger systems of n-place functions (see [1] and [7]). Eachhomomorphismofsuchabstractalgebrasintocorrespondingalgebrasofn- place functions is called a representation by n-place functions. Representations which are isomorphisms are called faithful. Let (Pi)i∈I be the family of representations of a Menger algebra (G,o) of rank n by n-place functions defined on sets (Ai)i∈I, respectively. By the union 2 of this family we mean the map P: g 7→ P(g), where g ∈ G, and P(g) is an n-place function on A= A defined by i iS∈I P(g)= P (g). [ i i∈I If Ai∩Aj =∅ for all i,j ∈I, i 6= j, then P is called the sum of (Pi)i∈I and is denotedbyP = P . Itisnotdifficulttoseethatthesumofrepresentationsis i iP∈I a representation, but the union of representations may not be a representation (see for example [1 − 7]). For every representation P: G → F(An,A) of an algebra (G,o) and every elementa∈GbyHa wedenotethesetofelementsofGcorrespondingtothese P n-place functions for which a is a fixed point, i.e., Ha ={g ∈G|P(g)(a,...,a)=a}. P Let G be a functional Menger system of rank n, x – an individual variable. By T (G) we denote the set of transformations t: x7→t(x) on G such that: n (a) x∈T (G), n (b) ift(x)∈T (G),thena[¯b| t(x)]∈T (G)andR t(x)∈T (G)foralla∈G, n i n i n ¯b∈Gn and i∈1,n, (c) T (G) contains only elements determined in (a) and (b). n Let us remind that a nonempty subset H of G is called • quasi-stable, if for all x∈G x∈H −→x[x...x]∈H, • f-quasi-stable, if for all x∈G x∈H −→x[x...x]fx∈H, • stable, if for all x,y ,...,y ∈G 1 n x,y ,...,y ∈H −→x[y ...y ]∈H, 1 n 1 n • f-stable, if for all x,y ∈G x,y ∈H −→xfy ∈H, • l-unitary, if for every x,y ∈G x[y...y]∈H ∧ y ∈H −→x∈H, 3 • v-unitary, if for all x,y ,...,y ∈G 1 n x[y ...y ]∈H ∧ y ,...,y ∈H −→x∈H, 1 n 1 n • a normal v-complex, if for all x,y ∈G, t∈T (G) n x,y ∈H ∧ t(x)∈H −→t(y)∈H, • an l-ideal, if for all x,y ,...,y ∈G 1 n (y ,...,y )∈Gn\(G\H)n −→x[y ...y ]∈H. 1 n 1 n A binary relation ρ⊂G×G is called • stable, if (x,y),(x ,y ),...,(x ,y )∈ρ−→(x[x ...x ],y[y ...y ])∈ρ 1 1 n n 1 n 1 n for all x,y,x ,y ∈G, i∈1,n, i i • l-regular, if (x,y)∈ρ−→(x[z ...z ],y[z ...z ])∈ρ 1 n 1 n for all x,y,z ∈G, i∈1,n, i • v-regular, if (x ,y ),...,(x ,y )∈ρ−→(z[x ...x ],z[y ...y ])∈ρ 1 1 n n 1 n 1 n for all x ,y ,z ∈G, i∈1,n, i i • i-regular, where i∈1,n, if (x,y)∈ρ−→(u[w¯| x],u[w¯| y])∈ρ i i for all x,y,u∈G, w¯ ∈Gn, • v-negative, if (x[y ...y ],y )∈ρ 1 n i for all x,y ,...,y ∈G and i∈1,n. 1 n On G we define two binary relations ζ and χ putting (x,y)∈ζ ←→x=y[R x...R x], (x,y)∈χ←→(R x,R y)∈ζ. 1 n 1 1 The first relation is a stable order, the second is an l-regular and v-negative quasi-order containing ζ (see [7]). For these two relations the following conditions are valid: x6y −→R x6R y, i∈1,n, x⊏y ←→R x6R y, i∈1,n, i i i i x⊏y ←→x[R y...R y]=x, (R x)[y ...y ]6y , i∈1,n, 1 n i 1 n i x[R y ...R y ]6x, R x=R R x, i,k ∈1,n, 1 1 n n i i k 4 where x6y ←→(x,y)∈ζ, and x⊏y ←→(x,y)∈χ. Let W be the empty set or an l-ideal which is an E-class of a v-regular equivalence relation E defined on a Menger algebra (G,o) of rank n. Denote by (Ha)a∈AE the family of all E-classes (uniquely indexed by elements of some set AE) such that Ha 6= W. Next, for every g ∈G we define on AE an n-place function P(E,W)(g) putting P(E,W)(g)(a1,...,an)=b←→g[Ha1...Han]⊂Hb, (3) where (a1,...,an) ∈ pr1P(E,W)(g) ←→ g[Ha1...Han]∩W = ∅, and Hb is an E-class containing all elements of the form g[h ...h ], h ∈H , i∈1,n. It is 1 n i ai not difficult to see that the map P(E,W): g 7→P(E,W)(g) satisfies the identity P(E,W)(g[g1...gn])=P(E,W)(g)[P(E,W)(g1)...P(E,W)(gn)], (4) i.e.,P(E,W)isarepresentationof(G,o)byn-placefunctions. Thisrepresentation will be called simplest. 3 Stabilizers Definition 1. A nonempty subset H of G is called a stabilizer of a functional MengersystemG (orafunctionalMengerf-algebraGf =(G,f,o,R1,...,Rn)) of rank n if there exists a representation P of G (respectively, Gf) by n-place functions on some set A, such that H =Ha for some point a∈A common for P all elements from H. Theorem 1. For a nonempty subset H of G to be a stabilizer of a functional Menger system G of rank n, it is necessary and sufficient to be a quasi-stable l- unitary normal v-complex contained in some subset U of G such that R U ⊂H, i R (G\U)⊂G\U and i x∈H ∧y ∈H ∧ t(x)∈U −→t(y)∈U, (5) x=y[R x...R x]∈U ∧ u[w¯| y]∈H −→u[w¯| x]∈H, (6) 1 n i i x=y[R x...R x]∈U ∧ u[w¯| y]∈U −→u[w¯| x]∈U (7) 1 n i i for all x,y ∈G, w¯ ∈Gn, t∈T (G) and i∈1,n, where the symbol u[w¯| ] may n i be empty.1 Proof. Necessity. Let Ha be a stabilizer of a for a functional Menger system Φ (Φ,O,R ,...,R ) of n-place functions. If f ∈Ha, i.e., f(a,...,a)=a, then 1 n Φ f[f...f](a,...,a)=f(f(a,...,a),...,f(a,...,a))=f(a,...,a)=a. Thus f[f...f]∈Ha. This means that Ha is quasi-stable. Φ Φ 1Ifu[w¯|i ]istheemptysymbol,thenu[w¯|ix]isequaltox. 5 Since, for f[g...g]∈Ha and g ∈Ha we have Φ Φ a=f[g...g](a,...,a)=f(g(a,...,a),...,g(a,...,a))=f(a,...,a), then f ∈Ha, therefore Ha is l-unitary. Φ Φ Moreover, if f(a¯)=g(a¯) for some f,g ∈Φ, where a¯=(a ,...,a ), then, as 1 n itisnotdifficulttosee,t(f)(a¯)=t(g)(a¯)foreveryt∈T (Φ). So,f,g,t(f)∈Ha n Φ implies t(g)∈Ha. Therefore Ha is a normal v-complex. Φ Φ It is clear that Ha ⊂ Ua = {f ∈ Φ|(a,...,a) ∈ pr f}, R Ua ⊂ Ha and Φ Φ 1 i Φ Φ R (Φ\Ua) ⊂ Φ\Ua. If t(f) ∈ Ua for some f,g ∈ Ha and t ∈ T (Φ), then i Φ Φ Φ Φ n (a,...,a) ∈ pr t(f), which, together with f(a,...,a) = a = g(a,...,a), gives 1 t(f)(a,...,a) = t(g)(a,...,a). Therefore, t(g) ∈ Ua. So, the condition (5) is Φ satisfied. To prove(6) assume f =g[R f...R f]∈Ua, i.e., f ⊂g andf ∈Ua. This 1 n Φ Φ implies (a,...,a) ∈ pr f and f(a,...,a) = g(a,...,a). So, for α[χ¯| g] ∈ Ha, 1 i Φ where α∈Φ and χ¯=(χ ,...,χ )∈Φn, we have 1 n a=α[χ¯| g](a,...,a)=α(χ¯(a,...,a)| g(a,...,a)) i i =α(χ¯(a,...,a)| f(a,...,a))=α[χ¯| f](a,...,a), i i where χ¯(a,...,a) is χ (a,...,a),...,χ (a,...,a). Thus α[χ¯| f] ∈ Ha, which 1 n i Φ completes the proof of (6). The proof of (7) is analogous. Sufficiency. Let H and U be two subsets of G satisfying all the conditions of the theorem. First we shall prove the following implications: x6y∧x∈H −→y ∈H, (8) x⊏y∧x∈U −→y ∈U. (9) Indeed, x 6 y means x = y[R x...R x]. Since x ∈ H, H ⊂ U and R U ⊂ H, 1 n i we haveR x∈H for everyi∈1,n. This, together with the fact that H is anl- i unitary normal v-complex, implies that H is v-unitary. So, y[R x...R x]∈H 1 n andR x∈H foreveryi∈1,n,whichbythev-unitarityofH givesy ∈H. This i proves (8). Now, if x ⊏ y and x ∈ U, then R x 6 R y, i.e., R x = (R y)[R x...R x]. 1 1 1 1 1 n From R U ⊂ H it follows R x ∈ H, so, applying the v-unitarity of H to i i (R y)[R x...R x] ∈ H we obtain R y ∈ H, whence we get R y ∈ U. Since 1 1 n 1 1 R (G\U)⊂G\U means that i R x∈U −→x∈U (10) i for every x ∈ G and i ∈ 1,n, from R y ∈ U it follows y ∈ U. This completes 1 the proof of (9). The set G\U is an l-ideal of G. In fact, the v-negativity of χ and x∈G\U imply u[w¯| x] ⊏ x, whence, according to (9), we conclude u[w¯| x] ∈ G\U for i i all i∈1,n, u∈G and w¯ ∈Gn. So, G\U is an l-ideal. Using this fact it is easy 6 to show that x6y∧x∈U ∧ t(y)∈H −→t(x)∈H, (11) x6y∧x∈U ∧ t(y)∈U −→t(x)∈U (12) for all x,y ∈G and t∈T (G). n On G we define two binary relations E and E putting H U E = (x,y)|(∀t∈T (G)) t(x)∈H ←→t(y)∈H , H n n (cid:16) (cid:17)o E = (x,y)|(∀t∈T (G)) t(x)∈U ←→t(y)∈U . U n n (cid:16) (cid:17)o These relations are v-regular equivalences. E = E ∩ E also is a v-regular H U equivalence. For this relation we have x[Ehy i...Ehy i]⊂Ehx[y ...y ]i (13) 1 n 1 n forallx,y ,...,y ∈G,whereEhy idenotesanequivalenceclassofE containing 1 n i y . Moreover,G\U is an equivalence class of E. i Also H is anE-class. To provethis factitis sufficientto verifythe following two conditions: g ∈H ∧ g ∈H −→g ≡g (E), (14) 1 2 1 2 g ≡g (E)∧ g ∈H −→g ∈H. (15) 1 2 1 2 Let g ,g ∈H and t(g )∈U for some t∈T (G). Then, from (5), it follows 1 2 1 n t(g ) ∈ U. Similarly, from g ,g ∈ H and t(g ) ∈ U, we conclude t(g ) ∈ U. 2 1 2 2 1 Hence, g ≡ g (E ). If g ,g ,t(g ) ∈ H, then, in view of the fact that H is 1 2 U 1 2 1 a normal v-complex, we have t(g ) ∈ H. Similarly, from g ,g ,t(g ) ∈ H we 2 1 2 2 deduce t(g )∈H. So, g ≡g (E ). Thus g ≡g (E). This proves (14). 1 1 2 H 1 2 Now let g ∈ H and g ≡ g (E). Then g ≡ g (E ), which means that for 1 1 2 1 2 H all t ∈ T (G) we have t(g ) ∈ H ←→ t(g ) ∈ H, whence g ∈ H ←→ g ∈ H. n 1 2 1 2 So, g ∈H, i.e., (15) is proved. Consequently, H is an E-class. 2 Let W =G\U. For every g ∈G we consider an n-place function P(E,W)(g) defined by (3). Since the map P(E,W): g 7→ P(E,W)(g) satisfies (4), it is a homomorphism with respect to the operation o. It satisfies also the identity P(E,W)(Rig)=RiP(E,W)(g). (16) Indeed, for every a¯=(a1,...,an)∈pr1P(E,W)(Rig), where H =Ehx i, x ∈U, i∈1,n, we have (R g)[x ...x ]∈U. Whence, ac- ai i i i 1 n cording to A , we obtain x [R g[x ...x ]...R g[x ...x ]] ∈ U. As G\U 7 i 1 1 n n 1 n is an l-ideal, the last condition implies R g[x ...x ] ∈ U for i ∈ 1,n. i 1 n Thus g[x ...x ] ∈ U, because g[x ...x ] 6∈ U implies x ∈ G\U. So, 1 n 1 n i a¯∈pr1P(E,W)(g). This proves pr1P(E,W)(Rig)⊂pr1P(E,W)(g). 7 To prove the converse inclusion let a¯ ∈ pr1P(E,W)(g), where Hai = Ehxii, x ∈ U for i ∈ 1,n. Then g[x ...x ] ∈ U, which, by R U ⊂ H ⊂ U, gives i 1 n k R g[x ...x ] ∈ U. Whence, by A , we get R (R )[x ...x ] ∈ U for k,i ∈ k 1 n 6 k i 1 n 1,n. From this, in view of (10), we deduce (R g)[x ...x ] ∈ U. Thus a¯ ∈ i 1 n pr1P(E,W)(Rig) for every i∈1,n. In this way we have proved pr1P(E,W)(Rig)=pr1P(E,W)(g)=pr1RiP(E,W)(g) (17) for every g ∈G. Let a¯ ∈ pr1P(E,W)(Rig), i.e., (Rig)[x1...xn] ∈ U, where Hai = Ehxii, x ∈ U, i ∈ 1,n. Applying the stability of ζ to (R g)[x ...x ] 6 x we i i 1 n i obtain t((R g)[x ...x ]) 6 t(x ) for every t ∈ T (G) and i ∈ 1,n. If i 1 n i n t((R g)[x ...x ]) ∈ H, then, according (8), we get t(x ) ∈ H. For t(x ) ∈ H, i 1 n i i in view of (11), from (R g)[x ...x ] 6 x and (R g)[x ...x ] ∈ U we deduce i 1 n i i 1 n t((R g)[x ...x ]) ∈ H. So, (R g)[x ...x ] ≡ x (E ). Similarly we can prove i 1 n i 1 n i H (R g)[x ...x ] ≡ x (E ). Thus (R g)[x ...x ] ≡ x (E) for every i ∈ 1,n. i 1 n i U i 1 n i Therefore P(E,W)(Rig)(a1,...,an)=ai. Consequently, P(E,W)(Rig)(a1,...,an)=(RiP(E,W)(g))(a1,...,an). This, together with (17), gives (16). From (4) and (16) it follows that P(E,W) is a representation of G by n-place functions. Observe that g ∈H ←→P(E,W)(g)(a,...,a)=a, (18) whereaisthiselementofAE whichisusedasindexoftheE-classH. Infact,for g ∈ H the quasi-stability of H implies g[g...g]∈ H. Whence, g[H...H]⊂ H becauseH isanE-classandtherelationE isv-regular. So,P(E,W)(g)(a,...,a)= a. Conversely, if g[H...H] ⊂ H, then g[h...h] ∈ H for every h ∈ H. From this, by the l-unitarity of H, we get g ∈H, which completes the proof of (18). To complete this proof we remind that any algebra G satisfying the axioms A −A , has a faithful representation by n-place functions [7]. Let P be this 1 7 1 representation. Then,asitisnotdifficulttoverify,P =P1+P(E,W) isafaithful representation of G for which H =Ha. This completes the proof. P Let G be a functional Menger system of rankn and H be some subset of G. We say that a subset X of G is C -closed, if for all a,b,c ∈ G, t ∈ T (G) the H n implication: a=b[R a...R a]∨a,b∈H, 1 n  t(a)[R1c...Rnc]=t(a), −→c∈X a,t(b)∈X  is valid. In the abbreviated form this implication can be written as (a6b∨a,b∈H) ∧ t(a)⊏c ∧ a,t(b)∈X −→c∈X. (19) 8 Let C (X) denotes the set of all c ∈ G for which there exist a,b ∈ G and H t∈T (G) such that the premise of (19) is satisfied. Further, let n ∞ m CH[X]= [ CH(X), m=0 0 m+1 m where CH(X)=X, CH(X)=CH(CH(X)) for every m=0,1,2,... m By induction we can prove that g ∈CH(X) if and only if the following system of conditions is fulfilled: (a =b [R a...R a]∨a ,b ∈H) ∧ t (a )[R g...R g]=t (a ) 1 1 1 n 1 1 1 1 1 n 1 1  2mi−V=11−1 ata222iii(+a=12i=b)2[Rib[2R1ia+1ia1.2[R.i..1R.a.n2Ria+ni1]a=.2.i].tR∨2in(aaa222iii,)+b,12]i∨∈aH2i,+1,b2i+1 ∈H,   (20)   i=2m2Vm−−11(ai ∈t2iX+1∧(at2ii(+b1i))[R∈1Xti()bi)...Rnti(bi)]=t2i+1(a2i+1)   where a ,b ∈G, t ∈T (G). k k k n Inthesequel,thesystemofconditions(20)willbedenotedbyM (X,m,g). H Theorem 2. Let G be a functional Menger system of rank n. A nonempty subset H of G is a stabilizer of G if and only if it is a quasi-stable l-unitary normal v-complex such that R H ⊂H for every i∈1,n and i x=y[R x...R x]∈C [H] ∧ u[w¯| y]∈H −→u[w¯| x]∈H (21) 1 n H i i for all x,y,u∈G, w¯ ∈Gn, i∈1,n, where the symbol u[w¯| ] may be empty. i Proof. Necessity. Let Ha be the stabilizer of a point a in a functional Menger Φ system(Φ,O,R ,...,R )ofn-place functions. Obviouslyitis a quasi-stablel- 1 n unitary normal v-complex of (Φ,O,R ,...,R ). If f ∈Ha, then f(a,...,a)= 1 n Φ a,whenceR f(a,...,a)=a,i.e.,R f ∈Ha. So,R Ha ⊂Ha foreveryi∈1,n. i i Φ i Φ Φ To prove (21), we shall consider f = g[R1f...Rnf] ∈ CHΦa[HΦa] and α[ω¯|ig] ∈ HΦa for some f,g,α ∈ Φ, ω¯ ∈ Φn, i ∈ 1,n, where CHΦa[HΦa] = ∞ m m mS=0 CHΦa (HΦa). Then f = g[R1f...Rnf] ∈CHΦa (HΦa) for some m ∈ N. But, m as it is easily to see by induction, ϕ ∈CHΦa (HΦa) implies (a,...,a) ∈ pr1ϕ. The above means that (a,...,a) ∈ pr f and (a,...,a) ∈ pr g[R f...R f]. 1 1 1 n Therefore f(a,...,a)=g[R f...R f](a,...,a) 1 n =g(R f(a,...,a),...,R f(a,...,a))=g(a,...,a). 1 n Moreover,for α[ω¯| g]∈Ha we have i Φ a=α[ω¯| g](a,...,a)=α(ω¯(a,...,a)| g(a,...,a)) i i =α(ω¯(a,...,a)| f(a,...,a))=α[ω¯| f](a,...,a), i i 9 where ω¯(a,...,a) denotes ω (a,...,a),...,ω (a,...,a). So, α[ω¯| f] ∈ Ha, 1 n i Φ which completes the proof of (21). Sufficiency. Let H satisfy all the conditions of the theorem. We prove that it satisfies also all the conditions of Theorem 1. Since by the assumption H is a quasi-stable l-unitary normal v-complex such that R H ⊂ H for every i i∈1,n,wemustprovethatitsatisfiestheconditions(5),(6),(7)andR U ⊂H, i R (G\U)⊂G\U for some U ⊂G containing H and all i∈1,n. i First we prove that H satisfies the condition (8). Indeed, if x 6 y and x ∈ H, then y[R x...R x] = x ∈ H and R x ∈ H, i ∈ 1,n. As H is an 1 n i l-unitary normal v-complex, H is a v-unitary subset. Therefore y ∈ H, which completes the proof of (8). ∞ m Now let U =CH[H]. Clearly H ⊂U. Since CH[H]= CH(H), to prove mS=0 that R U ⊂ H for every i ∈ 1,n, it is sufficient to show that for every m ∈ N i holds the inclusion m Ri(CH(H))⊂H. (22) 0 For m = 0 it is obvious because CH (H) = H and RiH ⊂ H for all i ∈ 1,n. Suppose that (22) is valid for some k ∈ N. We prove that it is valid for k+1. k+1 Let g ∈C (H). Then H k (a6b∨a,b∈H)∧ t(a)⊏g∧ a,t(b)∈CH(H) k forsomea,b∈G,t∈Tn(G). Froma,t(b)∈C(H),accordingtooursupposition, we get R a,R t(b)∈H. If a6b and a∈H, then, by (8), we haveb∈H. Thus i i a,b,R t(b) ∈ H. Whence R t(a) ∈ H because H is a normal v-complex. The i i condition t(a) ⊏ g implies R t(a) 6 R g, which, by (8), gives R g ∈ H. In a i i i similar way a,b ∈ H and t(a) ⊏ g proves R g ∈ H. Thus we have shown that i k+1 R (C (H))⊂H. So,the inclusion(22)isvalidform=k+1andconsequently i H for every m∈N. Therefore R U ⊂H for every i∈1,n. i To prove the inclusion R (G\U)⊂G\U observe that it is equivalent to the i condition (∀g ∈G)(g ∈G\U −→R g ∈G\U), i which can be written in the form (∀g ∈G)(R g ∈U −→g ∈U). (23) i In the case U =C [H] the last condition means that H m n (∀g ∈G)(∀m∈N) (cid:16)Rig ∈CH(H)−→(∃n∈N)g ∈CH(H)(cid:17). (24) m Let Rig ∈CH (H) for some g ∈ G and m ∈ N. Considering Rig = Ri(Rig) and (22), we conclude R g ∈ H. Thus R g 6 R g, R g ⊏ g and R g ∈ H. i i i i i 10