Abelization of join spaces of affine transformations of ordered field with proximity

@Applied General Topology c Universidad Polit´ecnica de Valencia Volume 6, No. 1, 2005 pp. 57-65 Abelization of join spaces of affine transformations of ordered field with proximity ˇ S´ arka Hoˇ skov´ a Abstract. Using groups of affine transformations of linearly ordered fields a certain construction of non-commutative join hypergroups is presented based on the criterion of reproducibility of semi-hypergroups which are determined by ordered semigroups. The aim of this paper is to construct the abelization of the non-commutative join space of affine transformations of ordered fields. A construction of commutative weakly associative hypergroup (Hν-group) is made and a proximity is defined on this structure. 2000 AMS Classification: 20F60, 20N20 Keywords: Transformation group, join space, abelization, hyperoperation, hyperstructures, weak associativity Transformation groups which represent the classical and developing discipline are situated in the intersection of several parts of mathematical structures. Transformation groups in the discrete approach create in a very natural way commutative hyperstuctures, which are in addition join spaces. On the other hand, groups of affine transformations of the field create naturally a transformation group on the supporting set of this field. More precisely: Let (F, +,·) be a field, A(F) be a group of affine transformations of Fof the form ϕa,b(x) = ax +b;x∈F, where the coefficient a∈Fis different from zero. Then we construct a discrete transformation group (X, T, π) by putting X=F;T=A(F) and the action πwe define by π(x;ϕa,b) = ϕa,b(x). Clearly the identity axiom and homomorphism axiom are satisfied. In this paper hyperstructures on groups of affine transformations of ordered fields are constructed. These affine transformations are represented by ordered pairs of elements of a given field. During the construction of final hyperstructures upper ends of the products of pairs of elements of a given field are used. This idea is adopted from the functorial assignment of a commutative hypergroup to an arbitrary transformation (discrete) group. We will describe this 58 ˇ S. Hoˇskov´a construction in more details. Let G= (X, T, π) be a transformation group, (i.e. X-phase set, T-phase group, πaction (projection): X×T→X). For any pair x, y ∈Xwe define x∗Gy=π(x, T )∪π(y, T ) = {π(x, t); t∈T} ∪ {π(y, t); t∈T}. It is easy to show that (X, ∗G) is an extensive commutative hypergroup— moreover a join space (see the definition below). In the paper [1] the authors studied the non-commutative join hypergroups of affine transformations of ordered fields. The aim of this paper is to abelize this structure using the construction described in papers [7, 14]. Recall that an ordering on a field (F, +, . ) (with the zero element 0Fand the unit 1F) is given by the choice of a set P⊆F(called the positive cone of the ordering) which satisfies the following axioms: (1) P+P⊆P , (2) P . P ⊆P , (3) P∪(−P) = F , (4) P∩(−P) = {0F}. As usual, we define a binary relation ≦Pon the set Fby x≦Py(resp. x <Py) if y−x∈P(resp. y−x∈˙ P, where ˙ P=Pr{0F}). The relation ≦P results in a total ordering of the elements of F. In detail, the relation ≦Pon Fis reflexive, anti symmetrical and transitive, compatible in the usual sense with commutative binary operations addition “+” and multiplication “ .”, and satisfies the law of trichotomy (for any pair x, y ∈Fexactly one of the following possibilities occurs: x=yor x <Pyor y <Px). Note that an ordered field (F, +, . , ≦P) has necessarily the characteristic equal to zero. As usual, P∗(S) denotes the system of all nonempty subsets of S. Let (F, +, . ) be a field of the characteristic zero. An affine transformation f:F→Fof the form f(x) = a . x +b,a, b ∈Fis uniquely represented by an ordered pair of its coefficients [a, b]∈F×F. We shall consider non-constant transformations only, i.e., transformations f(x) = a.x +bsatisfying the condition a6= 0F. So, let us denote A(F) = (Fr{0F})×Fand define a binary operation “·” on the set A(F) by the rule [a, b]·[c, d] = [a.c, a.d +b], which corresponds to the usual composition of affine transformations of F. It is easy to see that A(F),·is a non-commutative group with the identity [1F,0F] and inverse elements of the form [a, b]−1= [a−1,−a−1.b]. Denote by Kthe subset {[a, b]; a, b ∈F, a >P0F}of A(F) = (Fr{0F})×F. It is easy to see that (K, ·) is a subgroup of the group A(F),·. Define a binary relation “≦” on the set Kby the rule: [a, b]≦[c, d] for [a, b],[c, d]∈K whenever a=cand b≦Pd. Then evidently “≦” is an ordering on Kand we have Proposition 1 ([1]).If K={[a, b]; a, b ∈F, a >P0F}, then (K, ·)is an ordered group such that it is a normal subgroup of the group A(F),·. Abelization of join spaces 59 Recall that a hypergroupoid is a pair (H, ·) where His a (nonempty) set and ·:H×H→P∗(H) is a binary hyperoperation on the set H. If a·(b·c) = (a·b)·cfor all a, b, c ∈H , (associativity) , then (H, ·) is called a semihypergroup. A semihypergroup (H, ·) is said to be a hypergroup if the following axiom a·H=H=H·afor all a∈H , (reproduction axiom) , is satisfied—see e.g. [3],[4]. Here, for A, B ⊆H,A6=∅6=Bwe define A·B=S{a.b;a∈A, b ∈B}. Moreover, for subset Aand Bof H, it becomes convenient to use the relational notation A≈B(read Ameets B) to assert that Aand Bhave an element in common, that is, that A∩B6=∅([16], p. 79). A hypergroup (H, ·) is called a transposition hypergroup if it satisfies the transposition axiom: For all a, b, c, d ∈Hthe relation b\a≈c/d implies a·d≈b·c , where b\a={x∈H;a∈b·x},c/d ={x∈H;c∈x·d}. A commutative transposition hypergroup (H, ∗) is called a join space ([15]). The hyperoperation ⋆:H×H→P∗(H) is called weakly associative hyperoperation if a ⋆ (b ⋆ c)∩(a ⋆ b)⋆ c6=∅ for any triad a, b, c ∈H. Aweak semihypergroup (Hν-semigroup) is a set H(H6=∅) equipped with a weakly associative hyperoperation. AHν-semigroup is called a weak hypergroup (Hν-group) if moreover the reproduction axiom is satisfied for any a∈H([18]). Proposition 2 ([5] Theorem 1).Let (S, ·,≦)be an ordered semigroup. A binary operation ∗:S×S→P∗(S)defined by x∗y= [x . y)≦(= {t∈ S;x . y ≦t})for any pair x, y ∈Sis associative. Then we have 1◦The semi-hypergroup (S, ∗)is commutative if and only if the semigroup (S, ·)is commutative. 2◦For the ordered semigroup (S, ·,≦)the following conditions are equivalent: (i) for any pair of elements x, y ∈Sthere exists a pair z, z′∈Ssuch that y·z≦x,z′·y≦x, (ii) the semihypergroup (S, ∗)satisfies the reproduction condition (i.e. t∗S= S=S∗tfor any t∈S), hence it is a hypergroup. Lemma 1 ([7, 14]).Let (H, ·)be a hypergroupoid. Define a hyperoperation “⋆” on the diagonal ∆Has follows: [x, x]⋆[y, y] = D(x·y∪y·x) = [u, u]; u∈ x·y∪y·xfor any pair [x, x],[y, y]∈∆H. Then the following assertions hold: 1oFor any hypergroupoid (H, ·)we have that (∆H, ⋆)is a commutative hypergroupoid. 60 ˇ S. Hoˇskov´a 2oIf (H, ·)is a weakly associative hypergroupoid, then the hypergroupoid (∆H, ⋆) is weakly associative, as well. 3oIf (H, ·)is a quasi-hypergroup, the hypergroupoid (∆H, ⋆)also satisfies the reproduction law, i.e., it is a quasi-hypergroup. Define a hyperoperation ∗:K×K→P∗(K) by [a, b]∗[c, d] = [x, y]; [a, b]·[c, d]≦[x, y]=[a.c, y]; a.d +b≦Py. From Proposition 2 it is evident, that the hypergroupoid (K, ∗) is non-commutative hypergroup. The hypergroup (K, ∗) will be called to be determined by the ordered group (K, ·,≦). The hypergroup (K, ∗) is non-commutative, therefore we will abelize it. Let us define the set ∆K=n[a, b],[a, b]; [a, b]∈Koand a hyperoperation ⋆: ∆K×∆K→P∗(∆K) by [a, b],[a, b]⋆[c, d],[c, d]= (0.1) =n[x, y],[x, y]; [x, y]∈([a, b]∗[c, d]) ∪([c, d]∗[a, b])o =n[x, y],[x, y]; [x, y]∈[a.c, u]; a.d +b≦Pu∪[c.a, v]; c.b +d≦Pvo =n[x, y],[x, y];x=a.c ∧z≦y)o, where z= min{a.d +b;c.b +d}. Theorem 1. The hyperstructure (∆K, ⋆)is commutative weakly associative hypergroup, simply Hν-group. Proof. Recall that the field Fis commutative, so the multiplication is commutative, thus a.c =c.a. It is evident, that (∆K, ⋆)∼ =(M, ⋄), where M=[a, b, a, b]; [a, b]∈Kand [a1, b1, a1, b1]⋄[a2, b2, a2, b2] = [a, b, a, b]; a=a1·a2, z ≦Pb, z= min{a1·b2,+b1, a2,·b1+b2,}.Due to Lemma 1 it is evident that (∆K, ⋆) is the commutative weakly associative hypergroup—simply abelian Hν-group. To investigate the associativity law in more detail let us prove the weak associativity law using the concrete form of the structure (K, ∗). We want to verify that [a, b],[a, b]⋆[c, d],[c, d]⋆[p, q],[p, q]∩ [a, b],[a, b]⋆[c, d],[c, d]⋆[p, q],[p, q]6=∅. Abelization of join spaces 61 for any [a, b],[c, d],[p, q]∈K. For the next computation let us denote A=[a, b],[a, b]⋆[c, d],[c, d]⋆[p, q],[p, q], B=[a, b],[a, b]⋆[c, d],[c, d]⋆[p, q],[p, q], z1= min{a.d +b;c.b +d}, z2= min{c.q +d;p.d +q}. We get A=n[x, y],[x, y];x=a.c ∧z1≦Pyo⋆[p, q],[p, q]= =[ z1≦Py[a.c, y],[a.c, y]⋆[p, q],[p, q] =n[u, v],[u, v];u=a.c.p ∧w1≦Pvo, where w1= min{a.c.q +y, p.y +q}. As p∈F, p >P0F, we have w1= mina.c.q +y, p. min{a.d +b, c.b +d}+q= = mina.c.q + min{a.d +b, c.b +d},min{p.a.d +p.b, p.c.b +p.d}+q= = minmin{a.c.q +a.d +b, a.c.q +c.b +d}, min{p.a.d +p.b +q, p.c.b +p.d +q}= = min{a.c.q +a.d +b, a.c.q +c.b +d, p.a.d +p.b +q, p.c.b +p.d +q}. So, the set A=n[u, v],[u, v];u=a.c.p ∧w1≦Pvo.On the other hand B=[a, b],[a, b]⋆n[x, y],[x, y];x=c.p ∧z2≦Pyo =[ z2≦Py[a, b],[a, b]⋆[x, y],[x, y]=n[u, v],[u, v];u=a.c.p ∧w2≦Pvo, where w2= min{a.y +b, x.b +y}. Similarly, because a∈F, a >P0F w2= mina. min{c.q +d, p.d +q}+b, x.b + min{c.q +d;p.d +q}= = minmin{a.c.q +a.d, a.p.d +a.q}+b, min{x.b +c.q +d, x.b +p.d +q}= = minmin{a.c.q +a.d +b, a.p.d +a.q +b}, min{c.p.b +c.q +d, c.p.b +p.d +q}= = min{a.c.q +a.d +b, a.p.d +a.q +b, c.p.b +c.q +d, c.p.b +p.d +q}. So, the set B=n[u, v],[u, v];u=a.c.p ∧w2≦Pvo. Choose [u0, v0],[u0, v0]∈∆K,such that u0=a.c.p and v0≧Pmax{w1, w2}. Evidently, this pair of pairs belongs to A∩B. 62 ˇ S. Hoˇskov´a Remark 1. Using the previous calculations it is easy to check that the structure (∆K, ⋆)is never associative. To see it let us first consider F=Q. Necessarily, Q+⊂P. Choose, for example a= 1,b= 2,c= 3,d=−4,p= 5,q= −7. From the above notations we have w1= min{−23,−19,−17,3}=−23, w2= min{−23,−25,5,3}=−25. Thus the triple [a, b],[c, d],[p, q]fulfils only the weak associativity law. As it was mentioned earlier each ordered field Fcontains in itself a copy of the set Q. Thus each time it is possible to find a triple [a, b],[c, d],[p, q]∈ Q+×Qsuch that w16=w2and therefore the structure (∆K, ⋆)is only weakly associative. Remark 2. In the sense of the paper [14] it is possible to define a proximity on this structure, for example, in this way. Let A, B ⊂K, then A p B if and only if [A)≦∩[B)≦6=∅. where for M⊂Kwe define [M)≦=S m∈M [m)≦. We mean the proximity in the sense of the ˇ Cech monograph [2]: A relation pon the family of all subsets of the set His called a proximity on the set Hif psatisfies the following conditions: P1. ∅non p H P2. The relation pis symmetric, i.e., A, B ⊂H,A p B implies B p A . P3. For any pair of subset A, B ⊂H,A∩B6=∅implies A p B. P4. If A, B, C are subsets of Hthen (A∪B)p C if and only if either A p C or B p C. Recall that a triad (H, ·, pH)such that (H, ·)is a hypergroupoid and (H, pH) is a proximity space will be called a hypergroupoid with a proximity. If for any triad of elements x, y, z ∈H x·(y·z)pH(x·y)·z is valid, then the hyperoperation “·” is called proximally weakly associative–see e.g. [9],[14]. In fact, axioms P 1, P 2 and P 3 from the definition of proximity are obvious. It remains to show that the axiom P 4 is satisfied too. First let us prove the equality [A∪B)≦= [A)≦∪[B)≦, which will be helpful in the next calculations. 1. x∈L⇒ ∃ u∈A∪B:u≦x⇒ ∃ u∈A:u≦xor ∃u∈B:u≦x⇒x∈ [A)≦or x∈[B)≦ 2. x∈P⇒x∈[A)≦or x∈[B)≦⇒x∈[A∪B)≦ (By Lwe mean the left hand side of the equation and by Pthe right hand side.) Now we can prove the axiom P 4: If A, B, C are subsets of K, then (A∪B)p C if and only if either A p C or B p C. Abelization of join spaces 63 “⇒” (A∪B)p C ⇒[A∪B)≦∩[C)≦6=∅⇒[A)≦∪[B)≦∩[C)≦6=∅⇒A p C or B p C “⇐” A p C or B p C ⇒[A)≦∩[C)≦6=∅or [B)≦∩[C)≦6=∅. Since [A)≦⊆[A∪B)≦ and [B)≦⊆[A∪B)≦,we have [A∪B)≦∩[C)≦6=∅, i.e., (A∩B)p C. Thus we obtain Theorem 2. The hypergroup (K, ∗, p)is a hypergroup with proximity. Theorem 3. The hypergroupoid (∆K, ⋆)is a weakly associative and commutative transposition hypergroup, i.e., a weakly associative join space. Proof. First we will show that the reproduction axiom is fulfilled. Due to Theorem 2 the structure (K, ∗) is the hypergroup, thus [a, b]∗K=K=K∗[a, b] for any [a, b]∈K. If we define [a, b]⋆ [c, d] = [x, y]; [x, y]∈([a, b]∗[c, d]) ∪([c, d]∗[a, b]) =[x, y]; [x, y]∈ {[a.c, u]; a.d +b≦Pu} ∪ {[c.a, v]; c.b +d≦Pv} =[x, y]; x=a.c ∧z≦y), where z= min{a.d+b;c.b+d}, evidently [a, b]⋆ [c, d]⊃[a, b]∗[c, d]. Therefore K⊃[a, b]⋆ K⊃[a, b]∗K=K, which implies that K⋆ [a, b] = Kand similarly [a, b]⋆ K=K. From this we obtain that reproduction axiom holds in (∆K, ⋆). Second we will verify the transposition axiom. With respect to the definition of join space and (0.1) we get A=[b1, b2],[b1, b2][a1, a2],[a1, a2] =[x, y],[x, y]; [a1, a2]∈[b1, b2]⋆ [x, y] =[x, y],[x, y];a1=b1, .x ∧a2≧Pmin{b1.y +b2b2.x +y}= =[x, y],[x, y];x=a1.b−1 1∧y≦Pmax{(a2−b2).b−1 1, a2−b2.a1.b−1 1}, B=[c1, c2],[c1, c2][d1, d2],[d1, d2] =[x, y],[x, y]; [c1, c2]∈[x, y]⋆ [d1, d2] =[x, y],[x, y];c1=d1.x ∧c2≧Pmin{d1.y +d2, d2.x +y}= =[x, y],[x, y];x=c1.d−1 1∧y≦Pmax{(c2−d2).d−1 1, c2−d2.c1.d−1 1}, C=[a1, a2],[a1, a2]⋆[d1, d2],[d1, d2]= =[u, v],[u, v];u=a1.d1∧v≧Pmin{a1.d2+a2, a2.d1+d2}, D=[b1, b2],[b1, b2]⋆[c1, c2],[c1, c2]= =[u, v],[u, v];u=b1.c1∧v≧Pmin{b1.c2+b2, b2.c1+c2}. 64 ˇ S. Hoˇskov´a We have [x, y],[x, y]∈A∩B, i.e., A≈B, if and only if x=a1.b−1 1=c1.d−1 1 and y≦Pminmax{(a2−b2).b−1 1, a2−b2.a1.b−1 1}, max{(c2−d2).d−1 1, c2−d2.c1.d−1 1}. If A≈B, then necessarily a1.d1=b1.c1. Let us denote u0=a1.d1. For any v0 such that v0≧Pmaxmin{a1.d2+a2, a2.d1+d2},min{b1.c2+b2, b2.c1+c2} we obtain [u0, v0],[u0, v0]∈C∩Dwhich proves the transposition axiom.  Remark 3. It is easy to verify that under the assumption of the previous theorem even the following equivalence holds: b\a≈c/d if and only if a . d ≈b . c . In fact, one implication follows from Theorem 2. To obtain the converse one (using the notation from the proof of the mentioned theorem) suppose C≈D. Thus a1.d1=b1.c1. Let us denote x0=a1.b−1 1=c1.d−1 1. Then for an arbitrary y0such that y0≦Pminmax{(a2−b2).b−1 1, a2−b2.a1.b−1 1}, max{(c2−d2).d−1 1, c2−d2.c1.d−1 1}. we have [x0, y0],[x0, y0]∈A∩B, i.e., A≈B. References [1] J. Ber´anek, J. Chvalina, Noncommutative Join Hypergroups of Affine Transformations of Ordered Fields, Dept.Math.Report Series 10, Univ.of South Bohemia, (2002), 15–22. [2] E. ˇ Cech, Topological Spaces, revised by Z. Frol´ık and M. Katˇetov, (Academia, Praha 1966). [3] P. 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