Series with Commuting Terms in Topologized Semigroups
Series with Commuting Terms in Topologized Semigroups
Castejón, Alberto;Corbacho, Eusebio;Tarieladze, Vaja
2021-09-24 00:00:00
axioms Article 1, 1 2 Alberto Castejón *, Eusebio Corbacho and Vaja Tarieladze Departamento de Matemática Aplicada I, University of Vigo, 36310 Vigo, Spain; eusebio.corbacho@gmail.com Muskhelishvili Institute of Computational Mathematics, Georgian Technical University, Tbilisi 0159, Georgia; v.tarieladze@gtu.ge * Correspondence: acaste@uvigo.es Abstract: We show that the following general version of the Riemann–Dirichlet theorem is true: if every rearrangement of a series with pairwise commuting terms in a Hausdorff topologized semigroup converges, then its sum range is a singleton. Keywords: semigroup; group; topology; permutation; convergence MSC: Primary 54C35; Secondary 54E15 1. Introduction In 1827, Peter Lejeune-Dirichlet was the first to notice that it is possible to rearrange the terms of certain convergent series of real numbers so that the sum changes [1]. According to [2] (Ch. 2, §2.4), In 1833, Augustin-Louis Cauchy also noticed this in his “Resumes analytiques”. Later, in 1837, Dirichlet showed that this cannot happen if the series converges abso- lutely: if a series formed by absolute values of a term of series of real numbers converges, Citation: Castejón, A.; Corbacho, E.; then the series itself converges and every rearrangement also converges to the same sum. Tarieladze, V. Series with Commuting A series in which every rearrangement converges is called unconditionally convergent. Let Terms in Topologized Semigroups. us define the sum range of series as the set of all sums of all its convergent rearrangements. Axioms 2021, 10, 237. https://doi.org/ It is not clear in advance that an unconditionally convergent series of real numbers is 10.3390/axioms10040237 also absolutely convergent, and hence its sum range is a singleton. This is in fact true thanks to the following Riemann rearrangement theorem: if a convergent series of real numbers is Academic Editor: Sidney A. Morris not absolutely convergent, then some rearrangement is not convergent, and its sum range is the set of all real numbers. Received: 12 August 2021 Accepted: 13 September 2021 These results depend heavily on the structure of the set of real numbers. However, Published: 24 September 2021 the concepts of unconditional convergence and sum range make sense even in general topologized semigroups. An abelian version of the statement in the abstract appears in Publisher’s Note: MDPI stays neutral (unpublished) [3]. A non-abelian version for topological groups appears in [4]. with regard to jurisdictional claims in Section 2 focuses on ‘finite series’ and Section 3 treats the general case. Section 4 published maps and institutional affil- contains additional comments. iations. 2. Algebraic Part We write N for the set f1, 2, . . .g of natural numbers with its usual order and N := fk 2 N : k ng, n = 1, 2, . . . Copyright: © 2021 by the authors. n Licensee MDPI, Basel, Switzerland. A non-empty set, X, endowed with a binary operation + : X X ! X is called a This article is an open access article groupoid or a magma. For a groupoid, (X, +) , the value of + at (x , x ) 2 X X will be distributed under the terms and 1 2 denoted as x + x . conditions of the Creative Commons 1 2 Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Axioms 2021, 10, 237. https://doi.org/10.3390/axioms10040237 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 237 2 of 7 For a finite non-empty I N and a family (x ) of elements of a groupoid (X, +), i i2 I following Bourbaki, we define the (ordered) sum x 2 X (OS) å i i2 I inductively as follows: (1) If I consists of a single element, I = fjg, then x = x ; i j i2 I (2) If I has more than one element, j is the least element of I and I = Infjg, then x = x + x . å i j å i i2 I i2 I Note that: If I consists of two elements, then x = x + x , where j is the least element of I i2 I i j k and k is the last element of I; If I consists of three elements, then x = x + (x + x ), where again, j is the least i j m k i2 I element of I, k is the last element of I and j < m < k. If I = N , then instead of x we write also x . å å n i i i2 I i=1 A groupoid, (X, +), is a semigroup if its binary operation + is associative, i.e., for every (x , x , x ) 2 X X X we have x + (x + x ) = (x + x ) + x . 1 2 3 1 2 3 1 2 3 For a finite non-empty I N and a family (x ) of elements of a semigroup (X, +) i i2 I the above given definition of (OS) can be reformulated as follows: (1r) if I consists of a single element, I = fkg, then x = x , i2 I i k (2r) if I has more than one element, k is the last element of I and I = I nfkg, then x = x + x . å i å i k i2 I i2 I For a set I a bijection s : I ! I called a permutation of I; the set of all permutations of I is denoted by S( I). For a finite non-empty I N and a family (x ) of elements of a groupoid (X, +),we i i2 I define its sum range SR((x ) ) i i2 I as follows: SR((x ) ) := fs 2 X : 9s 2 S( I), s = x g . i i2 I å s(i) i2 I In a case where the multiplicative notation is applied for the binary operation, it would be natural to use the word ‘product’ instead of ‘sum’; ‘ordered product’ (OP) instead of ‘ordered sum’ (OS); ‘product range’ (PR) instead ‘sum range’ (SR) and instead of . Õ å Two elements, x and x , of a groupoid, (X, +), are said to commute (or to be per- 1 2 mutable) if x + x = x + x ; i.e., if SR (x ) is a singleton. 1 2 2 1 i i2N A family (x ) of elements of a groupoid (X, +) is commuting if for each i 2 I and i i2 I j 2 I, the elements x and x commute. i j An element a of a groupoid (X, +) is left cancellable if the left translation mapping x 7! a + x is injective; right cancellable is defined similarly. An element is cancellable if it is both left and right cancellable. Theorem 1 (Commutativity theorem). For a finite non-empty I N and a family (x ) of i i2 I elements of a semigroup (X, +) the following statements are true. (a) If (x ) is a commuting family, then SR((x ) ) is a singleton. i i2 I i i2 I (b) If SR (x ) is a singleton and either Card(I) 2 or for every i 2 I the element x is ( ) i i2 I i right (resp. left) cancellable, then (x ) is a commuting family. i i2 I Proof. (a) See [5] [Ch.1, §1.5, Theorem 2 (p. 9)]. Axioms 2021, 10, 237 3 of 7 (b) For the case Card(I) 2 the statement is evident. Now, let n = Card(I) > 2 and for every i 2 I the element x is right cancellable. Fix i, j 2 I, i 6= j, write I = Infi, jg. Also write I = fk , k , . . . , k g, where k < k < < k . Moreover, consider permu- 1 2 n 1 2 n tations s and p of I such that s(k ) = i, s(k ) = j, s(fk , . . . , k g) = I and p(k ) = j, 2 3 n 1 1 p(k ) = i, p(fk , . . . , k g) = I . As SR((x ) ) is a singleton, we can write: 2 3 n i i2 I ! ! x + x + x = x = x = x + x + x . r r i j å å s(i) å p(i) j i å 00 00 r2 I i2 I i2 I r2 I From this equality, as 00 x is right cancellable, we obtain x + x = x + x . r2 I r i j j i The case where Card(I) > 2 and for every i 2 I the element x is left cancellable is considered similarly. Our next claim is to find an analog of Theorem 1 when I = N. 3. Series A (formal) series corresponding to a sequence x = (x ) of elements of a groupoid n2N (X, +) is the sequence x . (S1) å k k2N n2N The ‘multiplicative’ counterpart is: a (formal) infinite product corresponding to a sequence x = (x ) of elements of a groupoid (X,) is the sequence n n2N x . (P1) Õ k k2N n2N We use the additive notation herein. Let (X, +) be a groupoid and t be a topology in X; such a triplet (X, +, t) will be called a topologized groupoid. A topologized groupoid (X, +, t) is a topological groupoid if its binary operation + is continuous as mapping from (X X, t t) to (X, t) (where t t stands for the prod- uct topology). A series corresponding to a sequence x = (x ) of elements of a topologized n2N groupoid (X, +, t) is said to be convergent in (X, +, t) if the sequence (S1) converges to an element s 2 X in the topology t; in such a case, we write s = x å k k=1 and call s a sum of the series. To a sequence x = (x ) of elements of a topologized groupoid (X, +, t), we n2N associate a subset P(x) of S(N) as follows: a permutation p : N ! N belongs to P(x) if and only if the series corresponding to (x ) is convergent in (X, +, t) and define the p(n) n2N sum range of the series corresponding to x = (x ) n n2N SR x ( ) as follows (cf. [6] (Definition 2.1.1)): SR(x) := ft 2 X : 9p 2 P(x), t = x g . p(k) k=1 It may happen that for a sequence x = (x ) the set P(x) is empty; in which case, n2N SR(x) = Æ as well. Axioms 2021, 10, 237 4 of 7 The series corresponding to x = (x ) is called unconditionally convergent (Bourbaki n n2N says commutatively convergent [7]) in (X, +, t) if P(x) = S(N) ; i.e., if for every permutation s : N ! N the series corresponding to x = (x ) is convergent s(n) n2N in (X, +, t). We proceed to our main result, extending to topologized semigroups the results for topological groups in [4] (Theorem 2 and Theorem 1). Theorem 2 (Commutativity Theorem 2). For a sequence x = (x ) of elements of a Hausdorff n2N topologized semigroup (X, +, t), the following statements are true. (a ) If the series corresponding to x is convergent in (X, +, t), x is a commuting family and SR x is not a singleton, then there is a permutation l : N ! N such that the series corresponding ( ) to x = (x ) is not convergent in (X, +, t). l n2N l(n) (a) If the series corresponding to x is unconditionally convergent in (X, +, t) and x = (x ) is a commuting family, then SR(x) is a singleton. n n2N (b) If SR(x) is a singleton, (X, +) is a group and for every n 2 N the left translation determined by x is sequentially continuous, then x = (x ) is a commuting family. n n n2N Proof. (a ). To prove (a ), denote by s the limit in (X, +, t) of the sequence (S1), i.e., (t) lim x = s . (1) å k k2N Since SR(x) is not a singleton, there is t 2 SR(x) such that t 6= s. Hence, there is a permutation p : N ! N such that the series corresponding to x = (x ) is p n2N p(n) convergent to t in (X, +, t), i.e., (t) lim x = t . (2) å p(k) k2N Construction of a permutation l : N ! N. Find and fix a strictly increasing sequence of natural numbers (m ) such that k k2N 1 = m , N p(N ) N , k = 1, 2, . . . (3) 1 m m m 2k 1 2k 2k+1 Now, define a mapping l : N ! N as follows: l(1) = 1; l(N n N ) = p(N )n N ; m m m m 2k 2k 1 2k 2k 1 l(N n N ) = N n p(N ), k = 1, 2, . . . (4) m m m m 2k+1 2k 2k+1 2k It is easy to see that l 2 S(N). From (3) and (4), we can conclude that l(N ) = N , k = 1, 2, . . . (5) m m 2k+1 2k+1 and l(N ) = p(N ) , k = 1, 2, . . . (6) m m 2k 2k From (5) and (6) together with Theorem 1(a) (which is applicable because x = (x ) n n2N is a commuting family), we conclude that the following relations are true: m m 2k+1 2k+1 x = x , k = 1, 2, . . . (7) å l(i) å i i=1 i=1 Axioms 2021, 10, 237 5 of 7 and m m 2k 2k x = x , k = 2, 3, . . . (8) å å l(i) p(i) i=1 i=1 The equality (7) implies: 2k+1 lim x = s , (9) å l(i) i=1 while the equality (8) implies: 2k lim x = t . (10) å l(i) i=1 From (9) and (10), since t 6= s and t is a Hausdorff topology, we conclude that ( x ) is not a convergent sequence. Therefore, we found a permutation l : N ! N l(i) n2N i=1 such that the series corresponding to x = (x ) is not convergent in (X, +, t) and l n2N l(n) (a ) is proved. (a) follows from (a ). (b) In view of Theorem 1(b), it is sufficient to show that for a fixed natural number n > 1 we find that SR((x ) ) is a singleton. i i2N We can suppose without loss of generality that the series corresponding to x is con- vergent in (X, +, t) to s 2 X. This implies: 0 1 @ A lim x + x = s . i i å å m>n i2N n i2N nN m n From this, since the left translations are continuous, we obtain: lim x = x + s . å i å i m>n i2N i2N nN m n n Now, fix an arbitrary permutation p : N ! N such that p(k) = k, k = n + 1, n + 2, . . . From the above equality, since the left translations are continuous, we can now write 0 1 @ A lim x + x = x + ( x + s) . å å i å å i p(i) p(i) m>n i2N i2N i2N n i2N nN n n m n Hence, since SR(x) is a singleton, we conclude: x + ( x + s) = s . å p(i) å i i2N i2N n n Therefore, x = x å å i p(i) i2N i2N n n and, as p is arbitrary, we prove that SR((x ) ) is a singleton. i i2N Remark 1. Theorem 2(a) for a Banach space was first proved in [8], where the term “B-space” was used and it was also noticed that this term is credited to M. Frechet. In [9], where the term ‘Banach space’ is already used, one finds a nice discussion of equivalent characterizations of unconditional convergence. Axioms 2021, 10, 237 6 of 7 4. Additional Comments 4.1. On Theorem 2 The statement (b) of Theorem 2 is not a complete converse of statement (a) of Theorem 2; in the case of Hausdorff topological groups, such a complete converse can be formulated as follows: If for a sequence x = (x ) of elements of a Hausdorff topological group X the set SR(x) n2N is a singleton, then the series corresponding to x is unconditionally convergent in X and x = (x ) is a commuting family. n2N Let us say that a Hausdorff topological group X has property (HM) if whenever for a sequence x = (x ) the set SR(x) is a singleton, then the series corresponding to x is n2N unconditionally convergent in X. The Riemann rearrangement theorem implies that X = R has property (HM). In [10], it was shown that if X is an infinite-dimensional Hilbert space, then X does not have property (HM); a similar result was obtained in [11] for infinite-dimensional Banach spaces. From the general result of [12], we conclude that the finite-dimensional real normed spaces, as well as the countable product of real lines R , have property (HM). 4.2. On Sum Ranges A subset A of a topological group X is a sum range if a sequence x = (x ) of n n2N elements of X exists such that A = SR(x). Known results and the history of the study of the structure of sum ranges in Banach spaces are found in [6]; see also, [12–18]. A subset A of a real vector space X is called affine if x 2 A, x 2 A, t 2 R, =) tx + (1 t)x 2 A . 1 2 1 2 It is known that: • A subset of a finite-dimensional real Banach space is a sum range if and only if it is affine (Steinitz’s theorem, see [6]); • A subset of a real nuclear Frechet space is a sum range if and only if it is closed and affine [13]; • Every closed affine subset of a separable real Frechet space can be a sum range (cf. [19], where the following question is left open: is every separable infinite-dimensional complete metrizable real topological vector space a sum range?); • An arbitrary finite subset of an infinite-dimensional Banach space can be a sum range [20]; • A non-analytic subset of an infinite-dimensional separable Banach space cannot be a sum range [21]; • A non-closed subset of an infinite-dimensional separable Banach space can be a sum range (see [6,22]; however, it is unknown whether a non-closed vector subspace of an infinite-dimensional separable Banach space can be a sum range [16]) . Finally, note that it would be interesting to: (1) Investigate, in connection with Theorem 2(a), the question of how rich the sum range SR(x) can be for a non-commuting sequence x = (x ) , the series corresponding n2N to which is unconditionally convergent; may happen that SR(x) = X? (2) Find a “semigroup version” of Theorem 2(b). Author Contributions: The material of the first two sections is the result of joint effort of A.C., E.C. and V.T. The third section is mainly on responsibility of the V.T. All authors have read and agreed to the published version of the manuscript. Funding: The third named author was partially supported by the Shota Rustaveli National Science Foundation grant no. DI-18-1429: “Application of probabilistic methods in discrete optimization and scheduling problems”. Data Availability Statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article. Axioms 2021, 10, 237 7 of 7 Acknowledgments: We are very grateful to our reviewers carefully reading the initial version of this note and for the suggested (not only) language improvements, which we have taken into account. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. References 1. Galanor, S. Riemann’s rearrangement Theorem. Math. Teach. 1987, 80, 675–681. [CrossRef] 2. Whittaker, E.T.; Watson, G.N. A Course of Modern Analysis, 4th ed.; Cambridge University Press: Cambridge, UK, 1927; [Russian translation by F. V. Shirokov, Moscow, 1962]. 3. Castejón, A.; Corbacho, E.; Tarieladze, V. Metric Monoids and Integration; Manuscript; Vigo, Spain, 1995; 268p. 4. McArthur, C.W. Series with Sums Invariant Under Rearrangement. Am. Math. Mon. 1968, 75, 729–731. [CrossRef] 5. Bourbaki, N. Algebra 1, Chapters 1–3; Hermann: Paris, France, 1974. 6. Kadets, M.; Kadets, V. Series in Banach Spaces: Conditional and Unconditional Convergence; Birkhauser: Basel, Switzerland, 1997; Volume 94. 7. Bourbaki, N. General Topology, Part 1, Chapters I–IV; Hermann: Paris, France, 1966. 8. 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