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 absolutely: if a series formed by absolute values of a term of series of real numbers converges, then the series itself converges and every rearrangement also converges to the same sum. A series in which every rearrangement converges is called unconditionally convergent. Let us define the sum range of series as the set of all sums of all its convergent rearrangements.
It is not clear in advance that an unconditionally convergent series of real numbers is 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 not absolutely convergent, then some rearrangement is not convergent, and its sum range is the set of all real numbers.
These results depend heavily on the structure of the set of real numbers. However, 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 An abelian version of the statement in the abstract appears in (unpublished) [
3]. A non-abelian version for topological groups appears in [
4].
2. Algebraic Part
We write
for the set
of natural numbers with its usual order and
A non-empty set, X, endowed with a binary operation is called a groupoid or a magma. For a groupoid, , the value of + at will be denoted as .
For a finite non-empty
and a family
of elements of a groupoid
, following Bourbaki, we define the (ordered) sum
inductively as follows:
(1) If I consists of a single element, , then ;
(2) If
I has more than one element,
j is the least element of
I and
, then
Note that:
If I consists of two elements, then , where j is the least element of I and k is the last element of I;
If I consists of three elements, then , where again, j is the least element of I, k is the last element of I and .
If , then instead of we write also .
A groupoid, , is a semigroup if its binary operation + is associative, i.e., for every we have .
For a finite non-empty and a family of elements of a semigroup the above given definition of (OS) can be reformulated as follows:
(1r) if I consists of a single element, , then ,
(2r) if
I has more than one element,
k is the last element of
I and
, then
For a set I a bijection called a permutation of I; the set of all permutations of I is denoted by .
For a finite non-empty
and a family
of elements of a groupoid
,we define its
sum range
as follows:
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’ instead of ‘ordered sum’ ; ‘product range’ instead ‘sum range’ and ∏ instead of ∑.
Two elements, and , of a groupoid, , are said to commute (or to be permutable) if ; i.e., if is a singleton.
A family of elements of a groupoid is commuting if for each and , the elements and commute.
An element a of a groupoid is left cancellable if the left translation mapping 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 and a family of elements of a semigroup the following statements are true.
If is acommutingfamily, then is a singleton.
If is a singleton and either or for every the element is right (resp. left) cancellable, then is acommutingfamily.
Proof. See [
5] [Ch.1, §1.5, Theorem 2 (p. 9)].
For the case
the statement is evident. Now, let
and for every
the element
is right cancellable. Fix
, write
. Also write
, where
. Moreover, consider permutations
and
of
I such that
and
. As
is a singleton, we can write:
From this equality, as is right cancellable, we obtain .
The case where and for every the element is left cancellable is considered similarly. □
Our next claim is to find an analog of Theorem 1 when .
3. Series
A (formal)
series corresponding to a sequence of elements of a groupoid
is the sequence
The ‘multiplicative’ counterpart is: a (formal)
infinite product corresponding to a sequence of elements of a groupoid
is the sequence
We use the additive notation herein.
Let be a groupoid and be a topology in X; such a triplet will be called a topologized groupoid.
A topologized groupoid is a topological groupoid if its binary operation + is continuous as mapping from to (where stands for the product topology).
A series corresponding to a sequence
of elements of a topologized groupoid
is said to be convergent in
if the sequence (S1) converges to an element
in the topology
; in such a case, we write
and call
s a sum of the series.
To a sequence
of elements of a topologized groupoid
, we associate a subset
of
as follows: a permutation
belongs to
if and only if the series corresponding to
is convergent in
and define
the sum range of the series corresponding to
as follows (cf. [
6] (Definition 2.1.1)):
It may happen that for a sequence the set is empty; in which case, as well.
The series corresponding to
is called
unconditionally convergent (Bourbaki says
commutatively convergent [
7]) in
if
i.e.,
if for every permutation the series corresponding to is convergent in .
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 of elements of a Hausdorff topologized semigroup , the following statements are true.
If the series corresponding to is convergent in , is acommutingfamily and is nota singleton, then there is a permutation such that the series corresponding to is not convergent in .
If the series corresponding to isunconditionally convergentin and is acommutingfamily, then is a singleton.
If is a singleton, is a group and for every the left translation determined by is sequentially continuous, then is acommutingfamily.
Proof. .
To prove
, denote by
s the limit in
of the sequence (S1), i.e.,
Since
is not a singleton, there is
such that
. Hence, there is a permutation
such that the series corresponding to
is convergent to
t in
, i.e.,
Construction of a permutation .
Find and fix a
strictly increasing sequence of natural numbers
such that
Now, define a mapping
as follows:
It is easy to see that .
From (
3) and (
4), we can conclude that
and
From (
5) and (
6) together with Theorem 1
(which is applicable because
is a commuting family), we conclude that the following relations are true:
and
The equality (
7) implies:
while the equality (
8) implies:
From (
9) and (
10), since
and
is a
Hausdorff topology, we conclude that
is not a convergent sequence. Therefore, we found a permutation
such that the series corresponding to
is not convergent in
and
is proved.
follows from .
In view of Theorem 1, it is sufficient to show that for a fixed natural number we find that is a singleton.
We can suppose without loss of generality that the series corresponding to
is convergent in
to
. This implies:
From this, since the left translations are continuous, we obtain:
Now, fix an arbitrary permutation
such that
From the above equality, since the left translations are continuous, we can now write
Hence, since
is a singleton, we conclude:
Therefore,
and, as
is arbitrary, we prove that
is a singleton. □
Remark 1. Theorem 2 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.