The Higman operations and embeddings of recursive groups

1.1 Higman’s embedding theorem

In 1961, Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it is recursively presented [7] (see definitions and notation in Section 2.1). In his work, Higman extensively uses specific recursively enumerable sets of integer sequences which in some sense “code” defining relations of groups. Our objective is to suggest an algorithm for construction of such integer sequences for certain types of groups. This allows us to list wide classes of groups for which Higman’s famous embedding construction can be constructive and effective.

The approach of [7] will be very briefly outlined in Section 3 below. For now, let us just mention the main steps of Higman’s construction to distinguish those parts to which our new algorithm concerns. A finitely generated group

with recursively enumerable relations r 1 , r 2 , … can be constructively embedded into a 2-generator group T = ⟨ b , c ∣ r 1 ′ , r 2 ′ , … ⟩ where the relations r 1 ′ = r 1 ′ ⁢ ( b , c ) , r 2 ′ = r 2 ′ ⁢ ( b , c ) , … are certain words on letters b , c , and they also are recursively enumerable. Then, for each r s ′ , s = 1 , 2 , … , a unique sequence f s of integers is compiled (see details in Section 3.2) so that the set { r 1 ′ , r 2 ′ , … } of relations is “coded” by means of a set B = { f 1 , f 2 , … } of such sequences. Since the transaction from relations set 𝑅 to sequences set ℬ is done via just a few constructive steps, the set ℬ also is recursively enumerable.

The tedious part of [7] is to show that ℬ is recursively enumerable if and only if ℬ can be constructed by some chain of special operators (H). And parallel to application of those operations, a respective benign subgroup is being constructed in the free group F 3 = ⟨ a , b , c ⟩ of rank three (see Section 2.5). As this process ends up on construction of ℬ, the respective benign subgroup A B is obtained inside F 3 .

In the final short step, the benign subgroup A B is used to get another benign subgroup in the free group F 2 = ⟨ b , c ⟩ of rank two. Then that new subgroup is used to embed the group 𝑇 (and hence also 𝐺) into a finitely presented group via the “The Higman Rope Trick” (see [12, p. 219] and [22]).

1.2 Our algorithm for construction of ℬ

In [7], Higman just relies on theoretical possibility for construction of ℬ via operations (H), without any examples of such construction for particular groups. This is understandable as the objective of the fundamental article [7] is much deeper, and for its purposes, it is sufficient to know that such a construction of ℬ is possible, provided that the set of Gödel numbers specifically constructed for the set 𝑅 is equal to the range of a certain partial recursive function described by Kleene’s characterization (see references in Section 2.1 below).

However, it is rather strange that, after Higman’s result, there was no attempt to explicitly find constructions of ℬ by operations (H) for particular groups (at least, we were unable to find them in the literature). Investigating the topic, we noticed that such construction may be a doable task for many classes of groups, such as the free abelian, metabelian, soluble, nilpotent groups, the additive group of rational numbers ℚ, the quasicyclic group C p ∞ , divisible abelian groups, etc. (see examples in Section 3.3).

We suggest an 𝐻-machine algorithm with some generic tools that allow to explicitly construct ℬ by operations (H) without any usage of Kleene’s characterization, at all. In Example 4.11, we show how simple it is to apply the algorithm (see Remark 4.12).

The advantage of this approach is that construction of the benign subgroup A B and, thus, of the explicit embedding of the given 𝐺 into a finitely presented group becomes a manageable procedure.

To shorten the routine of work with basic Higman operations, we introduce a few auxiliary operations which make the proofs not only shorter but also, we hope, more intuitive to understand (see notation in Section 2.4).

Another embedding aspect we touch upon is the manner by which the initial group 𝐺 is constructively embedded into a 2-generator group 𝑇, and how the relations of 𝑇 can be obtained from those of 𝐺. In the literature, there is no shortage in constructive embeddings of this type (in fact, the original method of [8] already allows that). However, for our purposes, we need a method which not only makes deduction of the relations of 𝑇 from the relations of 𝐺 a trivial automated task, but also preserves certain “patterns” in the relations (for illustration of the “patterns”, see [16] and also examples in Section 3.3 below).

2.1 Basic notation and references

For general group theory information, we refer to textbooks [19, 10, 21]. For background on free constructions, such as free products, free products with amalgamated subgroups, HNN-extension, we refer to [12, 4, 21]. See also the recent note [15], where we apply some methods related to free constructions. We use them here without restating the notation again. Information on varieties of groups can be found in Hanna Neumann’s monograph [18].

We will study recursive groups in the language of Higman operations (H). Recall that a recursive (or recursively presented) group 𝐺 is that possessing a presentation

with finite or countable set of generators 𝑋 and with a recursively enumerable set of defining relations 𝑅. That is to say, to each relation r i ∈ R , one can assign a Gödel number (see [7, Section 2] or [12, p. 218]) to interpret 𝑅 via a set of respective Gödel numbers, and then that set turns out to be the range of a partial recursive function. By Kleene’s characterization, a partial recursive function is that obtained from the zero function, the successor function and the identity function using the operations of composition, primitive recursion and minimization (see [6, 20, 5] for details).

Although Higman’s theorem is for finitely generated recursive groups, its analog holds for embeddings of countably generated recursive groups into finitely presented groups. For, a countably generated recursive group can first be effectively embedded into a finitely generated recursive group (see the remark proceeding the corollary in [7, p. 456]). Thus, in embedding procedures, we will not take care of the number of generators, as long as the relations are recursively enumerated.

2.2 Sets of integer-valued functions and sequences of integers

Denote by ℰ the set of all functions f : Z → Z with finite support

When 𝑚 is any positive integer such that sup ( f ) ⊆ { 0 , 1 , … , m − 1 } , then we can interpret 𝑓 as a sequence f = ( n 0 , … , n m − 1 ) of length 𝑚, assuming f ⁢ ( i ) = n i for each index i = 0 , … , m − 1 . The value f ⁢ ( i ) is called the 𝑖-th coordinate of 𝑓, or the coordinate of 𝑓 at the index 𝑖. Say, f = ( 0 , 0 , 7 , − 8 , 5 , 5 , 5 , 5 ) means that f ⁢ ( 2 ) = 7 , f ⁢ ( 3 ) = − 8 , f ⁢ ( i ) = 5 for i = 4 , … , 7 , and f ⁢ ( i ) = 0 for any i ≤ 1 or i ≥ 8 . Here the initial 0 is the 0th coordinate, and 7 is the 2nd coordinate.

Depending on the situation, we may interpret the same function by sequences of different length by adding some zeros to it. Say, the above function 𝑓 can be interpreted as the sequence f = ( 0 , 0 , 7 , − 8 , 5 , 5 , 5 , 5 , 0 , 0 , 0 ) with three “new” coordinates f ⁢ ( 8 ) = f ⁢ ( 9 ) = f ⁢ ( 10 ) = 0 . And the constant zero function f ⁢ ( i ) = 0 may equally well be interpreted as f = ( 0 ) or, say, as f = ( 0 , 0 , 0 , 0 ) .

2.3 The Higman operations

Start by two specific subsets of ℰ,

We are going to extensively use the following operations from [7]:

which we call Higman operations on subsets of ℰ. The first two operations are binary functions, and for any subsets A , B of ℰ, they are defined as just the intersection ι ⁢ ( A , B ) = A ∩ B and the union υ ⁢ ( A , B ) = A ∪ B of those sets. This notation differs a little from the original notation ι ⁢ A ⁢ B and υ ⁢ A ⁢ B of [7] which in our case would cause confusion when used in long formulas together with other operations.

The other Higman operations are unary functions defined on any subset 𝒜 of ℰ as follows.

• ρ ⁢ ( A ) consists of all f ∈ E for which there is a g ∈ A such that f ⁢ ( i ) = g ⁢ ( − i ) .

• σ ⁢ ( A ) consists of all f ∈ E for which there is a g ∈ A such that f ⁢ ( i ) = g ⁢ ( i − 1 ) .

• τ ⁢ ( A ) consists of all f ∈ E for which there is a g ∈ A such that f ⁢ ( 0 ) = g ⁢ ( 1 ) , f ⁢ ( 1 ) = g ⁢ ( 0 ) and f ⁢ ( i ) = g ⁢ ( i ) for i ≠ 0 , 1 .

• θ ⁢ ( A ) consists of all f ∈ E for which there is a g ∈ A such that f ⁢ ( i ) = g ⁢ ( 2 ⁢ i ) .

• ζ ⁢ ( A ) consists of all f ∈ E for which there is a g ∈ A such that f ⁢ ( i ) = g ⁢ ( i ) for i ≠ 0 .

• π ⁢ ( A ) consists of all f ∈ E for which there is a g ∈ A such that f ⁢ ( i ) = g ⁢ ( i ) for i ≤ 0 .

• 𝜁 and 𝜋 are called liberations of 𝒜 in the sense that 𝜁liberates the sequences on 0: for every g ∈ A , it adds to our set 𝒜 all the functions 𝑓 which accept any value at 0, but which coincide with 𝑔 elsewhere. And 𝜋liberates the sequences on positive integers: for every g ∈ A , the operation 𝜋 adds to 𝒜 all the functions which accept any values at positive indices, but which coincide with 𝑔 on zero and on all negative indices.

We may agree to apply the unary Higman operations to individual functions also: a set A = { f } may consist of a single function 𝑓 only, so notations like ρ ⁢ f , σ ⁢ f , τ ⁢ f , etc., should cause no confusion.

To get familiar with these operations, the reader may check examples and basic lemmas in [7, Section 2] or in [15, Section 2.2].

Following Higman [7], we denote by 𝒮 the set of all subsets of ℰ which can be obtained from 𝒵 and 𝒮 by any series of operations (H). The elements in 𝒮 play a key role in the study of recursively presented groups. One of our main tasks below is going to be the discovery of many “natural” generic types of subsets of ℰ inside 𝒮.

2.4 Extra auxiliary operations

Our proofs will be much simplified by some auxiliary operations, each of which is a combination of a few Higman operations on subsets 𝒜 of ℰ.

For a positive integer 𝑖, naturally denote by σ i ⁢ A = σ ⁢ ⋯ ⁢ σ ⁢ A the result of application of 𝜎 for 𝑖 times. Set the inverse σ − 1 = ρ ⁢ σ ⁢ ρ as follows: f ∈ σ − 1 ⁢ ( A ) when there is a g ∈ A such that f ⁢ ( i ) = g ⁢ ( i + 1 ) . This allows to define the negative powers of 𝜎. Setting σ 0 ⁢ A = A , we have the powers σ i for any integer i ∈ Z . Clearly, σ i just “shifts” a sequence g ∈ A by | i | steps to the right or to the left depending on the sign of 𝑖.

It is easy to verify that σ i ⁢ ζ ⁢ σ − i ⁢ A consists of all functions f ∈ A in which the 𝑖-th coordinate is liberated. For briefness, denote ζ i = σ i ⁢ ζ ⁢ σ − i . Moreover, for a finite subset S = { i 1 , … , i m } ⊆ Z , denote the result of application of ζ i 1 ⁢ ⋯ ⁢ ζ i m by ζ i 1 , … , i m or by ζ S . That is, ζ S ⁢ A is the set of all those functions f ∈ E for which there is some g ∈ A such that f ⁢ ( i ) = g ⁢ ( i ) for each i ∉ S .

Denote by π ′ ⁢ A = ρ ⁢ π ⁢ ρ ⁢ A the liberation of 𝒜 on all negative coordinates, i.e., the set of all functions f ∈ E for which there is some g ∈ A such that f ⁢ ( i ) = g ⁢ ( i ) for each i ≥ 0 . Denote by π i ⁢ A = σ i ⁢ π ⁢ σ − i ⁢ A the liberation of 𝒜 on all coordinates after the 𝑖-th coordinate and, similarly, denote by π i ′ ⁢ A = σ − i ⁢ π ′ ⁢ σ i ⁢ A the liberation of 𝒜 on all coordinates before the 𝑖-th coordinate. In this notation, the original Higman operation 𝜋 is nothing but π 0 .

For any integers k < l , set s = l − k − 1 . It is not hard to verify that the set τ k , l ⁢ A = σ k ⁢ ( τ ⁢ σ ) s ⁢ τ ⁢ ( σ − 1 ⁢ τ ) − s ⁢ σ − k ⁢ A consists of all modified functions of 𝒜 with 𝑘-th and 𝑙-th coordinates “swapped”. More precisely, τ k , l ⁢ A is the set of all functions f ∈ E for which there is some g ∈ A such that f ⁢ ( k ) = g ⁢ ( l ) , f ⁢ ( l ) = g ⁢ ( k ) and f ⁢ ( i ) = g ⁢ ( i ) for each i ≠ k , l . In this notation, the Higman operation 𝜏 is nothing but τ 0 , 1 .

Furthermore, since any permutation 𝛼 of a finite set 𝑆 has a transposition decomposition α = ( k 1 ⁢ l 1 ) ⁢ ⋯ ⁢ ( k m ⁢ l m ) , we may introduce the set

which can be obtained from 𝒜 by respective permutation of coordinates for all g ∈ A . Clearly, α ⁢ A is the set of all functions 𝑓 for which there is a g ∈ A such that f ⁢ ( i ) = g ⁢ ( α − 1 ⁢ ( i ) ) for any i ∈ Z .

For any finite set of indices S = { i 1 , … , i m } and for any A ⊆ E , define the extract ϵ S ⁢ A = ϵ i 1 , … , i m ⁢ A to be the 𝑚-tuples set { ( g ⁢ ( i 1 ) , … , g ⁢ ( i m ) ) ∣ g ∈ A } . This operation can be constructed using (H) as follows. Assume i 1 < ⋯ < i m for clarity, and denote S ′ = { i 1 , i 1 + 1 , … , i m − 1 , i m } \ S (the set of all integers from i 1 to i m except those in 𝑆). The set A 1 = ζ S ′ ⁢ π i 1 ′ ⁢ π i m ⁢ A consists of functions from 𝒜 with all coordinates outside 𝑆 liberated. And the set A 2 = ζ S ⁢ Z consists of all functions which accept any integer values on S = { i 1 , … , i m } and which are zero elsewhere. The intersection A 3 = ι ⁢ ( A 1 , A 2 ) consists of those functions 𝑓 for which there is some g ∈ A such that f ⁢ ( i ) = g ⁢ ( i ) when i ∈ S , and f ⁢ ( i ) = 0 elsewhere. To get ϵ S ⁢ A from A 3 , it remains to apply the appropriate permutation 𝛼 that re-distributes the coordinates of f ∈ A 3 at the indices i 1 , … , i m on the 0 , 1 , … , m − 1 (here 𝛼 is a permutation of the union { i 1 , … , i m ; 0 , 1 , … , m − 1 } ).

For any subset A ⊆ E , denote sup ( A ) = ⋃ { sup ( f ) ∣ f ∈ A } . The point-wise sum f + g of any functions f , g ∈ E is defined as

For any subsets A , B ⊆ E , their sum A + B is the set { f + g ∣ f ∈ A , g ∈ B } . The sum of three or more sets is defined in the same manner. We are going to use this operation for cases when sup ( A ) and sup ( B ) are disjoint finite sets.

The intersection of any three or more subsets, such as A , B , C ⊆ E , can be expressed by Higman operations as ι ⁢ ( ι ⁢ ( A , B ) , C ) . To have shorter notation, record this as ι 3 ⁢ ( A , B , C ) . Similarly, define intersections ι n and unions υ n . In [7], Higman denotes the same by ι 2 ⁢ A ⁢ B ⁢ C , but in our case, this would create confusion in long formulas with many operations.

Investigating the subsets of ℰ in 𝒮, in addition to standard operations (H), we may often use the introduced auxiliary operations. This will shorten the proofs without changing the actual set 𝒮, for we above have representation of each auxiliary operation via (H).

2.5 The benign subgroups

The concept of benign subgroups is the key group-theoretical notion used in [7] to connect the sets in 𝒮 with subgroups in free groups, needed in construction of embeddings into finitely presented groups. A subgroup 𝐻 in a finitely generated group 𝐺 is called a benign subgroup in 𝐺 if 𝐺 can be embedded in a finitely presented group 𝐾 with a finitely generated subgroup L ≤ K such that G ∩ L = H . For basic properties and examples of benign subgroups, we refer to [7, Section 3] or to [15, Sections 3.1 and 3.2].

Below, we reserve the letters K , L for these specific groups only. In particular, if we have K , L for an “old” group and then we construct a “new” group with a respective finitely presented overgroup and its finitely generated subgroup, we may again denote them by the same letters 𝐾 and 𝐿. Also, if we have two benign subgroups, say, H 1 and H 2 , we will denote the respective groups by K 1 , K 2 and L 1 , L 2 . The context will tell us for which benign subgroups they are being considered, and no misunderstanding will occur.

3.1 Embedding with “universal” words in a free group of rank 2

Any countable group 𝐺 is embeddable into a 2-generator group 𝑇 (see [8]). Higman’s embedding construction [7] starts by some effective embedding of 𝐺 into an appropriate 𝑇. In the recent note [16], we suggested a method of effective embedding of any countable group 𝐺 into a 2-generator group 𝑇 such that the defining relations of 𝑇 are straightforward to deduce from relations of 𝐺. In fact, the very first embedding construction [8] (based on free constructions) and some other embedding constructions (based on wreath products, group extensions, etc.) already allow finding the relations of 𝑇. However, we need a method that not only makes deduction of the relations of 𝑇 from those of 𝐺 an automated task, but also preserves certain pattern in them, as we will see a little later; see Remark 3.8.

Let a countable group 𝐺 be given as

where 𝐹 is a free group on a countable alphabet 𝐴 and where R ¯ = ⟨ r 1 , r 2 , … ⟩ F is the normal closure of the set of all defining relations

in 𝐹.

In the free group F 2 = ⟨ b , c ⟩ of rank 2, choose the words

i = 1 , 2 , … . The map γ : a i → a i ⁢ ( b , c ) defines a correspondence

obtained by replacing each a i s , j in r s by the word a i s , j ⁢ ( b , c ) , j = 1 , … , k s . In fact, 𝛾 defines an embedding of 𝐺 into the 2-generator group

given by the relations r s ′ ⁢ ( b , c ) , s = 1 , 2 , … , on letters b , c (see [16, Theorem 1.1]). If 𝑅 is recursively enumerable, then the set R ′ of all above relations r s ′ ⁢ ( b , c ) also is recursively enumerable. That is, 𝑇 is a recursive group, in case 𝐺 is.

And when 𝐺 is a torsion-free group, then (3.1) can be replaced by shorter words

Inserting these a ¯ i ⁢ ( b , c ) in r s , we get shorter words r s ′′ ⁢ ( b , c ) . Then γ : a i → a ¯ i ⁢ ( b , c ) defines an embedding of 𝐺 into the 2-generator group

(see [16, Theorem 3.2]).

3.2 The main construction of the Higman embedding

For free generators a , b , c , fix the free group F 3 = ⟨ a , b , c ⟩ in addition to the above mentioned F 2 = ⟨ b , c ⟩ . Denote by b i the conjugate b c i for any i ∈ Z . Then, for each function f ∈ E define the product

and the conjugate a f = a b f . Say, for f = ( 5 , 2 , − 1 ) , we have

For any subset ℬ of ℰ, introduce the subgroup A B = ⟨ a f ∣ f ∈ B ⟩ in F 3 . In particular, for the zero set B = Z , we get the subgroup

and for the set B = S , we get the subgroup

As is verified in [7, Lemma 4.4], A Z and A S are benign in F 3 , and the respective 𝐾, 𝐿 (check notation in Section 2.5) can easily be constructed for each of them.

The most part of [7] is occupied by proofs for [7, Theorems 3 and 4] which set up the environment in which recursion is studied by group-theoretical means. By [7, Theorem 4], a set ℬ is recursively enumerable in ℰ if and only if A B is benign in F 3 , and by [7, Theorem 3], ℬ is recursively enumerable in ℰ if and only if it belongs to 𝒮, i.e., it can be constructed from the basic sets 𝒵 and 𝒮 using the Higman operations (H). This means we can start from benign subgroups A Z and A S , and as ℬ is being built from 𝒵 and 𝒮 by some series of operations (H), the benign subgroup A B is being constructed step by step. Note that, after Section 2.4, we are free to also use the new auxiliary operations we suggested there.

Each relation r s ′ we constructed in Section 3.1 for our recursive 2-generator group T = ⟨ b , c ∣ R ′ ⟩ can be written as

and this presentation will be unique if we also require n 1 , … , n 2 ⁢ m ≠ 0 . Thus, r s ′ can be “coded” by the sequence of exponents f s = ( n 0 , n 1 , … , n 2 ⁢ m + 1 ) , and the elements b f = b f s and a f = a f s can be defined for these particular f = f s . Say, for b 3 ⁢ c ⁢ b − 1 ⁢ c 2 , we have f = ( 3 , 1 , − 1 , 2 ) and

The set B = { f 1 , f 2 , … } of all such sequences clearly is a subset of ℰ, and we can define the respective subgroup A B = ⟨ a f ∣ f ∈ B ⟩ = ⟨ a f s ∣ s = 1 , 2 , … ⟩ in F 3 . As we mentioned above, A B is benign in F 3 if and only if ℬ can be constructed from the basic sets 𝒵 and 𝒮 using the Higman operations. This launches the following massive procedure in [7]: the set ℬ is written as an output of a series of operations (H) started from 𝒵 and 𝒮. For each step, one of the following actions may be taken.

1. B 1 , B 2 are already given, and B 3 is obtained from them by any of the binary Higman operations ι , υ . Also given are the respective benign subgroups A B 1 , A B 2 in F 3 , together with the respective groups K 1 , K 2 and L 1 , L 2 (see the remark about notation in Section 2.5). Then A B 3 is also benign and can construct the respective K 3 and L 3 .

2. B 1 is already given, and B 2 is obtained from it by any of the unary Higman operations ρ , σ , τ , θ , ζ , π , ω m for m = 1 , 2 , … . Also given are the respective benign subgroup A B 1 in F 3 , together with the respective groups K 1 and L 1 . Then A B 2 is also benign, and we have a mechanism allowing us to construct the K 2 and L 2 .

If, for a group 𝐺 (or for groups of a given generic type), we are able to explicitly write the set ℬ and are able to tell how ℬ can be extracted from 𝒵 and 𝒮 by operations (H), then we have an embedding of F 3 into a finitely presented group 𝐾 with a finitely generated group 𝐿 such that F 3 ∩ L = A B .

The final part of the Higman embedding is far shorter. By the proofs of [7, Lemmas 5.1 and 5.2], the normal closure R ¯ = ⟨ R ′ ⟩ F 2 is benign in F 2 if and only if A B is benign in F 3 . The proofs of these lemmas also provide the finitely presented group 𝐾 with a finitely generated subgroup 𝐿 such that 𝐾 embeds F 2 , and also F 2 ∩ L = R ¯ . Then “the Higman Rope Trick” (see the end of [7, Section 5], [12, p. 219] or [22]) uses these 𝐾 and 𝐿 to embed T = ⟨ b , c ∣ R ′ ⟩ , and thus also 𝐺, into a finitely presented group using a free product with amalgamation and an HNN-extension.

That is, if we are able to explicitly write ℬ by the operations (H), then we can construct the explicit embedding of 𝐺 into a finitely presented group. This is what we are going to do in the rest of this article.

3.3 Examples, the structure of sequences in ℬ

Let us continue Example 3.1 by applying the constructions from the previous section for the group Z ∞ .

Examples similar to Example 3.2 and Example 3.4 are easy to construct for free soluble groups, for free nilpotent groups and, more generally, for other types of groups defined by commutator-based identities.

Further, since any divisible abelian group is a direct product of copies of ℚ and of some C p ∞ , it is not hard to use Example 3.5 and Example 3.7 to get sequences of similar formats for them also. Moreover, every abelian group is a subgroup in an abelian divisible group, so we get similar sequences for embeddings of any countable abelian group (provided that its embedding into a countable divisible abelian group is constructively, effectively given).

As we see now, construction of a set B ∈ S by operations (H) in many cases can be reduced to the following question: can we build a “machine” which constructs ℬ by performing the five operations listed above, i.e., by assigning fixed pre-given values to some coordinates, then copying those values to other coordinates, then assigning the opposites, the sums or differences of those values to some other coordinates? If yes, then constructive Higman embeddings are available for the considered types of groups.

In the next section, we will step by step collect a positive answer to this question. The reader may skip to Example 4.11 to see an application of the method.

4.1 Construction of sum of subsets with disjoint supports

For definition of the sum of subsets from 𝒮 and of other auxiliary operations, we refer to Section 2.4.

The analog of this lemma could be proved for the case of infinite supports, but we restrict to this case for simplicity.

4.2 Construction of ( n ) with restrictions on 𝑛

Denote by B + = { ( n ) ∣ n = 1 , 2 , … } the set of all functions with a single positive coordinate, and by B − = { ( n ) ∣ n = − 1 , − 2 , … } the set of all functions with a single negative coordinate. Their union B ± = { ( n ) ∣ n ∈ Z \ { 0 } } is the set of all functions with a single non-zero coordinate.

The reader may compare the above proof with the argument of [7, Lemma 2.1].

Let B k + or B k − denote the set of all ( n ) for which n > k or n < k , respectively.