On direct sums and quotient spaces of near-vector spaces

Abstract

In this paper we study the direct sums of subspaces and some constructions of quotient spaces of near-vector spaces, as defined by André. In particular, for near-vector spaces constructed by taking copies of finite fields, we characterise the quasi-kernels of their quotient spaces, find their cardinality and determine when they are regular. In the case of non-regular quotient spaces, we show how they decompose into maximal regular subspaces. We show how the theory of finite-dimensional near-vector spaces constructed from finite fields allows us to reconstruct near-vector spaces with certain quotient spaces.

1 Introduction

Near-vector spaces have less linearity than traditional vector spaces, in that one of the distributive laws does not hold in general. A few authors have tried to capture this. In [3] Beidleman defined a near-vector space in terms of near-ring modules, while in [10] Karzel defined a near-vector space in terms of a group and a double loop, so that it satisfies the left distributive law. Subsequently in 1974, André defined a third notion of a near-vector space in [1], based on some constructions of near-affine spaces (see [2]). André’s near-vector spaces have been extensively studied, see for example the recent papers [6, 11]. In this paper we will continue investigating these near-vector spaces.

In vector space theory it is natural to study the direct sum of subspaces and the quotient space of a vector space. In this paper we contribute to the study of these constructs for André’s near-vector spaces. Section 2 begins with the preliminary material we will need and ends with a new theorem which links the proper subspaces of a non-regular near-vector space to its maximal regular subspaces. In Sect. 3 we look at direct sums of subspaces and quotients of near-vector spaces. In Sect. 4 we focus specifically on constructions of near-vector spaces where copies of a finite field are taken. In Sect. 5 we show how an understanding of their construction allows us to construct near-vector spaces having quotient spaces with certain properties.

2 Preliminary material

We begin with some preliminary material. First, we define what a near-vector space is. Throughout this paper we will write $S^*$ for $S{\setminus }\{0\},$ where S is any set.

Definition 2.1

[1] A near-vector space is a pair (V, A) which satisfies the following conditions:

• (a) $(V,+)$ is a group and A is a set of endomorphisms of V;

• (b) A contains the endomorphisms 0, 1 and $-1$, where 1 is the identity endomorphism and $-1$ the endomorphism defined by $x(-1)=-x$ for all $x\in V$;

• (c) $A^{*}$ is a subgroup of the group (Aut(V), $\circ$);

• (d) If $x\alpha =x\beta$ with $x\in V$ and $\alpha ,\beta \in A$, then $\alpha =\beta$ or $x=0$, i.e. A acts fixed point free on V;

• (e) The quasi-kernel Q(V) of V, generates V as a group, i.e. for all $v\in V$, there exists $u_i\in Q(V)$ and $\lambda _i \in A,$ such that

  $$\begin{aligned} v=\displaystyle \sum _{i=1}^{n}u_i\lambda _i. \end{aligned}$$

  Here,

  $$\begin{aligned} Q(V)=\{x\in V\mid \forall \alpha , \beta \in A,\,\exists \gamma \in A \text { such that }x\alpha +x\beta =x\gamma \}. \end{aligned}$$

The elements of V are the vectors and the elements of A are called the scalars of the near-vector space, while the action of A on elements of V is called scalar multiplication. For convenience, we will write Q instead of Q(V) when there is no room for confusion.

Although in the definition above, it is not required for $(V,+)$ to be abelian, the fact that $-1 \in A$, in (b) above, ensures that it is.

Lemma 2.2

[1] Let (V, A) be a near-vector space. Then $(V,+)$ is an abelian group.

Lemma 2.3

[1] The quasi-kernel Q has the following properties:

• 1. $0 \in Q$;

• 2. For $u \in Q^{*}$, $\gamma$ in Definition 2.1(e), is uniquely determined by $\alpha$ and $\beta$;

• 3. If $u \in Q$ and $\lambda \in A$, then $u\lambda \in Q$, i.e. $uA \subseteq Q$;

• 4. If $u \in Q$ and $\lambda _{i} \in A$, $i=1,2,\ldots ,n$, then $\sum _{i = 1}^{n} u\lambda _{i} = u \eta \in Q$ for some $\eta \in A$;

• 5. If $u \in Q^{*}$ and $\alpha , \beta \in A$, there exists a $\gamma \in A$ such that $u \alpha - u \beta = u \gamma$.

By Lemma 2.3 (2.), we have that $\gamma$ in Q(V) is uniquely determined by $\alpha$ and $\beta .$ André used this to define an addition on A,  so that for any $u \in Q(V)^{*}$ we have that $(A,+_{u},\circ )$ is a near-field. For more on near-fields, we refer the reader to [12]. We can also use Lemma 2.3 (3.) to simplify the requirement that Q(V) generates V as a group, i.e. for all $v\in V$, there exists $q_i\in Q(V)$ such that

$$\begin{aligned} v=\displaystyle \sum _{i=1}^{n}q_i. \end{aligned}$$

Definition 2.4

[1]. Let (V, A) be a near-vector space, and let $u\in Q(V)^*$. Define the operator $+_u$ on A for all $\alpha ,\beta \in A$ by

$$\begin{aligned} u(\alpha +_u\beta )=u\alpha + u\beta . \end{aligned}$$

From this it is clear that for $\alpha ,\beta \in A, \alpha +_{u} \beta$ is the unique $\gamma$ in Lemma 2.3(2.).

André defined independence for vectors inside the quasi-kernel in terms of a dependence relation (see [1]) and later proved the following characterisation.

Proposition 2.5

[1] A subset M of Q(V) is independent if and only if

$$\begin{aligned} \sum _{i=1}^{n}u_i\lambda _i = 0, \end{aligned}$$

with $u_i \in M$ and $\lambda _i \in A$ for $i \in \{1,\ldots ,n\},$ implies that $\lambda _i = 0$ for $i \in \{1,\ldots ,n\}.$

The dimension of V,  denoted dim(V),  is uniquely determined by the cardinality of a basis of Q(V), called a basis of V. A basis for Q is defined to be a subset B of Q such that for each $u \in Q$ there exists $b_1,b_2,\ldots ,b_n \in B$ and $\lambda _1,\lambda _2,\ldots ,\lambda _n \in A$ such that $u = \sum _{i=1}^{n}b_i\lambda _i.$

Theorem 2.6

[1] Let (V, A) be a near-vector space and suppose $S\subseteq Q$ is an independent set. Then there exists a basis B of Q such that $S\subseteq B.$

As a simple corollary, because $\emptyset$ is independent, the following result holds.

Corollary 2.7

[1] Every near-vector space has a basis.

Finally, bases have the same dimension.

Theorem 2.8

[1] Let (V, A) be a near-vector space with $B, B'$ bases of Q. Then $|B| = |B'|.$

For further preliminary material, we refer the reader to [1].

The notion of a subspace of a near-vector space was first defined in [9].

Definition 2.9

[9] If (V, A) is a near-vector space and $W\subseteq V$ is such that W is the subgroup of $(V, +)$ generated additively by $XA = \{x\lambda \mid x \in X, \lambda \in A\}$, where X is an independent subset of Q(V), then we say that (W, A) is a subspace of (V, A), or simply W is a subspace of V if A is clear from the context.

It is important to note here that we do not think of the elements of A as functions, but rather as scalars acting on V.

Lemma 2.10

[7] If W is a subspace of V, then $Q(W) = W \cap Q(V)$.

For more on subspaces, we refer the reader to [9].

As we will see, regularity is central in the study of near-vector spaces. We begin by defining what a regular near-vector space is.

Definition 2.11

[1] A near-vector space is regular if any two vectors of $Q(V)^*$ are compatible, i.e. if for any two vectors u and v of $Q(V)^*$ there exists a $\lambda \in A^{*}$ such that $u + v\lambda \in Q(V)$.

It is clear that every vector space is regular, but this is not in general true for every near-vector space, as we will see. In addition, if $V=Q(V)$, V is regular, but the converse is not true in general (see Example 3.3 in [5]). It is easy to verify that if (V, A) is a regular near-vector space and W a subspace of V,  then W is regular.

Suppose we know that V is a non-regular near-vector space, can it somehow be written in terms of regular subspaces? André answered this question in the next important theorem, called the Decomposition Theorem.

Theorem 2.12

([1], The Decomposition Theorem) Every near-vector space (V, A) is the direct sum of regular subspaces $V_{j}$ ($j \in J$) such that each $u \in Q(V)^*$ lies in precisely one direct summand $V_{j}$. The subspaces $V_{j}$ are maximal regular subspaces of V.

In the above theorem, V is the internal direct sum of the $V_j$ ($j \in J$) and each $(V_j,A_j)$ is a near-vector space where $A_j$ is A restricted to $V_j.$ By the Uniqueness Theorem ( [1]) this decomposition is unique and is called the canonical decomposition of V. Thus it is evident that the study of near-vector spaces is largely reduced to the study of regular near-vector spaces and why André referred to the regular subspaces of a near-vector space as the building blocks of near-vector space theory. We begin with the smallest example of a near-vector space that is non-regular.

Example 2.13

Consider the near-vector space (V, A) where $V = ({\mathbb {Z}}_{5})^2$ and $A = {\mathbb {Z}}_{5}$. Suppose for all $(x_1, x_2) \in V$ and $\alpha \in A$, scalar multiplication is defined by

$$\begin{aligned} (x_1, x_2) \alpha = (x_1 \alpha , x_2 \alpha ^3). \end{aligned}$$

Here $Q(V)=\{(a,0)\mid a\in A\} \cup \{(0,b)\mid b\in A\}.$

For (1, 0) and $(0,1)\in Q(V)$ and for all $\lambda \in A^{*}$,

$$\begin{aligned} (1,0) + (0,1)\lambda&= (1,\lambda ^3) \notin Q(V). \end{aligned}$$

Hence, V is not regular and we have that V decomposes as

$$\begin{aligned} V&= V_1 \oplus V_2 \\&= \{(a,0)\mid a\in A\} \oplus \{(0,b)\mid b \in A\} . \end{aligned}$$

We will also need the concept of a near-vector space linear mapping and isomorphism.

Definition 2.14

[9] Let (V, A) and (W, A) be near-vector spaces over A. A function $T: V \rightarrow W$ is a linear mapping from V to W if

$$\begin{aligned} T(v_1 + v_2) = T(v_1) + T(v_2) \text{ for } \text{ all } v_1, v_2 \in V \end{aligned}$$

and

$$\begin{aligned} T(v\alpha ) = T(v) \alpha \text{ for } \text{ all } v \in V, \ \alpha \in A. \end{aligned}$$

Linear mappings preserve regularity.

Lemma 2.15

[9] Let T be a linear mapping from (V, A) into (W, A). If $V_j$ is a regular subspace of V, then $T(V_j)$ is a regular subspace of W.

Definition 2.16

[8] We say that two near-vector spaces $(V_1,A_1)$ and $(V_2,A_2)$ are isomorphic (written $(V_1,A_1)\cong (V_2,A_2)$) if there are group isomorphisms $\theta : (V_1,+) \rightarrow (V_2,+)$ and $\eta : (A_1^*,\cdot ) \rightarrow (A_2^*,\cdot )$ such that $\theta (x\alpha ) = \theta (x)\eta (\alpha )$ for all $x\in V_1$ and $\alpha \in A_1^*$. We express the isomorphism as a pair $(\theta ,\eta ).$

Theorem 2.17

[9] If the near-vector spaces $(V_1,A_1)$ and $(V_2,A_2)$ are isomorphic, say $(\theta ,\eta )$ is the isomorphism, then $\theta (Q(V_1))= Q(V_2)$.

Our focus will be on finite-dimensional near-vector spaces constructed using copies of finite fields. In [13] it was proved that finite-dimensional near-vector spaces can be characterised in the following way.

Theorem 2.18

([13], van der Walt’s theorem) Let $(G,+)$ be a group and let $A=D\cup \{0\}$, where D is a fixed point free group of automorphisms of G. Then (G, A) is a finite-dimensional near-vector space if and only if there exist a finite number of near-fields $F_1,\ldots ,F_m$, semigroup isomorphisms $\psi _i:\, (A,\circ )\rightarrow (F_i,\cdot )$, and an additive group isomorphism $\Phi :\, G\rightarrow F_1\oplus \cdots \oplus F_m$ such that if $\Phi (g)=(x_1,\ldots ,x_m)$, then $\Phi (g\alpha )=(x_1\psi _i(\alpha ),\ldots ,x_m\psi _m(\alpha ))$ for all $g\in G,$ $\alpha \in A.$

By Theorem 2.18, we can specify a finite-dimensional near-vector space over a finite field as follows. Let $F=GF(p^r)$, p a prime, $r\in {\mathbb {Z}}^{+},$ be a finite field. Put $V=F^m$ for $m\in {\mathbb {Z}}^+$ and let $I=\{1,\ldots ,m\}.$ Let $\psi _i:(F,\cdot ) \rightarrow (F,\cdot )$, for $i\in I$, be semi-group automorphisms. We define the scalar multiplication for all $\alpha \in F,~(x_i)\in V,~i\in I$ by

$$\begin{aligned} (x_1, \ldots , x_m)\alpha =(x_1\psi _1(\alpha ), \ldots ,x_m\psi _m(\alpha )). \end{aligned}$$

(2.1)

We recall that if $\psi _i:(F,\cdot )\rightarrow (F,\cdot )$ is a semigroup automorphism, then for all $\alpha \in F$ it has the form $\psi _i(\alpha )=\alpha ^q$, for some q such that gcd$(q,p^r-1)=1$. We will denote a specific instance of this construction by (V, F). For more on these constructions, we refer the reader to [4] and [9].

The following result by S.P. Sanon shows how we can construct regular near-vector spaces from copies of finite fields.

Lemma 2.19

[4] Let $F= GF(p^r)$ and $V=F^m$ be a near-vector space and $\psi _i$’s are automorphisms of $(F,\cdot )$ for $i \in I$. Then define scalar multiplication for all $(x_1,\ldots ,x_m)\in V$ and $\alpha \in F$ by

$$\begin{aligned} (x_1,\ldots ,x_m)\alpha =(x_1\psi _1(\alpha ),\ldots , x_m\psi _m(\alpha )). \end{aligned}$$

Then V is regular if and only if for all $i,j\in I$ and $\alpha \in GF(p^r)$, $\psi _i(\alpha )= \psi _j(\alpha ^{p^l})$, for some $l\in \{ 0,\ldots ,r-1\}$.

We use Lemma 2.19 to partition the set I as follows. Let $A_i=\{j\in I\mid \psi _i(\alpha )=\psi _j(\alpha ^{p^{l}})$ for some $l\in \{0,1,\ldots ,r-1\}\}$. The $A_t$, for $t\in K=\{1,\ldots ,k\}$ are called the blocks of the construction.

In [4] the following result was proved.

Lemma 2.20

[4] For the near-vector space defined above we have:

• 1. $Q(V)=\bigcup \nolimits _{t=1}^k {V}_t$ where ${V}_t=\{(0,0,\ldots ,a_1,0,a_2,0,\ldots ,a_s,0)\mid a_i\in F, a_i \text{ is } \text{ in } \text{ position } l \text{ with } l\in A_t\}, \text{ for } t\in K$;

• 2. Each of the $V_t,$ for $t \in K,$ is a regular subspace of V;

• 3. $V={V}_1 \oplus {V}_2 \oplus \cdots \oplus {V}_k$ is the canonical decomposition of V.

For this particular construction, regularity is equivalent to the quasi-kernel being the whole of V.

Theorem 2.21

[9] Let $F=GF(p^r)$ and $V=F^{m}$ be a near-vector space with scalar multiplication defined for all $\alpha \in F$ by

$$\begin{aligned} (x_1,\ldots ,x_m)\alpha =(x_1\psi _1(\alpha ),\ldots ,x_m\psi _m(\alpha )), \end{aligned}$$

where the $\psi _i's$ are automorphisms of $(F,\cdot )$ for $i\in I.$ Then the following are equivalent:

• 1. $Q(V) = V;$

• 2. V is regular;

• 3. For all $i,j \in \{1,\ldots ,m\}$ and $\alpha \in GF(p^r)$, $\psi _i(\alpha )=\psi _j(\alpha ^{p^{l}})$, for some $l\in \{0,1,\ldots ,r-1\}.$

By making use of Theorem 2.21 and the Decomposition Theorem we can prove the following new theorem, which links the proper subspaces of a non-regular near-vector space to its maximal regular subspaces. We will use this result in Sect. 4.

Theorem 2.22

Let $F=GF(p^r)$ and $V=F^{m}$ be a non-regular near-vector space with scalar multiplication defined for all $\alpha \in F$ by

$$\begin{aligned} (x_1,\ldots ,x_m)\alpha =(x_1\psi _1(\alpha ),\ldots ,x_m\psi _m(\alpha )), \end{aligned}$$

where the $\psi _i's$ are automorphisms of $(F,\cdot )$ for $i\in I$ and the canonical decomposition of V given by $V ={V}_1 \oplus {V}_2 \oplus \cdots \oplus {V}_k.$ If W is a proper subspace of V,  there are two possibilities:

• 1. W is regular, in which case $W \subseteq V_{j}$ for some $j \in K;$

• 2. W is non-regular, in which case the canonical decomposition of W is given by $W = {W}_1 \oplus {W}_2 \oplus \cdots \oplus {W}_t,$ where for each $s \in \{1,\ldots t\},$ we have that $W_{s}\subseteq V_{j}$ for some $j \in K.$

Proof

• 1. Suppose that W is regular and that $W \not \subseteq V_{j}$ for any $j \in K.$ Then $W \subseteq \oplus _{j' \in J'}V_{j'}$ for some $J^{'}\subset K,$ with $|J^{'}|\ge 2$ and $Q(W) \subset \cup Q(V_{j^{'}}).$ Now consider the vector x with 1’s in position $j' \in J'$ and zeros elsewhere. Then $x \in W,$ but $x \notin Q(W),$ a contradiction, by Theorem 2.21.

• 2. Follows from [1.] since every $W_s$ for $s \in \{1,\ldots t\},$ is a proper regular subspace of V.

$\square$

3 Some constructions of near-vector spaces

We begin with the direct sum of subspaces of a given near-vector space. The following theorem is not difficult to prove. We use it to define the direct sum of two subspaces.

Theorem 3.1

Let (V, A) be a near-vector space and $W_1$ and $W_2$ be subspaces of V. Then the following conditions are equivalent:

• 1. $W_1\cap W_2=\{ 0 \}$;

• 2. Every vector x in $W_1+W_2 =\{w_1+w_2\mid w_1\in W_1, w_2\in W_2\}$ is uniquely representable in the form $x=w_1+w_2$, with $w_1\in W_1$ and $w_2\in W_2$.

Definition 3.2

If (V, A) is a near-vector space which satisfies the conditions of Theorem 3.1, then $W_1+W_2$ is called the direct sum of $W_1$ and $W_2$ and is denoted by $W_1\oplus W_2$.

The following theorem exhibits a basis for $V=W_1\oplus W_2$ in terms of bases of the subspaces $W_1$ and $W_2$.

Theorem 3.3

Let (V, A) be a finite dimensional near-vector space with $W_1$ and $W_2$ subspaces of V such that $V = W_1 \oplus W_2$. If ${{{\mathcal {B}}}} = \{x_1,x_2.\ldots ,x_k\}$ is a basis for $W_1$ and ${{{\mathcal {C}}}}=\{y_1,y_2,\ldots ,y_m\}$ a basis for $W_2$, then ${{{\mathcal {K}}}} = \{x_1,x_2,\ldots ,x_k,y_1,y_2\ldots ,y_m\}$ is a basis for V. Thus $\text{ dim }(V) = \text{ dim }(W_1) + \text{ dim }(W_2)$.

Proof

We first need to show that ${{{\mathcal {K}}}}\subseteq Q(V).$ By Lemma 2.10, we have that $Q(W_i)\subseteq Q(V)$ for $i\in \{1,2\}$. Since ${{{\mathcal {B}}}}\subseteq Q(W_1)$ and ${{{\mathcal {C}}}}\subseteq Q(W_2)$, we have that both ${{{\mathcal {B}}}}$ and ${{{\mathcal {C}}}}$ are contained in Q(V). Hence ${{{\mathcal {K}}}}\subseteq Q(V)$.

By Theorem 3.1, for each $v\in Q(V)$, we can write $v=w_1+w_2$ for some $w_1\in W_1$ and $w_2\in W_2$. Then for some $\alpha _i,\beta _j \in A$, $x_i\in {{\mathcal {B}}}$, $y_{j} \in {{\mathcal {C}}}$,

$$\begin{aligned} v&= w_1+w_2\\&= \displaystyle \sum _{i=1}^{k}x_i\alpha _i+\sum _{j=1}^{m}y_j\beta _j, \end{aligned}$$

since ${{{\mathcal {B}}}}$ and ${{{\mathcal {C}}}}$ are bases for $W_1$ and $W_2$, respectively. Therefore, ${{{\mathcal {K}}}}$ generates Q(V). Since $W_1\cap W_2=\{0\}$ and ${{{\mathcal {K}}}} \subseteq Q(V),$ it is easy to show that the vectors in ${{{\mathcal {K}}}}$ are independent by Proposition 2.5. Thus ${{{\mathcal {K}}}}$ is a basis for Q(V),  and thus a basis of V. $\square$

It is routine to generalise Definition 3.2 and Theorem 3.3 for direct sums of more than two subspaces.

The following definition and result were included in [14].

Theorem 3.4

Let W be a subspace of a near-vector space (V, A) and $V/W=\{v+W\mid v\in V\}$. Then V/W (as an abelian group) under the operations

• (a) $(v_1 + W) + (v_2+W)= (v_1+v_2)+W$;

• (b) $(v_1+W)\alpha = v_1\alpha +W$;

is a near-vector space over A, called the quotient near-vector space over A.

However, an error was subsequently picked up in this proof by Dr Sophie Marques. A proof is given in a recent paper by Dr Marques and Ms Daniella Moore [11] (Theorem 4.2).

Both proofs showed that we get that a subset of Q(V/W) generates V/W.

Remark 3.5

Let W be a subspace of a near-vector space (V, A) and $V/W=\{v+W\mid v\in V\}$. It is not difficult to check that $Q(V)/W \subseteq Q(V/W)$ and that since Q(V) generates V,  we have that Q(V)/W will generate V/W. Thus $Q(V)/W \subseteq Q(V/W)$ generates V/W.

It is worth noting that when we define the quotient near-vector space V/W for a subspace W of V,  the near-vector space V/W is understood to be defined over a subset of A.

By using the fact that if (V, A) is a near-vector space, the action of A is well-defined on V/W,  it immediately follows that cosets that are equal have the same addition as defined in Definition 2.4.

Lemma 3.6

Let (V, A) be a near-vector space, W a subspace of V and $a+W\in Q(V)/W$. For $a,a' \in Q(V)^*$, if $a+W=a'+W$, then for all $\alpha ,\beta \in A$,

$$\begin{aligned} \alpha +_a \beta = \alpha +_{a'} \beta . \end{aligned}$$

We can construct a basis for (V/W, A). We note that Lemma 2.6 ensures that a basis of a subspace of a near-vector space can be extended to a basis of the near-vector space.

Theorem 3.7

Let (V, A) be a near-vector space and W a subspace of V. Suppose $\{w_1,\ldots ,w_m\}$ and $\{w_1,\ldots ,w_m, v_1, \ldots ,v_n\}$ are bases of W and V, respectively. Then

$$\begin{aligned} {{{\mathcal {B}}}} = \{v_1+W, \ldots ,v_n+W\} \end{aligned}$$

is a basis of V/W, i.e. $\dim V/W = \dim V - \dim W$.

Proof

${{{\mathcal {B}}}}\subseteq Q(V/W)$ by Remark 3.5, since $\{v_1,\ldots ,v_n\}\subseteq Q(V)$. Next we show that ${{{\mathcal {B}}}}$ generates Q(V/W). Let $x\in Q(V/W)$, then $x=v+W$ for some $v\in V$. But we can also write v as follows:

$$\begin{aligned} v=\displaystyle \sum _{j=1}^{m} w_j\eta _j+\displaystyle \sum _{i=1}^{n} v_i \varepsilon _i, \end{aligned}$$

where $\eta _j,\varepsilon _i\in A$ for $j\in J'=\{1,\ldots ,m\}$ and $i\in I'=\{1,\ldots ,n\}$. Therefore,

$$\begin{aligned} x&= v+W\\&= \displaystyle \sum _{j=1}^{m} w_j\eta _j+\displaystyle \sum _{i=1}^{n} v_i \varepsilon _i +W\\&= \displaystyle \sum _{i=1}^{n} v_i \varepsilon _i +W\\&= \displaystyle \sum _{i=1}^{n} (v_i+W) \varepsilon _i , \end{aligned}$$

since $\displaystyle \sum \nolimits _{j=1}^{m} w_j\eta _j\in W$ and V is abelian. Finally, we need to show that ${{{\mathcal {B}}}}$ is independent. Suppose $\sum v_i \alpha _i +W=0+W=W$, which implies that $\sum v_i \alpha _i\in W$. Then for some $\eta _j\in A$ we have that

$$\begin{aligned}&\sum v_i\alpha _i = \sum w_i \eta _j \\&\sum v_i\alpha _i+\sum w_i (-\eta _j) = 0. \end{aligned}$$

But $\{w_1,\ldots ,w_m, v_1, \ldots ,v_n\}$ is independent, so by Proposition 2.5, $\alpha _i=\eta _j=0$ for all $i\in I'$, $j\in J'$. $\square$

4 Direct sums and quotient spaces of constructions using finite fields

If we consider the construction (V, F), where $V=F^m$, F is a finite field, we can say more. Recall that $K = \{1,\ldots ,k\}.$

Proposition 4.1

Let (V, F) be a near-vector space where F is a finite field and W any subspace of V. If V is regular, then V/W is regular.

Proof

We show that $Q(V/W)=V/W,$ thus giving us that V/W is regular. Since we have that $Q(V/W)\subseteq V/W$, we show the other inclusion. Let $v+W\in V/W$, where $v\notin W$. Since V is regular, we have that $Q(V)=V$, and so $v\in Q(V)$. By Remark 3.5 we have that $v+W\in Q(V/W)$. This implies $V/W\subseteq Q(V/W)$ and thus we have that $Q(V/W)=V/W$.

$\square$

The converse of Proposition 4.1 is not true, in general, that is, V/W being regular does not imply that V is regular, as illustrated by the next example.

Example 4.2

Referring back to the near-vector space in Example 2.13, where $V = F^2$, where $F = {\mathbb {Z}}_{5}$. Suppose for all $(x_1, x_2) \in V$ and $\alpha \in F$, scalar multiplication is defined by:

$$\begin{aligned} (x_1, x_2) \alpha = (x_1 \alpha , x_2 \alpha ^3). \end{aligned}$$

The near-vector space V is not regular and decomposes as:

$$\begin{aligned} V&= V_1 \oplus V_2\\&= \{(a,0)\mid a\in F\} \oplus \{(0,b)\mid b,\in F\}. \end{aligned}$$

However, $V/V_2$ is regular and in fact, isomorphic to the vector space $(F^{2},F)$.

We now focus on the case where (V, F) is a non-regular near-vector space and W is an arbitrary subspace of V.

By Remark 3.5, we have that $Q(V)/W\subseteq Q(V/W)$ where V/W is a near-vector space over A. We can now show more.

Lemma 4.3

Let (V, F) be a non-regular near-vector space where F is a finite field such that its canonical decomposition is given by $V=V_1\oplus \cdots \oplus V_k.$ If W is a proper subspace of V,  then $Q(V/W)=Q(V)/W$.

Proof

By Remark 3.5, we have that $Q(V)/W\subseteq Q(V/W)$. Conversely, let $v'+W\in Q(V/W)$, where $v'+W\ne W$. Then for all $\alpha ,~ \beta \in F$ there exists an $\gamma \in F$ such that

$$\begin{aligned} (v'+W)\alpha + (v'+W)\beta&=(v'+W)\gamma \\ \implies (v'\alpha +v'\beta )+W&= v'\gamma + W. \end{aligned}$$

This implies that $v'\alpha +v'\beta -v'\gamma \in W$. By Theorem 2.22, there are two cases to consider, either $W \subseteq V_s$ for an $s \in K,$ or $W = \oplus _{t \in T}W_{t},$ where each $W_{t}\subseteq V_{s}$ for some $s\in K.$ We put $J = \{s\}$ for the first case, or $J = \{s \in K \,|\, W_t \subseteq V_{s}\}.$

Then $v' \in W' = \oplus _{j \in K{\setminus } J} V_j.$ We also have that $v'\alpha +v'\beta -v'\gamma \in W'$, since $W'$ is a near-vector space. But $W'\cap W=\{0\}$, so

$$\begin{aligned} v'\alpha +v'\beta -v'\gamma&= 0\\ v'\alpha +v'\beta&= v'\gamma . \end{aligned}$$

This means that $v'$ is ultimately in Q(V), which implies that $v'+W\in Q(V)/W$ and we have that $Q(V/W)\subseteq Q(V)/W$. Hence, $Q(V/W)=Q(V)/W$. $\square$

As stated before, for any near-vector space V defined over a finite field F,  we have that $Q(V)=V$ if and only if V is regular. The next theorem proves an analogous result for the near-vector space V/W defined over a finite field F, where W is a proper subspace of V.

Theorem 4.4

Let (V, F) be a non-regular near-vector space where F is finite field and the scalar multiplication is defined for all $(x_1,\ldots ,x_m)\in V$ and $\alpha \in F$ by

$$\begin{aligned} (x_1,\ldots x_m)\alpha = (x_1\psi _1({\alpha }),\ldots ,x_m\psi _m(\alpha )), \end{aligned}$$

where the $\psi _i$’s are automorphisms of $(F,\cdot )$ for $i\in I.$ Let $V=V_1\oplus \cdots \oplus V_k$ be the canonical decomposition of V and W a proper subspace of V. Then the following are equivalent:

• 1. V/W is regular;

• 2. $V/W \cong V_i \text{ for } \text{ some } i \in K;$

• 3. $Q(V/W)=V/W$.

Proof

By Theorem 2.22, either $W \subseteq V_s$ for an $s \in K,$ or $W = \oplus _{t \in T}W_{t},$ where each $W_{t}\subseteq V_{s}$ for some $s\in K.$ We put $J = \{s\}$ for the first case, or $J = \{s \in K \,|\, W_t \subseteq V_{s}\}.$ Now let $W' = \oplus _{j \in K{{\setminus }} J} V_j.$ By Proposition 4.5, $W'$ is a subspace of V. Define a mapping $\theta : W' \rightarrow V/W$ by $\theta (v) = v + W.$ Next let $\eta : (F,\cdot ) \rightarrow (F,\cdot )$ be the identity mapping. Then it is not difficult to check that $(\theta , \eta )$ is a near-vector space isomorphism.

1 $\implies$ 2: Suppose that V/W is regular. Then $W'$ must be regular by Theorem 2.15. Thus $W' \subseteq V_i$ for some $i \in K{\setminus } J,$ but W is the direct some of all the subspaces $V_j$ with $j \in K{\setminus } J,$ so $V_i \subseteq W',$ so $V_i = W'.$ This gives that $V/W \cong V_i.$

2 $\implies$ 3: Suppose that $V/W = \{v+W\mid v\in V_i \text{ for } \text{ some } i \in K{{\setminus }} J\}.$ We already have that $Q(V/W) \subseteq V/W.$ Now let $v + W \in V/W,$ then $v \in V_i = Q(V_i),$ for some $i \in K$ so that $v \in Q(V).$ Thus $V/W \subseteq Q(V/W)$ and thus $Q(V/W)= V/W.$

3 $\iff$ 1 follows from the fact that V/W is a near-vector space defined over a finite field. $\square$

We have encountered a case where the quotient space is non-regular. A natural question is what can we say about the canonical decomposition of the space. We take a look at the case where W is a direct sum of some, but not all, the maximal regular subspaces of V,  not necessarily picked consecutively.

Proposition 4.5

Let (V, F) be a non-regular near-vector space where F is a finite field such that its canonical decomposition is given by $V=V_1\oplus V_2\oplus \cdots \oplus V_k$. If $W=\displaystyle \bigoplus \nolimits _{j \in J} V_j$, for $J \subset K$, then W is a subspace of V and $Q(W)=\displaystyle \bigcup \nolimits _{j \in J}V_j.$

This proposition allows us to describe the quasi-kernel for these type of quotient spaces.

Lemma 4.6

Let (V, F) be a non-regular near-vector space where F is a finite field such that its canonical decomposition is given by $V=V_1\oplus \cdots \oplus V_k.$ If $W'= V_1\oplus V_2\oplus \cdots \oplus V_t$, where $t< k-1,$ then

$$\begin{aligned} Q(V/W')&= \displaystyle \bigcup _{i=t+1}^{k}(V_i+W)/W. \end{aligned}$$

Proof

By Proposition 4.5, $Q(W')=V_1\cup \cdots \cup V_t$ and by Theorem 4.4, $V/W'$ is not regular since $Q(V/W')\ne V/W'.$ Now making use of Lemma 4.3 and Lemma 2.20, the result follows. $\square$

In the following proposition we state what the canonical decomposition of $V/W'$ will be.

Proposition 4.7

Let (V, F) be a non-regular near-vector space where F is a finite field such that its canonical decomposition is given by $V=V_1\oplus \cdots \oplus V_k.$ If $W'= V_1\oplus V_2\oplus \cdots \oplus V_t$, where $t< k-1,$ then

$$\begin{aligned} V/W'=\displaystyle \bigoplus _{i=t+1}^{k}(V_i+W')/W' \end{aligned}$$

is the canonical decomposition of $V/W'$.

Proof

To show that $V/W'=\displaystyle \bigoplus \nolimits _{i=t+1}^{k}(V_i+W')/W'$ is the canonical decomposition of $V/W'$, we need to show that for $i\in \{t+1,\ldots ,k\}$, each $(V_i+W')/W'$ is a maximal regular subspace. Let $v+W'\in (V_i+W')/W'$, then for $\alpha ,\beta \in F$ we have for

$$\begin{aligned} (v+W')\alpha +(v+W')\beta&= (v\alpha + v\beta )+W'\\&= v\gamma +W' \end{aligned}$$

for $\gamma \in F$, since we have that $Q(V_i)=V_i$ for $i\in \{1,\ldots ,k\}$. Therefore for each $i\in \{t+1,\ldots k\}$, $(V_i+W')/W'$ is a subspace of $V/W'$ and hence, regular. Furthermore, cosets all have the same size, hence for each $i\in \{t+1,\ldots k\}$, $(V_i+W')/W'$ is maximal. $\square$

We end off with the cardinality of $Q(V/W')$.

Theorem 4.8

Let (V, F) be a non-regular near-vector space where F is a finite field such that its canonical decomposition is given by $V=V_1\oplus \cdots \oplus V_k.$ If $W'= V_1\oplus V_2\oplus \cdots \oplus V_t$, where $t< k-1,$ then

$$\begin{aligned} |Q(V/W')|=\displaystyle \sum _{i=t+1}^{k}|Q(V_i)^{*}|+1. \end{aligned}$$

Proof

Since $Q(V/W')=\displaystyle \bigcup \nolimits _{i=t+1}^{k}(V_i+W')/W'$ and the $V_i$’s all have only the zero-vector in common, the result follows immediately. $\square$

5 An algorithm for finding a near-vector space over a finite field for a given quotient space

We close the paper by looking at a reconstruction problem that shows how the theory of finite-dimensional near-vector spaces over finite fields can be used to construct a near-vector space whose quotient space has certain properties.

In [8] it is explained that suitable sequences can be used to construct the multiplicative automorphisms in van der Walt’s theorem (Theorem 2.18).

Definition 5.1

[8] A finite sequence of m integers $q_{1}, q_{2},\ldots , q_{m}$ is called suitable with respect to $F = GF(p^{n})$ if

• (a) $1\le q_{i} \le p^{n} -1$ and $gcd(q_i, p^{n}-1) = 1$ for all $i = 1,\ldots ,m$;

• (b) no $q_{i}$ can be replaced by a smaller $q_{i}^{'}$ that also satisfies (a) and such that $q_{i} \equiv q_{i}^{'} p^{l}$ (mod $p^{n} -1$) for some $l \in \{0, 1,\ldots ,n -1\}$.

Suitable sequences are always written in non-decreasing order: $q_{1}\le q_{2} \le \cdots \le q_{m}$. Hence, to obtain a suitable sequence with respect to $GF(p^{n})$, simply make a list of the smallest members of all the cosets determined by the subgroup $\langle p \rangle$ of the multiplicative group $U(p^{n} -1) =\{k \in {{\mathbb {Z}}} \,|\, 1\le k \le p^{n} -1 \text{ and } gcd(k, p^{n} -1) = 1\}$. Then select (possibly with repetition) any m members from this list, and write them down in non-decreasing order. Note that there will be $\phi (p^{n} - 1)/n$ elements in the list to choose from.

Given the following information:

• 1. V is an m-dimensional near-vector space, with $m \ge 2,$ that decomposes into k maximal regular subspaces, where $k \ge 2.$

• 2. V/W has dimension s,  with $1 \le s \le k-1.$

The algorithm:

• 1. Put $V=(GF(p^{n}))^{m}$ and $F=GF(p^{n}).$

• 2. Construct a partition $A_1, A_2,\ldots ,A_k$ of $\{1,\ldots ,m\}.$

• 3. For $i=1,\ldots ,k$, construct a suitable sequence $q_{1}\le q_{2} \le \cdots \le q_{k}$.

• 4. Now define

  $$\begin{aligned} (x_1,x_2,\ldots ,x_m)\alpha = (x_1\alpha ,x_2\alpha ^{s_1},\ldots ,x_m\alpha ^{s_{m-1}}), \end{aligned}$$

  where $\alpha ^{s_{j}} = \alpha ^{q_{i}}$ for $j \in A_{i}$, $i=1,\ldots ,k$ and $j \in \{1,\ldots ,m-1\}$.

• 5. (V, F) is a near-vector space (see Theorem 2.18).

• 6. Put $W=\displaystyle \bigoplus \nolimits _{j\in J} V_j$, where $J=\{1,\ldots ,m-s\}.$

We note that the choice of a different partition of $\{1,\ldots ,m\}$ in the second step will lead to a different near-vector space, not necessarily isomorphic to the first. Also, by replacing a given $q_{i}$ by a member of the same coset, we obtain a space isomorphic to the previous one (see Lemma 3.7, p. 58 in [8]).

Example 5.2

Suppose that $m = 4,\, k=3,\, s=1.$

We put $V= (GF(3^{3}))^{4}$ and $F= GF(3^{3})$. Then $A_{1} = \{1\}$, $A_{2} = \{2,3\}$, $A_{3} = \{4\}$ is a possible partition of $\{1,2,3,4\}$. The set of cosets determined by $\langle 3\rangle$ in the group $U(3^3-1)$ is given by $\{ \{1,3,9\}, \{5,15,19\}, \{7,21,11\},\{17,25,23\}\}$. The set of smallest elements $\{1,5,7,17\}$ in these cosets determines all possible suitable sequences. We list those with 1 as smallest element:

$$\begin{aligned} \{(1,1,1,1), (1,1,1,5), \ldots , (1,5,5,7), (1,5,5,17),\ldots ,(1,7,17,17), (1,17,17,17)\}. \end{aligned}$$

We require a sequence with two entries identical according to our partition, say (1, 5, 5, 17). Then (V, F) defines a near-vector space of dimension 4 where scalar multiplication is given by

$$\begin{aligned} (x_1,x_2,x_3,x_4)\alpha = (x_1\alpha ,x_2\alpha ^5,x_3\alpha ^5,x_4\alpha ^{17}). \end{aligned}$$

We have that $V = V_1 \oplus V_2 \oplus V_3,$ where $V_1 = \{(a,0,0,0) \mid a \in GF(3^{3})\}, V_2 = \{(0,b,c,0) \mid b,c \in GF(3^{3})\}, V_3= \{(0,0,0,d) \mid d \in GF(3^{3})\}.$ Now $J= \{1,2\},$ so $W = V_1 \oplus V_2$ and V/W has the required dimension.

6 Open problem

In this paper our main focus was quotient spaces of constructions using copies of finite fields and maximal regular subspaces. For future consideration, one could look at the quotient spaces using any subspace, as well as quotient spaces of constructions using copies of near-fields.