Binary convex fuzzy vector spaces over binary vector spaces and their applications

ABSTRACT

In this paper, we explore the concept of binary convex fuzzy vector spaces and binary convex fuzzy linear subspaces over binary vector spaces $F_2^n$ by formalizing their definition, and we found interesting results on their properties. We also studied binary fuzzy vector spaces, and binary convex fuzzy sets by establishing relevant properties. We accomplish these by redefining fuzzy subsets over $F_2^n$ in a new manner using the formula of the reliability of the channel to send messages correctly over noise channels to make the concepts applicable in natural and linguistic communication systems. We also defined binary convex fuzzy codes over binary convex fuzzy vector space and presented some properties. Furthermore, we realized that the properties discussed for binary fuzzy vector spaces are applied to binary convex fuzzy vector spaces. We used binary fuzzy vector spaces to formulate binary fuzzy codes and binary convex fuzzy vector spaces to define binary convex fuzzy codes. We also draw connections between binary fuzzy codes and binary convex fuzzy codes. The study of binary convex fuzzy vector spaces and binary fuzzy vector spaces is particularly relevant in the context of information transmission through noisy communication channels, where it allows for the exploration of fuzzy error-correcting/detecting codes and related properties to detect/correct errors that may occur during transmission.

KEYWORDS:

• fuzzy vector spaces
• convex fuzzy sets
• fuzzy dimensions
• convex fuzzy codes

1. Introduction

The study of classical set theory was started by Cantor in 1874 (Cantor, 1874). In the classical context, we represent membership of an object as either 1 (member) or 0 (not a member) of a set for collections of well-defined objects. So, we have only these two degrees of freedom. The principles of classical set theory have been widely applied in fields such as communication systems, electronics, computer science, and engineering. However, the real world is marked by uncertainty and many vague concepts that cannot be adequately expressed using classical set theory. The existence of these uncertain and ambiguous concepts in the natural world has led scholars to define fuzzy sets, a new type of set designed to represent uncertainty and vagueness, first introduced by (Zadeh, 1965). In this case, there is freedom in defining uncertain or vague concepts tailored to specific purposes, and the degree of membership falls within $[0,1]$ (which is the subset of real numbers). Zadeh also introduced the idea of fuzzy convex fuzzy sets in (Zadeh, 1965), which has since been further explored by various scholars (Lowen, 1980; Ammar, 1999; Wu & Cheng, 2004; Yuan & Lee, 2004).

Furthermore, scholars have been studying classical vector spaces to describe objects occurring in combinations. The concept of classical vector spaces was introduced by an Italian mathematician Peano in 1888 (Moore, 1995). While subsequent researchers expanded on Peano’s linear system and studied vector spaces more broadly. However, the world still needed to express uncertain and vague objects occurring in combinations. Katsaras and Liu were the pioneers in formulating the concept of fuzzy vector spaces in 1977(Katsaras & Liu, 1977), paving the way for extensive research in the concepts of fuzzy vector spaces (Lubczonok, 1990; Abdukhalikov et al., 1994; Kumar, 1992), fuzzy subspaces (Hedayati, 2009; Kumar, 1993), fuzzy linear spaces (Wenxiang & Tu, 1992), and fuzzy linear subspaces (Feng & Li, 2013). In modern times, the utilization of fuzzy set theory and fuzzy logic has become prevalent in diverse fields, including computer science (Gosain & Dahiya, 2016; Garibaldi, 2019), engineering (Aslansefat et al., 2020), communication systems Amudhambigai and Neeraja (2019); von Kaenel (1982); De Gang (2000); Gereme et al. (2023a); Gereme et al. (2023b), and beyond.

Binary convex codes are binary codes generated by patterns of intersections of collection of open convex sets in some Euclidean space Carina et al. (2019), with the property that every binary code is convex realizable (Franke & Muthiah, 2018), play crucial roles in linguistic communications, and image processing (Curto et al., 2017). Since binary fuzzy codes in (Gereme et al. (2023a); Gereme et al. (2023b) are extensions of classical binary codes (Blake & Mullin, 2014), paved the way for considering binary convex fuzzy codes as extensions of binary convex codes. To study further properties like dimension and decoding algorithms of binary fuzzy codes over binary fuzzy vector space described in Gereme et al. (2024), we need to study binary fuzzy vector spaces using the redefined fuzzy subsets in Gereme et al. (2023a). Similarly, to define and study the properties of binary fuzzy convex fuzzy codes, and binary convex fuzzy linear codes, we need to study fuzzy convex sets using the definitions in (Lowen, 1980; Zadeh, 1965), and fuzzy linear subspaces using the definitions in (Abdukhalikov et al., 1994). Since the concepts of binary classical vector spaces are used for the development of error-correction codes that are also used to correct/detect errors that occur during communications through noise channels (Gross, 2016), we are sure that studying binary convex fuzzy vector spaces (or binary fuzzy vector spaces) and related properties also gives a chance to study fuzzy codes and their properties including minimum distances, dimensions, and decoding, to use the codes for error correction/detection purpose in noise communication channels.

In summary, from the discussion of this paper, we realized that when a classical binary vector space is fuzzified, it may not be a binary fuzzy vector space. The fuzzified space to be a fuzzy vector space depends on the parameters α and γ in F₂. Also, in a binary case, fuzzy vector spaces and convex fuzzy vector spaces that we defined are equivalent. We believe studying binary convex fuzzy vector space or binary convex fuzzy linear subspaces gives a chance to define and study binary convex fuzzy codes, and binary convex codes, which have useful applications in image processing.

Throughout this paper, $F_t^n$ is $\mathit{n}-$dimensional vector space over a field F_t that contains t objects (i.e. $F_t=\{0,1,2,\dots ,\mathit{t}\}$). p and q are real numbers such that $0<\mathit{p}<\frac{1}{2}$ and $\mathit{p}+\mathit{q}=1$. And 0 and 1 represents $\mathit{n}-$tupled vectors $(0,0,\dots ,0)$ and $(1,1,\dots ,1)$, respectively. Also, $\mathit{wt}(\mathit{x}+y_i)=e_i$ (for $\mathit{i}=1,2,3,\dots ,2^n$) is the weight of $\mathit{x}+y_i$, and it is the number of 1’s in a vector $\mathit{x}+y_i$, where $\mathit{x},y_i\in F_2^n$ for $\mathit{n}-$dimensional binary vector space $F_2^n$.

2. Preliminaries

Definition 2.1.

Hedayati, (2009); Halmos, (2017)

A finite dimensional vector space $F_{\mathit{t}}^n$ (or also called linear space) is an abelian group with respect to addition modulo—t over a field F_t satisfying:

1. $(\mathit{\gamma }+\mathit{\alpha })\mathit{x}=\mathit{\gamma x}+\mathit{\alpha x},\mathrm{\forall x}\in F_t^n$ and $\mathrm{\forall \gamma },\mathit{\alpha }\in F_t$

2. $\mathit{\gamma }(\mathit{x}+\mathit{y})=\mathit{\gamma x}+\mathit{\gamma y},\mathrm{\forall x},\mathit{y}\in F_t^n$ and $\mathrm{\forall \gamma }\in F_t$

3. $(\mathit{\gamma \alpha })\mathit{x}=\mathit{\gamma }(\mathit{\alpha x}),\mathrm{\forall x}\in F_t^n$ and $\mathrm{\forall \gamma },\mathit{\alpha }\in F_t$

4. $1\mathit{x}=\mathit{x},\mathrm{\forall x}\in F_t^n$ and $1\in F_t$

Definition 2.2.

Halmos, (2017)

Let $\mathcal{Q}$ be a non-empty subset of $F_t^n$ over F_t. Then Q is called a linear subspace (or subspace), if $\mathit{\alpha x}+\mathit{\gamma y}\in \mathcal{Q},\mathrm{\forall }\mathit{x},\mathit{y}\in \mathcal{Q}$ and $\mathrm{\forall }\mathit{\alpha },\mathit{\gamma }\in F_t$.

Definition 2.3.

Gross, (2016)

Let $\mathcal{B}$ be a non—empty subset of $F_t^n$ over F_t. Then $\mathcal{B}$ is called linearly independent set if $\sum _{\mathit{i}=1}^{\mathit{n}}{\alpha }_{\mathit{i}}{x}_i=\mathbf{\text{0}}$ implies ${\alpha }_{\mathit{i}}=0,\mathrm{\forall i}$.

Definition 2.4.

Gross, (2016)

Let $\mathcal{B}$ be a non-empty subset of a vector space $F_t^n$ over F_t. Then $\mathcal{B}$ is called basis of $F_t^n$ if:

1. $\mathcal{B}$ is linearly independent set for $F_t^n$

2. $\mathcal{B}$ spans $F_t^n$

Proposition 2.5.

Axler, (1997)

Let $\mathcal{B}\subseteq F_t^n$, where $F_t^n$ be a vector space. Then the set $\mathcal{B}$ is the basis of $F_t^n$ if and only if every $\mathit{y}\in F_t^n$ can be written uniquely in the form $\mathit{y}={\alpha }_1{x}_1+{\alpha }_2{x}_2+\dots +{\alpha }_{\mathit{n}}{x}_n$, where ${\alpha }_{\mathit{i}}\in F_t$ and $x_i\in \mathcal{B}$.

Proof.

See Proposition 2.8 of (Axler, 1997).

Definition 2.6.

Halmos, (2017)

If $\mathcal{B}$ is basis for a vector space $F_t^n$ over F_t, then $|\mathcal{B}|$ is called the dimension of $F_t^n$.

Definition 2.7.

Blake & Mullin, (2014)

Every non-empty subset C of $F_2^n$ is a binary block code. The vectors in C are called codewords, and the weight of a codeword is the number of $1^{\prime }\mathit{s}$ in the binary $\mathit{n}-$dimensional vector. These codes are classical codes.

Definition 2.8.

Gereme et al., (2023)

Let $\mathit{\mu }:F_2^n⟶[0,1]$ defined by ${\mu }_x(y_i)=p^{e_i}{q}^{n-e_i}$, where $e_i=\mathit{wt}(\mathit{x}+y_i)$ for $\mathit{x},y_i\in F_2^n$. Then µ is a fuzzy set over $F_2^n$.

Definition 2.9.

Yang, (1995)

Let $\mathit{\mu }:F_t^n⟶[0,1]$ be a fuzzy subset of $F_t^n$. Then µ is called convex fuzzy set of $F_t^n$ if $\mathit{\mu }(\mathit{\alpha x}+(1-\mathit{\alpha })\mathit{y})\ge \mathit{min}\{\mathit{\mu }(\mathit{x}),\mathit{\mu }(\mathit{y})\},\mathrm{\forall x},\mathit{y}\in F_t^n$ and $\mathit{\alpha }\in F_t$.

Theorem 2.10.

Ammar, (1999)

If µ is a convex fuzzy set, then either µ_α is empty or convex set for every $\mathit{\alpha }\in [0,1]$ but the converse doesnot hold in general.

Proof.

See Theorem −2 of (Ammar, 1999).

Definition 2.11.

Lubczonok, (1990)

Let $\mathit{\mu }:F_t^n⟶[0,1]$ be a fuzzy subset of $F_t^n$. Then $f_t^n=\{\mathit{\mu }|\mathit{\mu }:F_t^n⟶[0,1]\}$ is called a fuzzy vector space of $F_t^n$ if $\mathit{\mu }(\mathit{\alpha x}+\mathit{\gamma y})\ge \mathit{min}\{\mathit{\mu }(\mathit{x}),\mathit{\mu }(\mathit{y})\},\mathrm{\forall x},\mathit{y}\in F_t^n$ and $\mathit{\alpha },\mathit{\gamma }\in F_t$

Definition 2.12.

Lubczonok, (1990)

Let $\mathcal{B}$ be a non-empty subset of $F_t^n$ over F_t, and let $f_t^n$ be a fuzzy vector space of $F_t^n$. Then $\mathcal{B}$ is called a fuzzy linear independent set for $f_t^n$, if:

1. $\mathcal{B}$ is linearly independent set for $F_t^n$.

2. $\mathit{\mu }(\sum _{\mathit{i}=1}^{\mathit{n}}{\gamma }_{\mathit{i}}{x}_{\mathit{i}})=\mathit{min}\{\mathit{\mu }({\gamma }_1{x}_1),\mathit{\mu }({\gamma }_2{x}_2),\dots ,\mathit{\mu }({\gamma }_{\mathit{n}}{x}_n)\}$, where $\{{\gamma }_{\mathit{i}}:\mathit{i}=1,2,\dots ,\mathit{n}\}\subset F_t$

Definition 2.13.

Lubczonok, (1990)

Let $f_t^n=\{\mathit{\mu }|\mathit{\mu }:F_t^n⟶[0,1]\}$ be a fuzzy vector space of $F_t^n$. Then the dimension of $f_t^n$ is : $\mathit{dim}(f_t^n)=\mathit{sup}\left(\sum _{\mathit{v}\in \mathcal{B}}\mathit{\mu }(\mathit{v})\right)$, where $\mathcal{B}$ is basis for $F_t^n$.

Proposition 2.14.

Lubczonok, (1990)

Let $f_t^n=\{\mathit{\mu }|\mathit{\mu }:F_t^n⟶[0,1]\}$ be a fuzzy vector space of $F_t^n$. Then $f_t^n$ has fuzzy basis.

Proof.

See Proposition 5.2 of (Lubczonok, 1990).

Proposition 2.15.

Lubczonok, (1990)

Let $f_t^n=\{\mathit{\mu }|\mathit{\mu }:F_t^n⟶[0,1]\}$ be a fuzzy vector space of $F_t^n$. If $\mathcal{B}$ is a fuzzy basis for $f_t^n$ and ${\mathcal{B}}^{\mathcal{\ast }}$ is any basis for $F_t^n$, then $\sum _{v\in {\mathcal{B}}^{\mathcal{\ast }}}\mathit{\mu }(\mathit{v})\le \sum _{\mathit{v}\in \mathcal{B}}\mathit{\mu }(\mathit{v})$.

Proof.

See Proposition 5.3 of (Lubczonok, 1990).

Definition 2.16.

Kumar, (1992)

Let $\mathcal{B}$ be a non-empty subset of a vector space $F_t^n$ over a field F_t, and let $f_t^n$ be a fuzzy vector space of a vector space $F_t^n$. Then $\mathcal{B}$ is called a fuzzy basis for $f_t^n$, if:

1. $\mathcal{B}$ is basis for $F_t^n$.

2. $\mathit{\mu }(\sum _{\mathit{i}=1}^{\mathit{n}}{\gamma }_{\mathit{i}}{x}_{\mathit{i}})=\mathit{min}\{\mathit{\mu }({\gamma }_1{x}_1),\mathit{\mu }({\gamma }_2{x}_2),\dots ,\mathit{\mu }({\gamma }_{\mathit{n}}{x}_n)\}$, where $\{{\gamma }_{\mathit{i}}:\mathit{i}=1,2,\dots ,\mathit{n}\}\subset F_t$.

Remark 1.

1. If $|\mathcal{B}|$ is a dimension of a fuzzy vector space $f_t^n$, then $|\mathcal{B}|$ also the dimension of $F_t^n$ (Shi & Huang, 2010).

2. If $\mathcal{B}$ and ${\mathbf{B}}^{\ast }$ are two fuzzy basis of a fuzzy vector space $f_t^n$, then $|\mathcal{B}|=|{\mathcal{B}}^{\mathcal{\ast }}|$ (Kumar, 1992).

Definition 2.17.

Nanda, (1991)

Let $F_t^n$ be $\mathit{n}-$dimensional linear space of a field F_t. Then the mapping $\mathit{\mu }:F_t^n⟶[0,1]$ is called a fuzzy linear subspace of $F_t^n$ if:

1. $\mathit{\mu }(\mathit{x}+\mathit{y})\ge \mathit{min}\{\mathit{\mu }(\mathit{x}),\mathit{\mu }(\mathit{y})\},\mathrm{\forall x},\mathit{y}\in F_t^n$

2. $\mathit{\mu }(\mathit{\gamma x})\ge \mathit{min}\{\mathit{\mu }(\mathit{\gamma }),\mathit{\mu }(\mathit{x})\},\mathrm{\forall \gamma }\in F_t$ and $\mathrm{\forall x}\in F_t^n$

Definition 2.18.

Zadeh, (1965)

Let $\mathit{\mu }:F_2^n⟶[0,1]$ be a fuzzy set. Then µ is a convex fuzzy set if and only if the set µ_t defined by ${\mu }_t=\{\mathit{x}|\mathit{\mu }(\mathit{x})\ge \mathit{t},\mathit{x}\in F_2^n\}$ is convex $\mathrm{\forall t}\in$ $(0,1]$.

Definition 2.19.

Brown, (1971)

The $\mathit{t}-$level sets (or $\mathit{t}-$cuts) of a fuzzy set $\mathit{\mu }:F_2^n⟶[0,1]$ is defined as

${\mu }_t=\{\mathit{x}\in F_2^n|\mathit{\mu }(\mathit{x})\ge \mathit{t}\}$, where $\mathit{t}\in (0,1]$.

Remark 2.

Zadeh, (1965)

Definition 2.18 is equivalent with:

A fuzzy set $\mathit{\mu }:F_2^n⟶[0,1]$ is convex fuzzy set if and only if $\mathit{\mu }(\mathit{\alpha x}+(1-\mathit{\alpha })\mathit{y})\ge \mathit{min}\{\mathit{\mu }(\mathit{x}),\mathit{\mu }(\mathit{y})\}$ for all $\mathit{x},\mathit{y}\in F_2^n$ and $\mathrm{\forall \alpha }\in [0,1]$.

Proposition 2.20.

Yuan and Lee, (2004)

$\mathit{\mu }:F_2^n⟶[0,1]$ is Zadeh’s convex fuzzy subset of $F_2^n$ if and only if ${\mu }_t=\{\mathit{x}\in F_2^n|\mathit{\mu }(\mathit{x})\ge \mathit{t}\}$ is convex subset of $F_2^n$ for any $\mathit{t}\in [0,1]$, where the empty set is seen as a convex subset.

We can combine Definition 2.18 and Definition 2.19 to have the following lemma.

Lemma 2.21.

Let $\mathit{\mu }:F_2^n⟶[0,1]$ defined by ${\mu }_x(y_i)=\mathit{t}$, where $\mathit{t}=p^{e_i}{q}^{n-e_i}$, $e_i=\mathit{wt}(\mathit{x}+y_i)$ and $\mathit{x},y_i\in F_2^n$. Then µ is a fuzzy convex set.

Proof.

Since $\mathit{t}=p^{e_i}{q}^{n-e_i}$ and $\mathit{p}\in (0,\frac{1}{2})$ such that $\mathit{p}+\mathit{q}=1$, $\mathit{t}\in (0,1)$.

Now, by Definition 2.18, $({\mu }_x{)}_t=\{y_i\in F_2^n|{\mu }_x(y_i)\ge \mathit{t}\}$ and it is a binary code over a vector space $F_2^n$. This implies $({\mu }_x{)}_t$ is a convex set.

Therefore, µ is a fuzzy convex set, because of Definition 2.19 and Definition 2.18.

3. Binary fuzzy vector spaces of $F_2^n$ over a Galois field F₂

3.1. Binary fuzzy vector spaces

Proposition 3.1.

Let $F_2^n$ be a binary vector space over a Galois field F₂ and $\mathit{\mu }:F_2^n\to [0,1]$ defined by ${\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}$, where $e_i=\mathit{wt}(\mathit{x}+y_i)$ and $\mathit{x},y_i\in F_2^n$. Then $f_2^n=\{{\mathit{\mu }}_x|{\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}\}$ is a binary fuzzy vector space whenever $\mathit{\alpha },\mathit{\gamma }\in F_2\mathit{such}\mathit{that}\mathit{\gamma }=1-\mathit{\alpha }$.

Proof.

Let $\mathit{x},y_1,y_2\in F_2^n$ be arbitrary.

Suppose : $\mathit{\alpha },\mathit{\gamma }\in F_2$ such that $\mathit{\gamma }=1-\mathit{\alpha }$.

Claim:$f_2^n=\{{\mathit{\mu }}_x|{\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}\}$ is a fuzzy vector space.

Case − 1 For α = 0

Since $\mathit{\gamma }=1-\mathit{\alpha },\mathit{\gamma }=1-0=1$. Then ${\mathit{\mu }}_x(\mathit{\alpha }{y}_1+\mathit{\gamma }{y}_2)={\mathit{\mu }}_x(\mathbf{0}+y_2)={\mathit{\mu }}_x\left(y_2\right)\ge \mathit{min}\{{\mathit{\mu }}_x(y_1),{\mathit{\mu }}_x(y_2)\},\text{allowbreak}\mathrm{\forall }\mathit{x},y_1,y_2\in F_2^n$.

Therefore, the definition of fuzzy vector space holds (see Definition 2.11).

Case − 2 For α = 1

Since $\mathit{\gamma }=1-\mathit{\alpha },\mathit{\gamma }=1-1=0$. Then ${\mathit{\mu }}_x(\mathit{\alpha }{y}_1+\mathit{\gamma }{y}_2)={\mathit{\mu }}_x(y_1+\mathbf{0})={\mathit{\mu }}_x\left(y_1\right)\ge \mathit{min}\{{\mathit{\mu }}_x(y_1),{\mathit{\mu }}_x(y_2)\},\mathrm{\forall x},y_1,y_2\in F_2^n.$

Therefore, the definition of fuzzy vector space holds (see Definition 2.11).

Hence, $f_2^n$ is a binary fuzzy vector space.

Theorem 3.2

Let $f_2^n$ be a binary fuzzy vector space. Then ${\mathit{\mu }}_x\left(y_1\right)\le {\mathit{\mu }}_x\left(y_2\right)$ if and only if $e_1\ge e_2$, where $e_1=\mathit{wt}(\mathit{x}+y_1)$ and $e_2=\mathit{wt}(\mathit{x}+y_2)$.

Proof.

The proof is similar with the proof of Theorem 2.1 of Gereme et al., (2023).

Corollary 3.3.

Let $f_2^n$ be a binary fuzzy vector space and let $e_1=\mathit{wt}(\mathit{x}+y_1),\mathrm{\forall x}\in F_2^n$. If $e_1=\mathit{n}$, then ${\mu }_x(y_1)\le {\mu }_x(y_i),\mathrm{\forall }{y}_i\in F_2^n$.

Corollary 3.4.

Let $f_2^n$ be a binary fuzzy vector space and let ${\mu }_x(y_1),{\mu }_x(y_2)\in f_2^n$ be arbitrary such that $y_1\ne y_2$. If $\mathit{x}=y_2$, then ${\mu }_x(y_1)<{\mu }_x(y_2)$.

Theorem 3.5

Let $f_2^n$ be a binary fuzzy vector space. Then ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}={\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$ if and only if $\mathrm{\exists y}\notin F_2^n$ such that $\mathit{t}\le {\mathit{\mu }}_x(\mathit{y})<\mathit{s}$.

Proof.

Let ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}},{\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$ with t < s be any two level sets of a binary fuzzy set µ with t < s.

$(⟹)$ Suppose ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}={\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$.

Claim: $\mathrm{\exists y}\notin F_2^n$ such that $\mathit{t}\le {\mathit{\mu }}_x(\mathit{y})<\mathit{s}$.

Assume there exists $\mathit{y}\in F_2^n$ such that $\mathit{s}\le {\mathit{\mu }}_x\left(\mathit{y}\right)<\mathit{t}$. Then $\mathit{y}\in {\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$ and $\mathit{y}\notin {\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$. Implies ${\left({\mathit{\mu }}_x\right)}_{\mathit{s}}\ne \mathit{y}\in {\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$, which contradicts with the supposition.

Hence, there is no $\mathit{y}\in F_2^n$ such that $\mathit{s}\le {\mathit{\mu }}_x\left(\mathit{y}\right)<\mathit{t}$.

($⟸)$ Conversely, suppose $\mathrm{\exists y}\notin F_2^n$ such that $\mathit{s}\le {\mathit{\mu }}_x\left(\mathit{y}\right)<\mathit{t}$.

Claim: ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}={\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$

From the supposition s < t. Then

(1) ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}\subseteq {\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$(1)

And also, let $\mathit{y}\in {\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$. Then ${\mathit{\mu }}_x(\mathit{y})\ge \mathit{s}$.

By the assumption there is no $\mathit{y}\in F_2^n$ such that ${\mathit{\mu }}_x\left(\mathit{y}\right)<\mathit{t}$. Then ${\mathit{\mu }}_x(\mathit{y})\ge \mathit{t}$. This implies $\mathit{y}\in {\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$. Therefore,

(2) ${\left({\mathit{\mu }}_x\right)}_{\mathit{s}}\subseteq {\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$(2)

From (1(1) ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}\subseteq {\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$(1) ) and (2(2) ${\left({\mathit{\mu }}_x\right)}_{\mathit{s}}\subseteq {\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$(2) ), we have ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}={\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$.

Proposition 3.6.

Let a non-empty subset $\mathcal{B}$ of $F_2^n$, and let µ be a fuzzy subset of $F_2^n$. If $\mathcal{B}$ is linearly independent set of $F_2^n$, then $\mathcal{B}$ is a fuzzy linearly independent set of a fuzzy vector space $f_2^n$.

Proof.

Let µ be a fuzzy subset of a binary vector space $F_2^n$

Let $f_2^n$ be a binary fuzzy vector space of $F_2^n$.

Suppose : $\mathrm{\varnothing }\ne \mathcal{B}\subseteq F_2^n$ is linearly independent set of $F_2^n$.

Claim: $\mathcal{B}$ is a fuzzy linearly independent set of a binary fuzzy vector space $f_2^n$.

Since $\mathcal{B}$ is linearly independent of $F_2^n$, it is enough to show ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)=\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_2\right),{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\dots ,{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}$.

Now, since ${\alpha }_i=0,\mathrm{\forall i}$, then ${\gamma }_i{y}_i=\mathbf{0},\mathrm{\forall }\mathit{i}$. Implies $\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i=\mathbf{0}$. Which gives ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)={\mathit{\mu }}_x\left(\mathbf{0}\right)$

And also, for each i, ${\mathit{\mu }}_x\left({\gamma }_i{y}_i\right)={\mathit{\mu }}_x\left(\mathbf{0}\right)$, which implies $\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_1\right),{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\dots ,{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}={\mu }_x(\mathbf{0})$

Therefore, ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)={\mathit{\mu }}_x\left(\mathbf{0}\right)=\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_1\right),\text{allowbreak}{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\dots ,{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}$.

Hence, ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)=\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_1\right),{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\text{allowbreak}\dots ,{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}$

Implies, $\mathcal{B}$ is fuzzy linearly independent set for a binary fuzzy vector space $f_2^n$.

Proposition 3.7.

Let a non-empty subset $\mathcal{B}$ of $F_2^n$, and let µ be a fuzzy subset of $F_2^n$. If $\mathcal{B}$ is a basis of $F_2^n$, then $\mathcal{B}$ is fuzzy basis of a fuzzy vector space $f_2^n$.

Proof.

Let µ be a fuzzy subset of a binary vector space $F_2^n$.

Let $f_2^n$ be a binary fuzzy vector space of $F_2^n$.

Suppose : $\mathrm{\varnothing }\ne \mathcal{B}\subseteq F_2^n$ is basis of $F_2^n$.

Claim: $\mathcal{B}$ is basis of a binary fuzzy vector space $f_2^n$.

Since $\mathcal{B}$ is basis of $F_2^n$, it is enough to show ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)=\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_2\right),{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\dots ,\text{allowbreak}{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}$.

Now, since $\mathcal{B}$ is basis of $F_2^n$, $\mathcal{B}$ is linearly independent set of $F_2^n$. Then ${\gamma }_i{y}_i=\mathbf{0},\mathrm{\forall }\mathit{i}$. Implies $\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i=\mathbf{0}$. Which gives ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)={\mathit{\mu }}_x\left(\mathbf{0}\right)$

And also, for each i, ${\mathit{\mu }}_x\left({\gamma }_i{y}_i\right)={\mathit{\mu }}_x\left(\mathbf{0}\right)$, which implies $\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_1\right),{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\dots ,{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}={\mu }_x(\mathbf{0})$.

Therefore, ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)={\mathit{\mu }}_x\left(\mathbf{0}\right)=\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_1\right),\text{allowbreak}{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\dots ,{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}$.

Hence, ${\mathit{\mu }}_x\left(\sum _{\mathit{i}=1}^n{\gamma }_i{y}_i\right)=\mathit{min}\{{\mathit{\mu }}_x\left({\gamma }_1{y}_1\right),{\mathit{\mu }}_x\left({\gamma }_2{y}_2\right),\dots ,\text{allowbreak}{\mathit{\mu }}_x\left({\gamma }_n{y}_n\right)\}$

Therefore, $\mathcal{B}$ is fuzzy basis for a binary fuzzy vector space $f_2^n$.

Proposition 3.8.

For a binary fuzzy vector space $f_2^n=\{{\mathit{\mu }}_x|{\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}\}$. Then $f_2^n$ has fuzzy basis and one of the basis is $\mathcal{B}=\{(1,0,0,\dots ,0),\text{allowbreak}(0,1,0,0,\dots ,0),\dots ,(0,0,\dots ,1)\}$ whenever $\mathcal{B}$ is fuzzy linearly independent set.

Proof.

Let $f_2^n=\{{\mathit{\mu }}_x|{\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}\}$ be a fuzzy vector space over $F_2^n$.

Let $\mathcal{B}$ is basis of $F_2^n$.

Suppose: $\mathcal{B}$ is fuzzy linearly independent set of $f_2^n$.

This implies $\mathcal{B}$ is a basis for $F_2^n$.

Since $f_2^n$ is constructed from the finite binary vector space $F_2^n$ and $f_2^n$ is a fuzzy vector space over $F_2^n$, $f_2^n$ is finite dimensional. Then $f_2^n$ has fuzzy basis, this is true because of Proposition 2.14.

Since $\mathcal{B}$ is basis for $F_2^n$ and $\mathcal{B}$ is fuzzy linearly independent set of $f_2^n$, $\mathcal{B}$ is fuzzy basis of $f_2^n$.

Since $\mathcal{B}=\{(1,0,0,\dots ,0),(0,1,0,0,\dots ,0),\dots ,(0,0,\dots ,\text{allowbreak}1)\}$ is basis for $F_2^n$ and $\mathcal{B}$ is fuzzy linearly independent set of $f_2^n$, then $\mathcal{B}=\{(1,0,0,\dots ,0),\text{allowbreak}(0,1,0,0,\dots ,0),\dots ,(0,0,\dots ,1)\}$ is one of the fuzzy basis for $f_2^n$.

Proposition 3.9.

Let µ be a fuzzy subset of $F_2^n$, and let $f_2^n$ be a fuzzy vector space over $F_2^n$. If a non-empty subset $\mathcal{B}$ of $F_2^n$ is the basis for $F_2^n$, then $\mathcal{B}$ is a fuzzy basis of $f_2^n$.

Proof.

Let µ be a fuzzy subset of a binary vector space $F_2^n$.

Let $f_2^n$ be a binary fuzzy vector space of $F_2^n$.

Suppose: $\mathrm{\varnothing }\ne \mathcal{B}\subseteq F_2^n$ be the basis of $F_2^n$.

Claim: $\mathcal{B}$ is basis of a binary fuzzy vector space $f_2^n$.

Since $\mathcal{B}$ is basis of $F_2^n$, it is enough to show $\mathcal{B}$ is linearly independent set over $f_2^n$.

Since $\mathcal{B}$ is basis over $F_2^n$, $\mathcal{B}$ is linearly independent over $F_2^n$. Which implies $\mathcal{B}$ is a fuzzy linearly independent set of $f_2^n$, by Proposition 3.6.

Hence, $\mathcal{B}$ is fuzzy basis for a binary fuzzy vector space $f_2^n$.

Definition 3.10.

For a binary fuzzy vector space $f_2^n$, $\mathit{dim}(f_2^n)=\sum _{\mathit{v}\in \mathcal{B}}(\mathit{\mu }(\mathit{v}))$, where $\mathcal{B}$ is a fuzzy basis of $f_2^n$.

Lemma 3.11.

Let $\mathcal{B}$ be a fuzzy basis for a fuzzy vector space $f_2^n$ over $F_2^n$. Then $\sum _{{\mathit{x}}_{\mathit{i}}\in F_2^n}{\mu }_{x_i}(\mathit{y})=1$ for each $\mathit{y}\in \mathcal{B}$.

Proof.

Let $\mathit{y}\in \mathcal{B}$ be arbitrary and $x_i\in F_2^n$. Then there are 2ⁿ possibilities for x_i. From these possible vectors of x_i, one possible vector has weight 0, one vector also has weight n, $(\genfrac{}{}{0pt}{}{n}{1})$ vectors have weight 1, $(\genfrac{}{}{0pt}{}{n}{2})$ vectors have weight 2, and so on.

The fuzzy subset µ is defined as ${\mu }_{x_i}(\mathit{y})=p^{e_i}{q}^{n-e_i}$ for $\mathit{p}+\mathit{q}=1$, $\mathit{p}\in (0,\frac{1}{2})$, and $e_i=\mathit{wt}(x_i+\mathit{y})$. The weight of $x_i+\mathit{y}$ runs from 0 up to n. There is one possibility of $e_i=0$, there is also one possibility in which $e_i=\mathit{n}$, $(\genfrac{}{}{0pt}{}{n}{1})$ possibilities in which $e_i=1$, $(\genfrac{}{}{0pt}{}{n}{2})$ possibilities in which $e_i=2$ and so on.

Therefore $\sum _{{\mathit{x}}_{\mathit{i}}\in F_2^n}{\mu }_{x_i}(\mathit{y})=(\genfrac{}{}{0pt}{}{n}{0})p^0{q}^{\mathit{n}-0}+(\genfrac{}{}{0pt}{}{n}{1})p^1{q}^{\mathit{n}-1}+(\genfrac{}{}{0pt}{}{n}{2})p^2{q}^{\mathit{n}-2}+\dots +(\genfrac{}{}{0pt}{}{n}{n})p^{\mathit{n}}{q}^{\mathit{n}-\mathit{n}}$. This implies

$\sum _{{\mathit{x}}_{\mathit{i}}\in F_2^n}{\mu }_{x_i}(\mathit{y})=\sum _{\mathit{k}=0}^{\mathit{n}}(\genfrac{}{}{0pt}{}{n}{k})p^k{q}^{\mathit{n}-\mathit{k}}=(\mathit{p}+\mathit{q}{)}^n=1$

Lemma 3.12.

Let $\mathcal{B}$ be a fuzzy base for a fuzzy vector space $f_2^n$ over $F_2^n$. Then $\mathcal{B}$ has n members.

Proof.

Since $\mathcal{B}$ is a fuzzy base for $f_2^n$, $\mathcal{B}$ is also a base for $F_2^n$. In the classical case, we know that $\mathit{dim}(F_2^n)=\mathit{n}$ and it is equivalent with the number of elements of $\mathcal{B}$.

Therefore, $\mathit{n}(\mathcal{B})=\mathit{n}$.

Proposition 3.13.

Let $f_2^n$ be a binary fuzzy vector space of $F_2^n$ with fuzzy basis $\mathcal{B}$. Then $\mathit{dim}(f_2^n)=\mathit{n}$.

Proof.

Let $\mathcal{B}$ be the fuzzy basis for a binary fuzzy vector space $f_2^n$. By Lemma 3.12, there are n possible elements for $\mathcal{B}$.

By Definition 3.10, $\mathit{dim}(f_2^n)=\sum _{{\mathit{y}}_{\mathit{i}}\in \mathcal{B}}({\mu }_x(y_i))$ for all $\mathit{x}\in F_2^n$, where $\mathcal{B}$ is a fuzzy basis of $f_2^n$.

Using Lemma 3.11 and Lemma 3.12, $\mathit{dim}(f_2^n)=\sum _{{\mathit{y}}_{\mathit{i}}\in \mathcal{B}}({\mu }_x(y_i))=\mathit{n}\ast 1=\mathit{n}$.

Proposition 3.14.

For a binary fuzzy vector space $f_2^n=\{{\mathit{\mu }}_x|{\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}\}$. If $\mathcal{B}$ is a fuzzy basis for $f_2^n$, then it is a basis for $F_2^n$ and $\mathit{dim}(f_2^n)=\mathit{dim}(F_2^n)$.

Proof.

The prove is the direct outcome of Proposition 3.8 and Definition 3.10.

3.2. Binary convex fuzzy vector spaces

Proposition 3.15.

Let $\mathit{\mu }:F_2^n⟶[0,1]$ defined by ${\mu }_x(y_i)=p^{e_i}{q}^{n-e_i}$, where $e_i=\mathit{wt}(\mathit{x}+y_i)$ for $\mathit{x},y_i\in F_2^n$. µ is a convex fuzzy set if and only if ${\mu }_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)\ge \mathit{min}\{{\mu }_x(y_1),{\mu }_x(y_2)\}$ for $\mathit{\gamma }\in F_2$.

Proof.

Let ${\mu }_x(y_1)=p^{e_1}{q}^{n-e_1}=t_1$ and ${\mu }_x(y_2)=p^{e_2}{q}^{n-e_2}=t_2$, where $e_1=\mathit{wt}(\mathit{x}+y_1)$ and $e_2=\mathit{wt}(\mathit{x}+y_2)$.

Let $\mathit{t}=\mathit{min}\{t_1,t_2\}$.

$(\Rightarrow )$ Suppose: µ is a convex fuzzy set.

Claim: ${\mu }_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)\ge \mathit{min}\{{\mu }_x(y_1),{\mu }_x(y_2)\}$ for $\mathit{\gamma }\in F_2$.

Since µ is convex fuzzy set, $({\mu }_x{)}_t$ is a convex set.

Let$y_1,y_2\in ({\mu }_x{)}_{\mathit{t}}$. Then $\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2\in ({\mu }_x{)}_t$. Which implies ${\mu }_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)\ge \mathit{t}=\mathit{min}\{t_1,t_2\}=\mathit{min}\{{\mu }_x(y_1),{\mu }_x(y_2)\}$.

$(\Leftarrow )$ Conversely suppose: ${\mu }_x(\mathit{\alpha y}+(1-\mathit{\alpha })\mathit{z})\ge \mathit{min}\{{\mu }_x(\mathit{y}),{\mu }_x(\mathit{z})\}$ for $\mathit{\alpha }\in F_2^n$.

Claim: µ is convex fuzzy set.

Case–1: For γ = 0

Now, ${\mu }_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)={\mu }_x(y_2)=t_2\ge \mathit{min}\{t_1,t_2\}=\mathit{min}\{{\mu }_x(y_1),{\mu }_x(y_2)\}=\mathit{t}$. Which implies $\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2\in ({\mu }_x{)}_t$. Implies $({\mu }_x{)}_t$ is convex set.

Therefore, µ is convex fuzzy set, since $({\mu }_x{)}_t$ is convex set for $\mathit{t}\in (0,1)$.

Case–2: For γ = 1

Now, ${\mu }_x(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})={\mu }_x(y_1)=t_1\ge \mathit{min}\{t_1,t_2\}=\mathit{min}\{{\mu }_x(y_1),{\mu }_x(y_2)\}=\mathit{t}$. Which implies $\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2\in ({\mu }_x{)}_t$. Implies $({\mu }_x{)}_t$ is convex set.

Therefore, µ is convex fuzzy set, since $({\mu }_x{)}_t$ is convex set for $\mathit{t}\in (0,1)$

Hence, µ is a convex fuzzy set.

Lemma 3.16.

Let $\mathit{\mu }:F_2^n⟶[0,1]$ defined by ${\mu }_x(y_i)=p^{e_i}{q}^{n-e_i}$. Then $({\mu }_x{)}_{t_1}\subseteq ({\mu }_x{)}_{t_2}$ whenever $t_1\ge t_2$.

Proof.

Suppose: $t_1\ge t_2$.

Let $\mathit{z}\in ({\mu }_x{)}_{t_1}$. Then ${\mu }_x(\mathit{z})\ge t_1\ge t_2$. Which implies ${\mu }_x(\mathit{z})\ge t_2$. Implies $\mathit{z}\in ({\mu }_x{)}_{t_2}$.

Therefore, $({\mu }_x{)}_{t_1}\subseteq ({\mu }_x{)}_{t_2}$.

Proposition 3.17.

Let $\mathit{\mu }:F_2^n⟶[0,1]$ defined by ${\mu }_x(y_i)=p^{e_i}{q}^{n-e_i}$. Then $({\mu }_x{)}_t=F_2^n$ whenever $\mathit{x}+y_i=$1 for $\mathit{t}=p^{e_i}{q}^{n-e_i}$ and fixed x.

Proof.

Suppose: $\mathit{x}+y_i=$1 for $\mathit{t}=p^{e_i}{q}^{n-e_i}$, where $\mathit{x},y_i\in F_2^n$ and 1 is all 1’s vector of length n.

Since $\mathit{t}=p^{e_i}{q}^{n-e_i}$, $\mathit{x}+y_i=\mathbf{\text{1}}$ and $e_i=\mathit{wt}(\mathit{x}+y_i)$, then $e_i=\mathit{n}$. Implies $\mathit{t}=p^{e_i}=p^n$. Which implies $({\mu }_x{)}_t=\{y_i\in F_2^n|{\mu }_x(y_i)\ge \mathit{t}\}$.

We left to show there is no other $p^{e_i}{q}^{n-e_i}$ which is less than t.

For $e_i=0$. Then $p^{e_i}{q}^{n-e_i}=q^n>p^n$, since p < q.

For $\mathit{x}\ne y_i$.

Let $t_1=p^{e_i}{q}^{n-e_i}$.

Since $0<e_i\le \mathit{n}$ for $\mathit{x}\ne y_i$, $\mathit{t}<t_1$. This is true because of Theorem 3.2.

Therefore, $({\mu }_x{)}_{t_1}=\{y_i\in F_2^n|{\mu }_x(y_i)\ge t_1\}$. Which gives $({\mu }_x{)}_{t_1}\subseteq ({\mu }_x{)}_{\mathit{t}}$, because of Lemma 3.16.

Since $0\le e_i\le \mathit{n}$, there is no $p^{e_i}{q}^{n-e_i}$ that is less than t.

Hence, $({\mu }_x{)}_t=F_2^n$.

Proposition 3.18.

Let $\mathit{\mu }:F_2^n⟶[0,1]$ defined by ${\mu }_x(y_i)=p^{e_i}{q}^{n-e_i}$, where $\mathit{x},y_i\in F_2^n$, and $e_i=\mathit{wt}(\mathit{x}+y_i)$. Then $({\mu }_x{)}_t$ contains a single element z whenever x = z for $\mathit{t}=p^{e_i}{q}^{n-e_i}$, and fixed x.

Proof.

Suppose: x = z for $\mathit{t}=p^{e_i}{q}^{n-e_i}$, where $\mathit{x},\mathit{z}\in F_2^n$.

Since $\mathit{t}=p^{e_i}{q}^{n-e_i}$, x = z for $y_i=\mathit{z}$ and $e_i=\mathit{wt}(\mathit{x}+\mathit{z})$, $e_i=0$. Implies $\mathit{t}=q^{\mathit{n}}$. Which implies $({\mu }_x{)}_t=\{y_i\in F_2^n|{\mu }_x(y_i)\ge \mathit{t}\}$.

We left to show there is no other $y_i\in F_2^n$ other than z which gives qⁿ and qⁿ is the greatest value. Therefore,

1. Since $F_2^n$ has 2ⁿ distinct elements, for a fixed $\mathit{x}\in F_2^n$, we have only a single $y_i\in F_2^n$ say z such that x = z.

2. For $\mathit{x}\ne y_i$, $e_i\ne 0$ and $p^{e_i}{q}^{n-e_i}=(\frac{p}{q}{)}^{e_i}{q}^n<q^n$, since $\frac{p}{q}<1$.

Hence, $({\mu }_x{)}_t=\{\mathit{z}\}$, where $\mathit{z}\in F_2^n$.

Proposition 3.19.

A finite intersection of binary convex fuzzy sets is again binary convex.

Proof.

Let $F_2^n$ be a binary vector space over a Galois field F₂ and $\mathit{\mu }:F_2^n\to [0,1]$ defined by ${\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}$, where $e_i=\mathit{wt}(\mathit{x}+y_i)$ and $\mathit{x},y_i\in F_2^n$.

Let $\mathit{V}=\bigcap _{\mathit{j}=1}^n{\mathit{\mu }}_j$, where${\mathit{\mu }}_j$ is a binary convex fuzzy set for each j.

Claim: V is a binary convex fuzzy set.

Let $y_1,y_2\in F_2^n$ be arbitrary. Then

${({\mathit{\mu }}_j)}_{\mathit{x}}(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)\ge \mathit{min}\{{({\mathit{\mu }}_j)}_{\mathit{x}}\left(y_1\right),\text{allowbreak}{({\mathit{\mu }}_j)}_{\mathit{x}}\left(y_2\right)\}$ for $\mathit{\gamma }\in F_2$ and $y_1,y_2\in F_2^n$, since ${\mathit{\mu }}_j$ is a binary convex fuzzy set for each j.

And also, $\mathit{V}=\mathit{min}\{{\mathit{\mu }}_j:\mathit{j}=1,2,\dots ,\mathit{n}\}$. Then $V_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)=\mathit{min}{\left({\mathit{\mu }}_j\right)}_{\mathit{x}}(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)\ge$ $\mathit{min}\{\mathit{min}\{{({\mathit{\mu }}_j)}_{\mathit{x}}\left(y_1\right),{({\mathit{\mu }}_j)}_{\mathit{x}}\left(y_2\right)\}\}=\mathit{min}\{{\left({\mathit{\mu }}_j\right)}_{\mathit{x}}\left(y_1\right),\text{allowbreak}{\left({\mathit{\mu }}_j\right)}_{\mathit{x}}\left(y_2\right)\}$

Therefore, V is a binary convex fuzzy set.

Definition 3.20.

A binary fuzzy vector space $f_2^n$ in which every elements are binary convex fuzzy sets $\mathit{\mu }:F_2^n\to [0,1]$ defined by ${\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}$ is called binary convex fuzzy vector space, where $e_i=\mathit{wt}(\mathit{x}+y_i)$, and $\mathit{x},y_i\in F_2^n$.

Proposition 3.21.

$f_2^n$ is binary fuzzy vector vector spaces if and only if $f_2^n$ is binary convex fuzzy vector space.

Proof.

The prove is the direct outcome of Definition 3.20 and Proposition 3.1.

Remark 3.

The theorems, propositions and colloraries stated for binary fuzzy vector space also works for binary convex fuzzy vector space. We stated the theorems, propositions, corollaries and lemmas below without proof (since the proofs are similar with proves that we follow in subsection 3.1).

Theorem 3.22

Let $f_2^n$ be a binary convex fuzzy vector space. Then ${\mathit{\mu }}_x\left(y_1\right)\le {\mathit{\mu }}_x\left(y_2\right)$ if and only if $e_1\ge e_2$, where $e_1=\mathit{wt}(\mathit{x}+y_1)$ and $e_2=\mathit{wt}(\mathit{x}+y_2)$.

Proof.

The proof is similar with the proof of Theorem 2.1 of Gereme et al., (2023).

Corollary 3.23.

Let $f_2^n$ be a binary convex fuzzy vector space and let $e_1=\mathit{wt}(\mathit{x}+y_1),\mathrm{\forall x}\in F_2^n$. If $e_1=\mathit{n}$, then ${\mu }_x(y_1)\le {\mu }_x(y_i),\mathrm{\forall }{y}_i\in F_2^n$.

Corollary 3.24.

Let $f_2^n$ be a binaryconvex fuzzy vector space, and let ${\mu }_x(y_1),{\mu }_x(y_2)\in f_2^n$ be arbitrary such that $y_1\ne y_2$. If $\mathit{x}=y_2$, then ${\mu }_x(y_1)<{\mu }_x(y_2)$.

Theorem 3.25

Let $f_2^n$ be a binary fuzzy vector space. Then ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}={\left({\mathit{\mu }}_x\right)}_{\mathit{s}}$ if and only if $\mathrm{\exists y}\notin F_2^n$ such that $\mathit{t}\le {\mathit{\mu }}_x(\mathit{y})<\mathit{s}$.

Proof.

The proof is similar with the proof of Theorem 3.5.

Proposition 3.26.

Let a non-empty subset $\mathcal{B}$ of $F_2^n$, and let µ be a fuzzy subset of $F_2^n$. If $\mathcal{B}$ is linearly independent set of $F_2^n$, then $\mathcal{B}$ is a fuzzy linearly independent set of a binary convex fuzzy vector space $f_2^n$.

Proof.

The proof is similar with the proof of Theorem 3.6.

Proposition 3.27.

Let a non-empty subset $\mathcal{B}$ of $F_2^n$, and let µ be a fuzzy subset of $F_2^n$. If $\mathcal{B}$ is a basis of $F_2^n$, then $\mathcal{B}$ is fuzzy basis of binary convex fuzzy vector space $f_2^n$.

Proof.

The proof is similar with the proof of Theorem 3.7.

Proposition 3.28.

For a binary convex fuzzy vector space $f_2^n=\{{\mathit{\mu }}_x|{\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}\}$. Then $f_2^n$ has fuzzy basis and one of the basis is $\mathcal{B}=\{(1,0,0,\dots ,0),(0,1,0,0,\dots ,0),\dots ,(0,0,\dots ,1)\}$ whenever $\mathcal{B}$ is fuzzy linearly independent set.

Proof.

The proof is similar with the proof of Theorem 3.8.

Proposition 3.29.

Let µ be a fuzzy subset of $F_2^n$, and let $f_2^n$ be a binary convex fuzzy vector space over $F_2^n$. If a non-empty subset $\mathcal{B}$ of $F_2^n$ is the basis for $F_2^n$, then $\mathcal{B}$ is a fuzzy basis of a binary convex fuzzy vector space $f_2^n$.

Proof.

The proof is similar with the proof of Theorem 3.9.

Lemma 3.30.

Let $\mathcal{B}$ be a fuzzy basis for a binary convex fuzzy vector space $f_2^n$ over $F_2^n$. Then $\sum _{{\mathit{x}}_{\mathit{i}}\in F_2^n}{\mu }_{x_i}(\mathit{y})=1$ for each $\mathit{y}\in \mathcal{B}$.

Proof.

The proof is similar with the proof of Theorem 3.11.

Lemma 3.31.

Let $\mathcal{B}$ be a fuzzy base for a binary convex fuzzy vector space $f_2^n$ over $F_2^n$. Then $\mathcal{B}$ has n members.

Proof.

The proof is similar with the proof of Theorem 3.12.

Proposition 3.32.

Let $f_2^n$ be a binary convex fuzzy vector space of $F_2^n$ with fuzzy basis $\mathcal{B}$. Then $\mathit{dim}(f_2^n)=\mathit{n}$.

Proof.

The proof is similar with the proof of Theorem 3.13.

Proposition 3.33.

For a binary convex fuzzy vector space $f_2^n=\{{\mathit{\mu }}_x|{\mathit{\mu }}_x(y_i)=p^{e_i}{q}^{n-e_i}\}$. If $\mathcal{B}$ is a fuzzy basis for $f_2^n$, then it is a basis for $F_2^n$ and $\mathit{dim}(f_2^n)=\mathit{dim}(F_2^n)$.

Lemma 3.34.

Let $f_2^n$ be a binary convex fuzzy vector space, and If ${\mathit{\mu }}_x\left(y_1\right)>{\mathit{\mu }}_x\left(y_2\right)$, then ${\mathit{\mu }}_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)=\mathit{min}\{{\mathit{\mu }}_x\left(y_1\right),{\mathit{\mu }}_x\left(y_2\right)\}$, where $\mathit{x},y_1,y_2\in F_2^n,\mathit{and}\mathit{\gamma }\in F_2\mathrm{\setminus }\{1\}$.

Proof.

Suppose: If ${\mathit{\mu }}_x\left(y_1\right)>{\mathit{\mu }}_x\left(y_2\right),\mathrm{\forall x},y_1,y_2\in F_2^n$ and $\mathit{\gamma }\in F_2\mathrm{\setminus }\{1\}$.

Implies ${\mathit{\mu }}_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)={\mu }_x(y_2)=\mathit{min}\{{\mathit{\mu }}_x\left(y_1\right),\text{allowbreak}{\mathit{\mu }}_x\left(y_2\right)\}$ for $\mathit{x},y_1,y_2\in F_2^n$, Since γ = 0 and $1-\mathit{\gamma }=1$.

Lemma 3.35.

Let $f_2^n$ be a binary convex fuzzy vector space. Then ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$ for $\mathit{t}\in [0,1]$ is a binary convex set of $F_2^n$, $\mathrm{\forall x},y_i\in F_2^n$.

Proof.

Since $f_2^n$ is a binary convex fuzzy vector space, µ is a binary convex fuzzy set by Definition 3.20.

By Proposition 2.20, ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$ for $\mathit{t}\in [0,1]$ is a binary convex set of $F_2^n$, $\mathrm{\forall x},y_i\in F_2^n$.

Theorem 3.36

Let $f_2^n$ be a binary convex fuzzy vector space. Then the following statements are equivalent for $\mathrm{x},\mathrm{y}\in F_2^n$:

1. µ is a binary convex fuzzy set of $f_2^n$.

2. ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$, $\mathit{t}\in \mathit{Im\mu }$, is a binary convex set of $f_2^n$.

Proof.

Suppose $(1⟹2)$.

By Lemma 3.35 the condition is true.

Suppose $(2⟹1)$.

Let $\mathit{y},\mathit{z}\in$ ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$ be arbitrary. Then ${\mathit{\mu }}_x(\mathit{y})\ge \mathit{t}$ and ${\mathit{\mu }}_x(\mathit{z})\ge \mathit{t}$.

Assume, ${\mathit{\mu }}_x\left(\mathit{y}\right)=t_1$ and ${\mathit{\mu }}_x\left(\mathit{z}\right)=t_2$ such that $\mathit{t}=\mathit{min}\{t_1,t_2\}$.

Since ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$ is a binary convex set, $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in {\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$, for $\mathit{\gamma }\in [0,1]$.

Therefore, ${\mathit{\mu }}_x(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\ge \mathit{t}=\mathit{min}\{{\mathit{\mu }}_x\left(\mathit{y}\right),{\mathit{\mu }}_x\left(\mathit{z}\right)\}$.

Hence, µ is a binary convex fuzzy set of $f_2^n$.

Proposition 3.37.

Let ${\mathcal{X}}_C$ be a characteristic function of a subset $\mathit{C}\subseteq F_2^n$. Then C is a binary convex set of $F_2^n$ if and only if ${\mathcal{X}}_C$ is a binary convex fuzzy subset of a binary convex fuzzy vector space $f_2^n$.

Proof.

Let $F_2^n$ be a binary vector space and a non-empty set $\mathit{C}\subseteq F_2^n$.

Let ${\mathcal{X}}_C(\mathit{y})=\{\begin{array}{l}1,\mathit{if}\mathit{y}\in \mathit{C}\\ 0,\mathit{if}\mathit{y}\notin \mathit{C}\end{array}$ be a characteristic function of a subset C.

$(⟹)$ Suppose C is a binary convex set of $F_2^n$.

Claim: ${\mathcal{X}}_C$ is a binary convex fuzzy subset of $f_2^n$.

Let $\mathit{y},\mathit{z}\in \mathit{C}$, then $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in \mathit{C}.$ This implies ${\mathcal{X}}_C(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})=1\ge \mathit{min}\{{\mathcal{X}}_C\left(\mathit{y}\right),{\mathcal{X}}_C\left(\mathit{z}\right)\}$. Therefore, ${\mathcal{X}}_C$ is a binary convex fuzzy subset of $f_2^n$.

($⟸)$ Conversely, suppose ${\mathcal{X}}_C$ is a binary convex set of $F_2^n$.

Claim: C is a binary convex subset of $f_2^n$.

Let $\mathit{y},\mathit{z}\in \mathit{C}$ be arbitrary. Then ${\mathcal{X}}_C\left(\mathit{y}\right)=1$ and ${\mathcal{X}}_C\left(\mathit{z}\right)=1$.

Since ${\mathcal{X}}_C$ is a binary binary convex fuzzy subset of $f_2^n$,

(3) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\ge \mathit{min}\{{\mathcal{X}}_{\mathit{C}}(\mathit{y}),{\mathcal{X}}_{\mathit{C}}(\mathit{z})\}=\mathit{min}\{1,1\}=1$(3)

But,

(4) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\le 1$(4)

From (3(3) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\ge \mathit{min}\{{\mathcal{X}}_{\mathit{C}}(\mathit{y}),{\mathcal{X}}_{\mathit{C}}(\mathit{z})\}=\mathit{min}\{1,1\}=1$(3) ) and (4(4) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\le 1$(4) ), we have ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})=1$. Which implies $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in \mathit{C}$.

Therefore, $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in \mathit{C},\mathrm{\forall y},\mathit{z}\in \mathit{C}$ and $\mathit{\gamma },(1-\mathit{\gamma })\in F_2$. Implies C is a binary convex subset of a binary convex fuzzy vector space $f_2^n$.

3.3. Binary convex fuzzy linear subspaces

Proposition 3.38.

Let $f_2^n$ be a binary convex fuzzy vector space. Then µ is binary convex fuzzy linear subspace of $F_2^n$ whenever ${\mathit{\mu }}_x\left(\mathbf{0}\right)\ge {\mathit{\mu }}_x\left(y_i\right),\mathrm{\forall x},y_i\in F_2^n$.

Proof.

Let $y_1,y_2\in F_2^n$ be arbitrary and $\mathit{\gamma }\in F_2$.

Suppose: ${\mathit{\mu }}_x\left(\mathbf{0}\right)\ge {\mathit{\mu }}_x\left(y_i\right),\mathrm{\forall x},y_i\in F_2^n$.

Claim: µ is binary convex fuzzy linear subspace of $F_2^n$.

Case − 1 : For γ = 1

${\mathit{\mu }}_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)={\mathit{\mu }}_x(y_1+\mathbf{0})={\mathit{\mu }}_x(y_1)\ge \mathit{min}\{{\mathit{\mu }}_x(y_1),{\mathit{\mu }}_x(y_2)\}$.

And also, ${\mathit{\mu }}_x\left(\mathit{\gamma }{y}_i\right)={\mathit{\mu }}_x(y_i)={\mathit{\mu }}_x(y_i)$.

Therefore, µ is a binary convex linear subspace of $F_2^n$.

Case − 2 : For γ = 0

${\mathit{\mu }}_x(\mathit{\gamma }{y}_1+(1-\mathit{\gamma })y_2)={\mathit{\mu }}_x(\mathbf{0}+y_2)={\mathit{\mu }}_x\left(y_2\right)\ge \mathit{min}\{{\mathit{\mu }}_x(y_1),{\mathit{\mu }}_x(y_2)\}$.

And also, ${\mathit{\mu }}_x\left(\mathit{\gamma }{y}_i\right)\ge {\mathit{\mu }}_x(\mathbf{0})\ge {\mathit{\mu }}_x(y_i)$.

Therefore, µ is a binary convex linear subspace of $F_2^n$.

Hence, µ is a binary convex linear subspace of $F_2^n$.

Remark 4.

A binary convex fuzzy linear subspace µ must contain the zero vector with its corresponding degree of freedom which is greater or equal to all degree of freedoms.

Corollary 3.39.

Let $f_2^n$ be a binary convex fuzzy vector space. If ${\mathit{\mu }}_x\left(\mathbf{0}\right)=1,\mathrm{\forall }\mathit{x},y_i\in F_2^n$, then µ is convex fuzzy linear subspace of $F_2^n$.

Theorem 3.40

Let $f_2^n$ be a binary convex fuzzy vector space. Then ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$ for $\mathit{t}\in [0,1]$ is a binary convex linear subspace of $F_2^n$ whenever ${\mathit{\mu }}_x\left(\mathbf{0}\right)\ge {\mathit{\mu }}_x\left(y_i\right),\mathrm{\forall x},y_i\in F_2^n$.

Proof.

Suppose: ${\mathit{\mu }}_x\left(\mathbf{0}\right)\ge {\mathit{\mu }}_x\left(y_i\right),\mathrm{\forall x},y_i\in F_2^n$.

This implies ${\mathit{\mu }}_x\left(\mathbf{0}\right)\ge \mathit{t},\mathrm{\forall t}\in [0,1]$. Then $\mathbf{0}\in ({\mu }_x{)}_t$, implies $({\mu }_x{)}_t\ne \mathrm{\varnothing }$ and $({\mu }_x{)}_t\subseteq F_2^n$.

Now, $\mathit{\mu }$ is a binary fuzzy convex set of $F_2^n$, since $f_2^n$ be a binary convex fuzzy vector space. By Proposition 3.38 µ is a binary fuzzy linear subspace of $F_2^n$, since ${\mathit{\mu }}_x\left(\mathbf{0}\right)\ge {\mathit{\mu }}_x\left(y_i\right),\mathrm{\forall x},y_i\in F_2^n$.

Since µ is binary convex fuzzy linear subspace, ${\mu }_{\mathit{x}}(\mathit{\gamma y})\ge {\mu }_{\mathit{x}}(\mathit{y}),\mathrm{\forall x},\mathit{y}\in F_2^n$ and $\mathit{\gamma }\in F_2$.

Let $t_1,t_2\in [0,1]$ such that ${\mu }_x(\mathit{y})=t_1$ and ${\mu }_x(\mathit{z})=t_2$.

Since µ is binary convex linear fuzzy subspace of $F_2^n$, ${\mu }_x(\mathit{\gamma y})\ge {\mu }_x(\mathit{y}),\mathrm{\forall x},\mathit{y}\in F_2^n$.

Let $\mathit{t}=\mathit{min}\{t_1,t_2\}$. Then ${\mu }_x(\mathit{\gamma y}+(1-\mathit{\gamma z}))\ge \mathit{min}\{{\mu }_x(\mathit{y}),{\mu }_x(\mathit{z})\}=\mathit{min}\{t_1,t_2\}=\mathit{t}$. This implies $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in ({\mu }_x{)}_t$. Since $\mathit{\gamma },(1-\mathit{\gamma })\in F_2$ and $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in ({\mu }_x{)}_t$, $({\mu }_x{)}_t$ is linear subspace of $F_2^n$. Also, by Definition 2.19, $({\mu }_x{)}_t$ is convex set of a binary vector space $F_2^n$.

Hence, ${\left({\mathit{\mu }}_x\right)}_{\mathit{t}}$ is a binary convex linear subspace of $f_2^n$.

Proposition 3.41.

Let ${\mathcal{X}}_C$ be a characteristic function of a subset $\mathit{C}\subseteq F_2^n$. Then, C is a binary convex linear subspace of $F_2^n$ if and only if ${\mathcal{X}}_C$ is a binary convex fuzzy linear subspace of a binary convex fuzzy vector space $f_2^n$.

Proof.

Let $F_2^n$ be a binary vector space and a non-empty set $\mathit{C}\subseteq F_2^n$.

Let ${\mathcal{X}}_C(\mathit{y})=\{\begin{array}{l}1,\mathit{if}\mathit{y}\in \mathit{C}\\ 0,\mathit{if}\mathit{y}\notin \mathit{C}\end{array}$ be a characteristic function of a subset C.

$(⟹)$ Suppose C is a binary convex linear subspace of $F_2^n$.

Claim: ${\mathcal{X}}_C$ is a binary convex fuzzy linear subspace of a binary convex fuzzy vector space $f_2^n$.

Let $\mathit{y},\mathit{z}\in \mathit{C}$, then $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in \mathit{C}.$ This implies ${\mathcal{X}}_C(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})=1\ge \mathit{min}\{{\mathcal{X}}_C\left(\mathit{y}\right),{\mathcal{X}}_C\left(\mathit{z}\right)\}$.

And also, ${\mathcal{X}}_C\left(\mathit{\gamma y}\right)=\{\begin{array}{l}{\mathcal{X}}_C\left(\mathbf{0}\right),\mathit{if}\mathit{\gamma }=0\\ {\mathcal{X}}_C\left(\mathit{y}\right),\mathit{if}\mathit{\gamma }=1\end{array}\mathrm{\forall y}\in \mathit{C}\subseteq F_2^n$. Then ${\mathcal{X}}_C\left(\mathit{\gamma y}\right)\ge {\mathcal{X}}_C\left(\mathit{y}\right)$. Therefore, ${\mathcal{X}}_C$ is a binary convex fuzzy linear subspace of $f_2^n$.

($⟸)$ Conversely, suppose ${\mathcal{X}}_C$ is a binary convex fuzzy linear subspace of $F_2^n$.

Claim: C is a binary convex linear subspace of $f_2^n$.

Let $\mathit{y},\mathit{z}\in \mathit{C}$ be arbitrary. Then ${\mathcal{X}}_C\left(\mathit{y}\right)=1$ and ${\mathcal{X}}_C\left(\mathit{z}\right)=1$.

Since ${\mathcal{X}}_C$ is a binary convex fuzzy linear subspace of $f_2^n$,

(5) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\ge \mathit{min}\{{\mathcal{X}}_{\mathit{C}}(\mathit{y}),{\mathcal{X}}_{\mathit{C}}(\mathit{z})\}=\mathit{min}\{1,1\}=1$(5)

But,

(6) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\le 1$(6)

From (5(5) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\ge \mathit{min}\{{\mathcal{X}}_{\mathit{C}}(\mathit{y}),{\mathcal{X}}_{\mathit{C}}(\mathit{z})\}=\mathit{min}\{1,1\}=1$(5) ) and (6(6) ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})\le 1$(6) ), we have ${\mathcal{X}}_{\mathit{C}}(\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z})=1$. Which implies $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in \mathit{C}$.

Therefore, $\mathit{\gamma y}+(1-\mathit{\gamma })\mathit{z}\in \mathit{C},\mathrm{\forall y},\mathit{z}\in \mathit{C}$ and $\mathit{\gamma },(1-\mathit{\gamma })\in F_2$. Implies C is a binary convex linear subspace of $F_2^n$.

3.4. Binary convex fuzzy codes over binary convex fuzzy vector spaces

Considering p to be the probability of a binary symmetric channel (BSC) not receiving a sent codeword correctly and e to be the weight of the error patterns between the received and possible sent classical codewords, the defined convex fuzzy subset is taken as a codeword over a binary convex fuzzy vector space $f_2^n$. Standing from this logical argument, we defined binary convex fuzzy codes over a binary convex fuzzy vector space $f_2^n$ and discussed some basic properties of these codes. The concepts need further discussion about these codes and their properties to use them for error—correction/detection process over a noise BSC.

Definition 3.42.

Let ς be a non-empty subset of a binary fuzzy vector space $f_2^n$. Then ς is called a binary fuzzy code over a binary fuzzy vector space $f_2^n$. The members of ς are also called fuzzy codewords Gereme et al., (2024).

Definition 3.43.

Let ς be a non-empty subset of a binary convex fuzzy vector space $f_2^n$. Then ς is called a binary convex fuzzy code over a binary convex fuzzy vector space $f_2^n$. The members of ς are also called convex fuzzy codewords.

Remark 5.

1. The binary fuzzy code over a binary fuzzy vector space containing every binary fuzzy linear subspaces is a binary convex fuzzy code.

2. Every binary convex fuzzy code is a binary fuzzy code but the converse is not true, since a fuzzy code can be defined as any non-empty subset of a binary fuzzy space $f_2^n$ that may not be binary fuzzy vector spaces.

Proposition 3.44.

Let $\mathit{C}\subseteq F_2^n$ be a convex code over $F_2^n$. Then the characteristic function ${\mathcal{X}}_C$ is a binary convex fuzzy code over a binary convex fuzzy vector space $f_2^n$.

Proof.

The proof is the direct outcome of Proposition 3.41.

Proposition 3.45.

Let ς be a binary convex fuzzy code formed by a classical code C of length n over $F_2^n$. Then $({\mu }_x{)}_t=\mathit{C}$ whenever e_i is equal with the maximum weight of C.

Proof.

Suppose: e_i is equal with the maximum weight of C.

Since $e_i=\mathit{wt}(\mathit{x}+y_i)$ for $\mathit{x},y_i\in \mathit{C}$ and the maximum of C is the maximum weight between two distinct codewords of C, $\mathit{t}=p^{e_i}{q}^{n-e_i}$ is smaller (because of Theorem 3.22).

Assume t_i be other $p^{e_i}{q}^{n-e_i}$’s such that $e_i>\mathit{wt}(\mathit{x}+y_i)$ for maximum $\mathit{wt}(\mathit{x}+\mathit{y})$ with $\mathit{x},y_i\in \mathit{C}$. Then $\mathit{t}<t_i$. By Lemma 3.16, $({\mu }_x{)}_{t_i}\subseteq ({\mu }_x{)}_t\mathrm{\forall }\mathit{i}$. Implies $({\mu }_x{)}_t$ the maximum set that contains all subsets of C. Therefore, $({\mu }_x{)}_t=\mathit{C}$.

Proposition 3.46.

Let ς be a convex fuzzy code over a binary convex fuzzy vector space $f_2^n$. Then $({\mu }_x{)}_t$ is a binary code over $F_2^n$ for any $\mathit{x}\in F_2^n$, where $\mathit{t}=p^{e_i}{q}^{n-e_i}$ and ${\mu }_x\in \mathit{\varsigma }$.

Proof.

Let ς be a convex fuzzy code over a binary convex fuzzy vector space $f_2^n$.

Let ${\mu }_x\in \mathit{\varsigma }$ be arbitrary for $\mathit{x}\in F_2^n$.

Claim: $({\mu }_x{)}_t$ is a binary code over $F_2^n$ for $\mathit{t}=p^{e_i}{q}^{n-e_i}$.

Since ς is convex fuzzy code, ${\mu }_x\in \mathit{\varsigma }$ is a fuzzy codeword which is also a convex fuzzy set. By Lemma 3.35, $({\mu }_x{)}_t$ for $\mathit{t}\in (0,1)$ is a binary convex set of $F_2^n$. Also, By Definition 2.7, $({\mu }_x{)}_t$ is a binary code over $F_2^n$.

4. Conclusions

In the results discussed in this paper, it was observed that when classical binary vector spaces are fuzzified, they may not necessarily qualify as binary fuzzy vector spaces. The distinction of the fuzzified space as a fuzzy vector space is contingent upon the parameters α and γ in F₂. Furthermore, in the case of binary fuzzy vector spaces and the convex fuzzy vector spaces that we have defined, they are found to be equivalent. As a consequence, the facts and properties established for binary fuzzy vector spaces apply to binary convex fuzzy vector spaces. We utilize the concepts of binary convex fuzzy vector spaces for the formulation of binary convex fuzzy codes. Following the introduction of binary convex fuzzy codes, some key properties of these codes were presented. This article serves as a fundamental reference for investigating the properties of binary convex fuzzy codes like fuzzy distances, and fuzzy dimensions for their utilization in neural communication. However, this article formulates these properties and the possibility of generating convex (or neural codes) as an open problem.

5. Open problems

We believe, binary convex fuzzy codes that we defined over a binary convex fuzzy vector space $f_2^n$ can be applicable in error correction/detection process for neural networks. In the classical case, binary codes are convex realizable (Franke & Muthiah, 2018). Also, neural codes can be expressed using binary strings (Curto et al., 2017) and convex sets are used to study neural codes (Gambacini et al., 2022; Lienkaemper et al., 2017), there are an open questions that we raised. The questions are:

1. is it possible to generate convex codes and neural codes from the binary convex fuzzy codes over a binary convex fuzzy vector space? If it is possible, applying convex codes in neural networks will become more simple than the way that is used in the neural communication systems.

2. what are the facts of the properties such as dimensions, fuzzy distances (Euclidean and Hamming distances), and decoding algorithms of binary convex fuzzy codes, for their application in error correction/detection and image processing?

Disclosure statement

There is no any conflict of interest between the authors of this manuscript.

Funding

No funding source is available for this thesis.

Additional information

Notes on contributors

Mezgebu Manmekto Gereme

Mezgebu Manmekto Gereme Email: [email protected], Department of Mathematics, Bahir Dar University, College of Science, Bahir Dar, Ethiopia