Picture Fuzzy Subring of a Crisp Ring

Abstract

In this paper, the concept of picture fuzzy subring of a crisp ring is introduced and some related basic results are studied. Also, some properties of picture fuzzy subring under classical ring homomorphism are investigated.

1 Introduction

Fuzzy set was introduced by Zadeh [1] as an extension of the concept of classical set theory to deal with uncertainty in human life. Later on several researchers applied fuzzy set theory in different fields. Fuzzy group was introduced by Rosenfeld [2], and fuzzy invariant subgroups and fuzzy ideals were studied by Liu [3]. Fuzzy ideals and quotient fuzzy rings were investigated by Ren [4]. As a generalization of fuzzy set theory, intuitionistic fuzzy set theory was propounded by Atanassov [5]. Based on this idea of intuitionistic fuzzy set proposed by Atanassov, intuitionistic fuzzy subgroup was introduced by Biswas [6]. Notion of intuitionistic fuzzy ring was propounded by Hur et al. [7]. Further works on intuitionistic fuzzy subring and intuitionistic fuzzy ideals were done by Banerjee and Basnet [8]. Including more possible types of uncertainty, picture fuzzy set was introduced by Cuong [9] which is a generalization of intuitionistic fuzzy set. It is necessary to mention that in intuitionistic fuzzy set, each element of the set of universe has two components namely measure of membership and measure of non-membership, whereas in picture fuzzy set, each element of the set of universe has three components namely measure of positive membership, measure of neutral membership and measure of negative membership. As the time goes, several works were done by several researchers using picture fuzzy set in different directions [10,11,12,13].

In this paper, we introduce the concept of picture fuzzy subring of a crisp ring and study some basic results related to it. Also, we investigate some properties of picture fuzzy subring under classical ring homomorphism.

2 Preliminaries

In the current section, we recapitulate some basic concepts of intuitionistic fuzzy set (IFS), intuitionistic fuzzy subring (IFSR), picture fuzzy set (PFS) and some operations on picture fuzzy sets (PFSs).

Definition 1

[5] Let A be the set of universe. Then an IFS P over A is defined as $P=\{(a, \mu _P, v_P):a\in A\}$, where $\mu _P(a)\in [0,1]$ is the measure of membership and $v_P(a)\in [0,1]$ is the measure of non-membership of a in P satisfying the condition $0\leqslant \mu _P(a)+v_P(a)\leqslant 1$ for all $a\in A$.

Definition 2

[8] Let $(R,+,\cdot )$ be a crisp ring. Then an IFS $P=\{(a,\mu _P(a), v_P(a)):a\in R\}$ in R is said to be IFSR of R if the below stated conditions are fulfilled.

• (i) $\mu _P(a-b)\geqslant \mu _P(a)\wedge \mu _P(b)$, $v_P(a-b)\leqslant v_P(a)\vee v_P(b)$

• (ii) $\mu _P(a\cdot b)\geqslant \mu _P(a)\wedge \mu _P(b)$, $v_P(a\cdot b)\leqslant v_P(a)\vee v_P(b)$ for all $a,b\in R$.

Definition 3

[9] Let A be the set of universe. Then a PFS P over A is defined as $P=\{(a,\mu _P(a), \eta _P(a), v_P(a)):a\in A\}$, where $\mu _P(a)\in [0,1]$ is the measure of positive membership, $\eta _P(a)\in [0,1]$ is the measure of neutral membership and $v_P(a)\in [0,1]$ is the measure of negative membership of a in P satisfying the condition $0\leqslant \mu _P(a)+\eta _P(a)+v_P(a)\leqslant 1$ for all $a\in A$.

Definition 4

[9] Let $P=\{(a,\mu _P(a), \eta _P(a), v_P(a)):a\in A\}$ and $Q=\{(a,\mu _Q(a), \eta _Q(a), v_Q(a)):a\in A\}$ be two PFSs over the universe A. Then

• (i) $P\subseteq Q$ iff $\mu _P(a)\leqslant \mu _Q(a)$, $\eta _P(a)\leqslant \eta _Q(a)$ and $v_P(a)\geqslant v_Q(a)$ for all $a\in A$.

• (ii) $P=Q$ iff $\mu _P(a)=\mu _Q(a)$, $\eta _P(a)=\eta _Q(a)$ and $v_P(a)=v_Q(a)$ for all $a\in A$.

• (iii) $P\cup Q =\{(a, \max (\mu _P(a),\mu _Q(a)), \min (\eta _P(a),\eta _Q(a))$,

  $\min (v_P(a), v_Q(a))): a\in A\}$.

• (iv) $P\cap Q=\{(a, \min (\mu _P(a),\mu _Q(a)), \min (\eta _P(a),\eta _Q(a))$,

  $\max (v_P(a), v_Q(a))): a\in A\}$.

Definition 5

Let $P=\{(a_1, \mu _P(a_1), \eta _P(a_1), v_P(a_1)):a_1\in A_1\}$ and $Q=\{(a_2, \mu _Q(a_2), \eta _Q(a_2), v_Q(a_2)):a_2\in A_2\}$ be two PFSs in $A_1$ and $A_2$, respectively. Then Cartesian product of P and Q is the PFS $P\times Q$ =$\{((a,b), \mu _{P\times Q}((a,b)), \eta _{P\times Q}((a,b)), v_{P\times Q}((a,b)))\}$, where $\mu _{P\times Q}((a,b))=\mu _P(a)\wedge \mu _Q(b)$, $\eta _{P\times Q}((a,b))$ $=\eta _P(a)\wedge \eta _Q(b)$ and $v_{P\times Q}((a,b))=v_P(a)\vee v_Q(b)$ for all $(a,b)\in A_1\times A_2$.

Definition 6

Let $h:A_1\rightarrow A_2$ be a surjective mapping and $P=\{(a_1,\mu _P(a_1), \eta _P(a_1), v_P(a_1)):a_1\in A_1\}$ be a PFS in $A_1$. Then image of P under the map h is the PFS $h(P)=\{(a_2, \mu _{h(P)}(a_2), \eta _{h(P)}(a_2), v_{h(P)}(a_2)):a_2\in A_2\}$, where $\mu _{h(P)}(a_2)=\underset{a_1\in h^{-1}(a_2)}{\vee }{\mu _P(a_1)}$, $\eta _{h(P)}(a_2)=\underset{a_1\in h^{-1}(a_2)}{\wedge }{\eta _P(a_1)}$ and $v_{h(P)}(a_2)=\underset{a_1\in h^{-1}(a_2)}{\wedge }{v_P(a_1)}$ for all $a_2\in A_2$.

Definition 7

Let $h:A_1\rightarrow A_2$ be a mapping and $Q=\{(a_2,\mu _Q(a_2), \eta _Q(a_2), v_Q(a_2)):a_2\in A_2\}$ be a PFS in $A_2$. Then inverse image of Q under the map h is the PFS $h^{-1}(Q)=\{(a_1, \mu _{h^{-1}(Q)}(a_1), \eta _{h^{-1}(Q)}(a_1), v_{h^{-1}(Q)}(a_1)):a_1\in A_1\}$, where $\mu _{h^{-1}(Q)}(a_1)=\mu _Q(h(a_1))$, $\eta _{h^{-1}(Q)}(a_1)=\eta _Q(h(a_1))$ and $v_{h^{-1}(Q)}(a_1)=v_Q(h(a_1))$ for all $a_1\in A_1$.

Definition 8

Let $P=\{(a,\mu _P, \eta _P, v_P):a\in A\}$ be a PFS over the universe A. Then $(\theta ,\phi ,\psi )$-cut of P is the crisp set in A denoted by $C_{\theta ,\phi ,\psi }(P)$ and is defined as $C_{\theta ,\phi ,\psi }(P)=\{a\in A : \mu _P(a)\geqslant \theta , \eta _P(a)\geqslant \phi , v_P(a)\leqslant \psi \}$, where $\theta , \phi , \psi \in [0,1]$ with the condition $0\leqslant \theta + \phi +\psi \leqslant 1.$

Proposition 1

[13] Let $P=\{(a_1,\mu _P(a_1), \eta _P(a_1), v(a_1))\!:a_1\in A_1\} \,and\,Q\,=\{(a_2, \mu _P(a_2), \eta _P(a_2), v_P(a_2))\!:a_2\in A_2\}$ be two PFSs over the sets of universe $A_1$ and $A_2$, respectively. Also, let $h:A_1\rightarrow A_2$ be a mapping. Then the followings hold.

• (i) $C_{\theta ,\phi , \psi }(P)\subseteq C_{\theta ,\phi , \psi }(Q)$ whenever $P\subseteq Q$.

• (ii) $C_{\theta ,\phi , \psi }(P\cap Q)=C_{\theta ,\phi , \psi }(P)\cap C_{\theta ,\phi , \psi }(Q)$

• (iii) $C_{\theta ,\phi , \psi }(P\cup Q)\supseteq C_{\theta ,\phi , \psi }(P)\cup C_{\theta ,\phi , \psi }(Q)$

• (iv) $C_{\theta ,\phi , \psi }(P\times Q)=C_{\theta ,\phi , \psi }(P)\times C_{\theta ,\phi , \psi }(Q)$

• (v) $h^{-1}((C_{\theta , \phi , \psi }(Q))=C_{\theta , \phi , \psi }(h^{-1}(Q))$

Throughout the paper, we write PFS $P=\{(a, \mu _P(a), \eta _P(a), v_P(a)): a\in A\}$ as $P=(\mu _P, \eta _P, v_P)$.

3 Picture Fuzzy Subring

Let us define picture fuzzy subring (PFSR) generalizing the concept of IFSR.

Definition 9

Let $(R, +, \cdot )$ be a crisp ring and $P=(\mu _P, \eta _P, v_P)$ be a PFS in R. Then P is said to be PFSR of R if the below stated conditions are meet.

• (i) $\mu _P(a-b)\geqslant \mu _P(a)\wedge \mu _P(b)$, $\eta _P(a-b)\geqslant \eta _P(a)\wedge \eta _P(b)$ and $v_P(a-b)\leqslant v_P(a)\vee v_P(b)$,

• (ii) $\mu _P(a\cdot b)\geqslant \mu _P(a)\wedge \mu _P(b)$, $\eta _P(a\cdot b)\geqslant \eta _P(a)\wedge \eta _P(b)$ and $v_P(a\cdot b)\leqslant v_P(a)\vee v_P(b)$ for all $a,b\in R$.

Example 1

Let us consider the crisp ring $R=(Z, +, \cdot )$ and a PFS $P=(\mu _P, \eta _P, v_P)$ in R defined by

$$\begin{aligned} \mu _P(a)= & {} \left\{ \begin{array}{ll} 0.4, &{} \mathrm{when }\;\; a=0\\ 0.2, &{} \mathrm{when }\;\;a\ne 0 \end{array}\right. \\ \eta _P(a)= & {} \left\{ \begin{array}{ll} 0.4, &{} \mathrm{when }\;\; a=0\\ 0.15, &{} \mathrm{when }\;\;a\ne 0 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} v_P(a)=\left\{ \begin{array}{ll} 0.2, &{} \mathrm{when }\;\; a=0\\ 0.3, &{} \mathrm{when }\;\;a\ne 0. \end{array}\right. \end{aligned}$$

It can be shown that P is a PFSR of R.

Proposition 2

Let $(R, +,\cdot )$ be a ring and $P=(\mu _P, \eta _P, v_P)$ be a PFSR of R. Then

• (i) $\mu _P(0)\geqslant \mu _P(a)$, $\eta _P(0)\geqslant \eta _P(a)$ and $v_P(0)\leqslant v_P(a)$

• (ii) $\mu _P(-a)=\mu _P(a)$, $\eta _P(-a)=\eta _P(a)$ and $v_P(-a)=v_P(a)$ for all $a\in R$, where 0 is the additive identity in R and $-a$ is the additive inverse of a.

Proof

 

• (i) Since P is a PFSR of R therefore

  $$\begin{aligned} \mu _P(0)=\mu _P(a-a)&\geqslant \mu _P(a)\wedge \mu _P(a)=\mu _P(a),\\ \eta _P(0)=\eta _P(a-a)&\geqslant \eta _P(a)\wedge \eta _P(a)=\eta _P(a)\\ {\text{and}}\,\,v_P(0)=v_P(a-a)&\leqslant v_P(a)\vee v_P(a)\,\,{\text{for all}}\,\,a\in R. \end{aligned}$$

Thus, $\mu _P(0)\geqslant \mu _P(a)$, $\eta _P(0)\geqslant \eta _P(a)$ and $v_P(0)\leqslant v_P(a)$ for all $a\in R$.

• (ii) For all $a\in R$, we have,

  $$\begin{aligned} \mu _P(-a)=\mu _P(0-a)&\geqslant \mu _P(0)\wedge \mu _P(a)\\&=\mu _P(a)\,{\text{[by (i)]}}\\ \eta _P(-a)=\eta _P(0-a)&\geqslant \eta _P(0)\wedge \eta _P(a)\\&=\eta _P(a)\,{\text{[by (i)]}}\\ {\text{and}}\,\,v_P(-a)=v_P(0-a)&\leqslant v_P(0)\vee v_P(a)\\&=v_P(a)\,{\text{[by (i)]}}. \end{aligned}$$

Thus, $\mu _P(-a)\geqslant \mu _P(a)$, $\eta _P(-a)\geqslant \eta _P(a)$ and $v_P(-a)\leqslant v_P(a)$ for $a\in R$.

Now, replacing a by $-a$ we get, $\mu _P(a)\geqslant \mu _P(-a)$, $\eta _P(a)\geqslant \eta _P(-a)$ and $v_P(a)\leqslant v_P(-a)$ for all $a\in R$. Consequently, $\mu _P(-a)=\mu _P(a)$, $\eta _P(-a)=\eta _P(a)$ and $v_P(-a)=v_P(a)$ for all $a\in R$.

$\square$

Proposition 3

Let $(R,+, \cdot )$ be a crisp ring and $P=(\mu _P, \eta _P, v_P)$ be a PFSR of R. Then $C_{\theta , \phi , \psi }(P)$ is a crisp subring of R, provided that $\mu _P(0)\geqslant \theta$, $\eta _P(0)\geqslant \phi$ and $v_P(0)\leqslant \psi$, where 0 is the additive identity in the ring R.

Proof

Clearly, $C_{\theta , \phi , \psi }(P)$ is non-empty.

Let $a,b\in C_{\theta , \phi , \psi }(P)$. Then $\mu _P(a)\geqslant \theta$, $\eta _P(a)\geqslant \phi$, $v_P(a)\leqslant \psi$ and $\mu _P(b)\geqslant \theta$, $\eta _P(b)\geqslant \phi$, $v_P(b)\leqslant \psi$. Since P is a PFSR of R therefore

$$\begin{aligned} \mu _P(a-b)&\geqslant \mu _P(a)\wedge \mu _P(b)\geqslant \theta \wedge \theta =\theta ,\\ \eta _P(a-b)&\geqslant \eta _P(a)\wedge \eta _P(b)\geqslant \phi \wedge \phi =\phi \\ {\text{and}}\,\,v_P(a-b)&\leqslant v_P(a)\vee v_P(b)\leqslant \psi \vee \psi =\psi . \end{aligned}$$

Thus, $\mu _P(a-b)\geqslant \theta$, $\eta _P(a-b)\geqslant \phi$ and $v_P(a-b)\leqslant \psi$.

It follows that $a-b\in C_{\theta , \phi , \psi }(P)$.

Also, since P is a PFSR of R therefore

$$\begin{aligned} \mu _P(a\cdot b)&\geqslant \mu _P(a)\wedge \mu _P(b)\geqslant \theta \wedge \theta =\theta ,\\ \eta _P(a\cdot b)&\geqslant \eta _P(a)\wedge \eta _P(b)\geqslant \phi \wedge \phi =\phi \\ {\text{and}}\,\,v_P(a\cdot b)&\leqslant v_P(a)\vee v_P(b)\leqslant \psi \vee \psi =\psi . \end{aligned}$$

Thus, $\mu _P(a\cdot b)\geqslant \theta$, $\eta _P(a\cdot b)\geqslant \phi$ and $v_P(a\cdot b)\leqslant \psi$.

It follows that $a\cdot b\in C_{\theta , \phi , \psi }(P)$. Consequently, $C_{\theta , \phi , \psi }(P)$ is a crisp subring of R. $\square$

Proposition 4

Let $(R, +, \cdot )$ be a crisp ring and $P=(\mu _P, \eta _P, v_P)$ be a PFS in R. Then P is a PFSR of R if all $(\theta , \phi , \psi )$-cuts of P are crisp subrings of R.

Proof

Let $a,b \in R$. Also, let $\theta =\mu _P(a)\wedge \mu _P(b)$, $\phi =\eta _P(a)\wedge \eta _P(b)$ and $\psi =v_P(a)\vee v_P(b)$. Clearly, $\theta \in [0,1]$, $\phi \in [0,1]$ and $\psi \in [0,1]$ with $0\leqslant \theta +\phi +\psi \leqslant 1$.

It is observed that

$$\begin{aligned} \mu _P(a)&\geqslant \mu _P(a)\wedge \mu _P(b)=\theta ,\\ \eta _P(a)&\geqslant \eta _P(a)\wedge \eta _P(b)=\phi \\ {\text{and}}\,\,v_P(a)&\leqslant v_P(a)\vee v_P(b)=\psi \end{aligned}$$

Thus, $\mu _P(a)\geqslant \theta$, $\eta _P(a)\geqslant \phi$ and $v_P(a)\leqslant \psi$.

Also, we have

$$\begin{aligned} \mu _P(b)&\geqslant \mu _P(a)\wedge \mu _P(b)=\theta ,\\ \eta _P(b)&\geqslant \eta _P(a)\wedge \eta _P(b)=\phi \\ {\text{and}}\,\,v_P(b)&\leqslant v_P(a)\vee v_P(b)=\psi \end{aligned}$$

Thus, $\mu _P(b)\geqslant \theta$, $\eta _P(b)\geqslant \phi$ and $v_P(b)\leqslant \psi$.

It follows that $a,b \in C_{\theta , \phi , \psi }(P)$. Since $C_{\theta , \phi , \psi }(P)$ is a crisp subring of R therefore $a-b$ and $a\cdot b \in C_{\theta , \phi , \psi }(P)$.

This yields

$$\begin{aligned} \mu _P(a-b)&\geqslant \theta =\mu _P(a)\wedge \mu _P(b),\\ \eta _P(a-b)&\geqslant \phi =\eta _P(a)\wedge \eta _P(b)\\ v_P(a-b)&\leqslant \psi =v_P(a)\vee v_P(b)\\ {\text{and}}\,\,\mu _P(a\cdot b)&\geqslant \theta =\mu _P(a)\wedge \mu _P(b),\\ \eta _P(a\cdot b)&\geqslant \phi =\eta _P(a)\wedge \eta _P(b)\\ v_P(a\cdot b)&\leqslant \psi =v_P(a)\vee v_P(b). \end{aligned}$$

Since a, b are arbitrary elements of R therefore $\mu _P(a-b)\geqslant \mu _P(a)\wedge \mu _P(b)$, $\eta _P(a-b)\geqslant \eta _P(a)\wedge \eta _P(b)$, $v_P(a-b)\leqslant v_P(a)\vee v_P(b)$ and $\mu _P(a\cdot b)\geqslant \mu _P(a)\wedge \mu _P(b)$, $\eta _P(a\cdot b)\geqslant \eta _P(a)\wedge \eta _P(b)$, $v_P(a\cdot b)\leqslant v_P(a)\vee v_P(b)$ for all $a,b\in R$.

Consequently, P is a PFSR of R. $\square$

Proposition 5

Let $(R, +, \cdot )$ be a crisp ring and $L=(\mu _L, \eta _L, v_L)$ be a PFSR of R. Then $\mu _L(ra) \geqslant \mu _P(a)$, $\eta _L(ra) \geqslant \eta _L(a)$ and $v_L(ra) \leqslant v_L(a)$ for all $a\in R$ and for all integers r.

Proof

Case 1: Let r be a positive integer. Let $P(r): \mu _L(ra) \geqslant \mu _L(a)$, $\eta _L(ra) \geqslant \eta _L(a)$ and $v_L(ra) \leqslant v_L(a)$ for all $a \in R$. Here, P(1) is trivially true. Since L is a PFSR of R therefore $\mu _L(a^2)= \mu _L(a\cdot a) \geqslant \mu _L(a) \wedge \mu _L(a)= \mu _L(a)$, $\eta _L(a^2)= \eta _L(a\cdot a) \geqslant \eta _L(a) \wedge \eta _L(a)= \eta _L(a)$ and $v_L(a^2)= v_L(a\cdot a) \leqslant v_L(a) \vee v_L(a)=v_L(a)$. Therefore, P(2) is true. Let us assume that P(r) is true for $r=m$, i.e. P(m) is true. Then $\mu _L(ma) \geqslant \mu _L(a)$, $\eta _L(ma) \geqslant \eta _L(a)$ and $v_L(ma) \leqslant v_L(a)$ for all $a\in R$. Since L is a PFSR of R therefore for all $a\in R$,

$$\begin{aligned} \mu _L((m+1)a)&=\mu _L(ma+a)\\&\geqslant \mu _L(ma) \wedge \mu _L(a)\\&\geqslant \mu _L(a) \wedge \mu _L(a)=\mu _L(a),\\ \eta _L((m+1)a)&=\eta _L(ma+a)\\&\geqslant \eta _L(ma)\wedge \eta _L(a)\\&\geqslant \eta _L(a) \wedge \eta _L(a)=\eta _L(a)\\ {\text{and}}\,\,v_L((m+1)a)&=v_L(ma+a)\\&\leqslant v_L(ma)\vee v_L(a)\\&\leqslant v_L(a) \vee v_L(a)=v_L(a). \end{aligned}$$

Therefore P(r) is true for all positive integers r.

Case 2: Let r is a negative integer. Also, let $s=-r$. Since r is a negative integer therefore $r\leqslant -1$ which implies that $s\geqslant 1$, i.e. s is a positive integer. Now, $\mu _L(ra)= \mu _L(-sa)=\mu _L(sa) \geqslant \mu _L(a)$, $\eta _L(ra)= \eta _L(-sa) =\eta _L(sa) \geqslant \eta _L(a)$ and $v_L(ra)=v_L(-sa)=v_L(sa) \leqslant v_L(a)$ [by Proposition 2 and Case 1]

Case 3: When $r=0$ then we see that P(r) is trivially true because it is known from Proposition 2 that $\mu _L(0)\geqslant \mu _L(a)$, $\eta _L(0)\geqslant \eta _L(a)$ and $v_L(0)\leqslant v_L(a)$ for all $a\in R$. $\square$

Proposition 6

Let $(R,+, \cdot )$ be a crisp ring and $P=(\mu _P, \eta _P, v_P)$ be a PFSR of R. If a be the additive generator of R with $a\in C_{\theta , \phi , \psi }(P)$ then $C_{\theta , \phi , \psi }(P)=R$.

Proof

We know that $C_{\theta , \phi , \psi }(P)\subseteq R$. Let a be the additive generator of R with $a\in C_{\theta , \phi , \psi }(P)$. Then $\mu _P(a)\geqslant \theta$, $\eta _P(a)\geqslant \phi$ and $v_P(a)\leqslant \psi$. Also, let $t\in R$. Since R is an additive cyclic group therefore $t=pa$ for some integer p. Now, we have,

$$\begin{aligned} \mu _P(t)=\mu _P(pa)&\geqslant \mu _P(a)\,{\text{[by Proposition}}\,5]\\&\geqslant \theta ,\\ \eta _P(t)=\eta _P(pa)&\geqslant \eta _P(a)\,{\text{[by Proposition}}\,5]\\&\geqslant \phi \\ {\text{and}}\,v_P(t)=v_P(pa)&\leqslant v_P(a)\,{\text{[by Proposition}}\,5]\\&\leqslant \psi . \end{aligned}$$

Thus, we get $\mu _P(t)\geqslant \theta$, $\eta _P(t)\geqslant \phi$ and $v_P(t)\leqslant \psi$. Therefore, $t\in R\Rightarrow t\in C_{\theta , \phi , \psi }(P)$ which yields $R\subseteq C_{\theta , \phi , \psi }(P)$. Consequently, $C_{\theta , \phi , \psi }(P)=R$. $\square$

Proposition 7

Let $P=(\mu _P, \eta _P, v_P)$ and $Q=(\mu _Q, \eta _Q, v_Q)$ be two PFSRs of a ring $(R, +, \cdot )$. Then $P\cap Q$ is a PFSR of R.

Proof

It is known from Proposition 1 that $C_{\theta , \phi , \psi }(P\cap Q)=C_{\theta , \phi , \psi }(P)\cap C_{\theta , \phi , \psi }(Q)$. Since P and Q are PFSRs therefore by Proposition 3, $C_{\theta , \phi , \psi }(P)$ and $C_{\theta , \phi , \psi }(Q)$ are crisp subrings of R. Also, it is known that the intersection of any two crisp subrings is a crisp subring. As a result, $C_{\theta , \phi , \psi }(P\cap Q)$ is a crisp subring of R. Consequently, by Proposition 4, $P\cap Q$ is a PFSR of R. $\square$

Proposition 8

Let P and Q be two PFSRs of a ring $(R, +, )$. Then $P\cup Q$ is a PFSR of R if either $P\subseteq Q$ or $Q\subseteq P$.

Proof

Case 1: Let $P\subseteq Q$. Then $\mu _P(a)\leqslant \mu _Q(a)$, $n_P(a)\leqslant n_Q(a)$ and $v_P(a)\geqslant v_Q(a)$ for all $a\in R$. Therefore, $\mu _{P\cup Q}(a)=\mu _P(a)\vee \mu _Q(a)=\mu _Q(a)$, $\eta _{P\cup Q}(a)=\eta _P(a)\wedge \eta _Q(a)=\eta _P(a)$ and $v_{P\cup Q}(a)=v_P(a)\wedge v_Q(a)=v_Q(a)$ for all $a\in R$. It is observed that $Q\subseteq P\cup Q\Rightarrow C_{\theta , \phi , \psi }(Q)\leqslant C_{\theta , \phi , \psi }(P\cup Q)$ [by Proposition 1].

Let $a\in C_{\theta , \phi , \psi }(P\cup Q)$. Then $\mu _{P\cup Q}(a)\geqslant \theta$, $\eta _{P\cup Q}(a)\geqslant \phi$ and $v_{P\cup Q}(a)\leqslant \psi$, i.e. $\mu _Q(a)\geqslant \theta$, $\eta _P(a)\geqslant \phi$ and $v_Q(a)\leqslant \psi$, i.e. $\mu _Q(a)\geqslant \theta$, $\eta _Q(a)\geqslant \phi$ and $v_Q(a)\leqslant \psi$. Thus, $a\in C_{\theta , \phi , \psi }(Q)$. Therefore, $C_{\theta , \phi , \psi }(P\cup Q)\subseteq C_{\theta , \phi , \psi }(Q)$. Thus, finally, $C_{\theta , \phi , \psi }(P\cup Q)=C_{\theta , \phi , \psi }(Q)$. Since P and Q are PFSRs of R therefore by Proposition 3, $C_{\theta , \phi , \psi }(P)$ and $C_{\theta , \phi , \psi }(Q)$ are crisp subrings of R. As a result, $C_{\theta , \phi , \psi }(P\cup Q)$ is a crisp subring of R. Consequently, by Proposition 4, $P\cup Q$ is a PFSR of R.

Case 2: When $Q\subseteq P$, it can be proved in the similar way that $P\cup Q$ is a PFSR of R.

The converse of the above proposition does not necessarily hold which is clear from the following example, i.e. if P and Q are two PFSRs of a crisp ring R then $P\cup Q$ is a PFSR of R does not necessarily imply that either $P\subseteq Q$ or $Q\subseteq P$. $\square$

Example 2

Let us consider the ring $(R, +, \cdot )$ and the PFSR $P=(\mu _P, \eta _P, v_P)$ of R given in Example 1. Also, a PFSR $Q=(\mu _Q, \eta _Q, v_Q)$ of R is defined as follows.

$$\begin{aligned} \mu _Q(a)= & {} \left\{ \begin{array}{ll} 0.35, &{} \mathrm{when }\;\; a=0\\ 0.3, &{} \mathrm{when }\;\;a\ne 0 \end{array}\right. \\ \eta _Q(a)= & {} \left\{ \begin{array}{ll} 0.35, &{} \mathrm{when }\;\; a=0\\ 0.2, &{} \mathrm{when }\;\;a\ne 0 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} v_Q(a)=\left\{ \begin{array}{ll} 0.1, &{} \mathrm{when }\;\; a=0\\ 0.4, &{} \mathrm{when }\;\;a\ne 0. \end{array}\right. \end{aligned}$$

Thus, $P\cup Q$ is given by

$$\begin{aligned} \mu _{P\cup Q}(a)= & {} \left\{ \begin{array}{ll} 0.4, &{} \mathrm{when }\;\; a=0\\ 0.3, &{} \mathrm{when }\;\;a\ne 0 \end{array}\right. \\ \eta _{P\cup Q}(a)= & {} \left\{ \begin{array}{ll} 0.35, &{} \mathrm{when }\;\; a=0\\ 0.15, &{} \mathrm{when }\;\;a\ne 0 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} v_{P\cup Q}(a)=\left\{ \begin{array}{ll} 0.1, &{} \mathrm{when }\;\; a=0\\ 0.3, &{} \mathrm{when }\;\;a\ne 0. \end{array}\right. \end{aligned}$$

Here, $P\cup Q$ is a PFSR of R but neither $P\subseteq Q$ nor $Q\subseteq P$.

Proposition 9

Let $P=(\mu _P, \eta _P, v_P)$ and $Q=(\mu _Q, \eta _Q, v_Q)$ be two PFSRs of a ring $(R, +, \cdot )$. Then $P\times Q$ is a PFSR of $R\times R$.

Proof

It is known from Proposition 1 that $C_{\theta , \phi , \psi }(P\times Q)=C_{\theta , \phi , \psi }(P)\times C_{\theta , \phi , \psi }(Q)$. Since P and Q are PFSRs of R therefore by Proposition 3, $C_{\theta , \phi , \psi }(P)$ and $C_{\theta , \phi , \psi }(Q)$ are crisp subrings of R. Also, it is known that the Cartesian product of two crisp subrings is a crisp subring. As a result, $C_{\theta , \phi , \psi }(P\times Q)$ is a crisp subring of $R\times R$. Consequently, by Proposition 4, $P\times Q$ is a PFSR of $R\times R$. $\square$

4 Ring Homomorphism of Picture Fuzzy Subring

In the current section, we study some important properties of PFSR under classical ring homomorphism.

Proposition 10

Let $(R_1, +, \cdot )$ and $(R_2, +, \cdot )$ be two crisp rings and $Q=(\mu _Q, \eta _Q, v_Q)$ be a PFSR of $R_2$. Then for a ring homomorphism, $h:R_1\rightarrow R_2$, $h^{-1}(Q)$ is a PFSR of $R_1$.

Proof

Let $h^{-1}(Q)=(\mu _{h^{-1}(Q)}, \eta _{h^{-1}(Q)}, v_{h^{-1}(Q)})$.

Also, let $a, b\in C_{\theta , \phi , \psi }(h^{-1}(Q))$. Then

$$\begin{aligned} \mu _{h^{-1}(Q)}(a)\geqslant \theta , \eta _{h^{-1}(Q)}(a)\geqslant \phi , v_{h^{-1}(Q)}(a)\leqslant \psi \end{aligned}$$

${\text{and}}\,\mu _{h^{-1}(Q)}(b)\geqslant \theta , \eta _{h^{-1}(Q)}(b)\geqslant \phi , v_{h^{-1}(Q)}(b)\leqslant \psi .$

This implies,

$$\begin{aligned} \mu _Q(h(a))\geqslant \theta , \eta _Q(h(a))\geqslant \phi , v_Q(h(a))\leqslant \psi \end{aligned}$$

${\text{and}}\,\mu _Q(h(b))\geqslant \theta , \eta _Q(h(b))\geqslant \phi , v_Q(h(b))\leqslant \psi .$

$$\begin{aligned} {\text{Therefore}},\mu _Q(h(a-b))&=\mu _Q(h(a)-h(b))\\&\geqslant \mu _Q(h(a))\wedge \mu _Q(h(b))\\&\geqslant \theta \wedge \theta =\theta , \\ \eta _Q(h(a-b))&=\eta _Q(h(a)-h(b))\\&\geqslant \eta _Q(h(a))\wedge \eta _Q(h(b))\\&\geqslant \phi \wedge \phi =\phi ,\\ v_Q(h(a-b))&=v_Q(h(a)-h(b))\\&\leqslant v_Q(h(a))\vee v_Q(h(b))\\&\leqslant \psi \vee \psi =\psi \\ {\text{and}}\,\mu _Q(h(a\cdot b))&=\mu _Q(h(a)\cdot h(b))\\&\geqslant \mu _Q(h(a))\wedge \mu _Q(h(b))\\&\geqslant \theta \wedge \theta =\theta , \\ \eta _Q(h(a\cdot b))&=\eta _Q(h(a)\cdot h(b))\\&\geqslant \eta _Q(h(a))\wedge \eta _Q(h(b))\\&\geqslant \phi \wedge \phi =\phi ,\\ v_Q(h(a\cdot b))&=v_Q(h(a)\cdot h(b))\\&\leqslant v_Q(h(a))\vee v_Q(h(b))\\&\leqslant \psi \vee \psi =\psi \\&\quad {\text{[as}}\,h\,{\text{is a ring homomorphism]}}. \end{aligned}$$

Thus, $h(a-b)$ and $h(a\cdot b)\in C_{\theta , \phi , \psi }(Q)$.

This implies, $a-b$ and $a\cdot b\in h^{-1}(C_{\theta , \phi , \psi }(Q))=C_{\theta , \phi , \psi }(h^{-1}(Q))$.

Thus, $C_{\theta , \phi , \psi }(h^{-1}(Q))$ is a crisp subring of $R_1$. Therefore, by Proposition 4, $h^{-1}(Q)$ is PFSR of $R_1$. $\square$

Proposition 11

Let $(R_1, +,\cdot )$ and $(R_2, +, \cdot )$ be two crisp rings and $P=(\mu _P, \eta _P, v_P)$ be a PFSR of $R_1$. Then for a bijective ring homomorphism $h:R_1\rightarrow R_2$, h(P) is a PFSR of $R_2$.

Proof

Let $h(P)=(\mu _{h(P)}, \eta _{h(P)}, v_{h(P)})$.

$$\begin{aligned} {\text {Then}}\quad \mu _{h(P)}(b_1)&=\underset{a_1\in h^{-1}(b_1)}{\vee }{\mu _P(a_1)},\\ \text { }\eta _{h(P)}(b_1)&=\underset{a_1\in h^{-1}(b_1)}{\wedge }{\eta _P(a_1)}\\ {\text {and}}\,\, v_{h(P)}(b_1)&=\underset{a_1\in h^{-1}(b_1)}{\wedge }{v_P(a_1)}. \end{aligned}$$

Since h is bijective therefore $h^{-1}(b_1)$ must be a singleton set. So, for each $b_1\in R_2$, there exists an unique $a_1\in R_1$ such that $a_1=h^{-1}(b_1)$, i.e. $h(a_1)=b_1$. Thus, in this case, $\mu _{h(P)}(b_1)=\mu _{h(P)}(h(a_1))=\mu _P(a_1)$, $\eta _{h(P)}(b_1)=\eta _{h(P)}(h(a_1))=\eta _P(a_1)$ and $v_{h(P)}(b_1)=v_{h(P)}(h(a_1))=v_P(a_1)$.

Let $b\in C_{\theta , \phi , \psi }(h(P))$. Then

$$\begin{aligned} \mu _{h(P)}(b)\geqslant \theta , \eta _{h(P)}(b)\geqslant \phi \hbox { and }v_{h(P)}(b)\leqslant \psi . \end{aligned}$$

That is,

$\mu _{h(P)}(h(a))\geqslant \theta , \eta _{h(P)}(h(a))\geqslant \phi$ and $v_{h(P)}(h(a))\leqslant \psi$ [where $b=h(a)$ for unique $a\in R_1$].

That is, $\mu _{P}(a)\geqslant \theta , \eta _{P}(a)\geqslant \phi$ and $v_{P}(a)\leqslant \psi$.

This gives, $a\in C_{\theta , \phi , \psi }(P)$.

This implies, $h(a)\in h(C_{\theta , \phi , \psi }(P))$.

That is, $b\in h(C_{\theta , \phi , \psi }(P))$.

Therefore, $C_{\theta , \phi , \psi }(h(P))\subseteq h(C_{\theta , \phi , \psi }(P))$.

Now, let $d\in h(C_{\theta , \phi , \psi }(P))$. Then there exists an unique $c\in C_{\theta , \phi , \psi }(P)$ such that $d=h(c)$. Therefore,

$$\begin{aligned} \mu _{P}(c)\geqslant \theta , \eta _{P}(c)\geqslant \phi \hbox { and }v_{P}(c)\leqslant \psi . \end{aligned}$$

That is,

$$\begin{aligned} \mu _{h(P)}(h(c))\geqslant \theta , \eta _{h(P)}(h(c))\geqslant \phi \hbox { and }v_{h(P)}(h(c))\leqslant \psi . \end{aligned}$$

That is,

$$\begin{aligned} \mu _{h(P)}(d)\geqslant \theta , \eta _{h(P)}(d)\geqslant \phi \hbox { and }v_{h(P)}(d)\leqslant \psi . \end{aligned}$$

This gives, $d\in C_{\theta , \phi , \psi }(h(P))$.

Therefore, $h(C_{\theta , \phi , \psi }(P))\subseteq C_{\theta , \phi , \psi }(h(P))$.

Thus, finally, it is obtained that $C_{\theta , \phi , \psi }(h(P)) =h(C_{\theta , \phi , \psi }(P))$.

Let us suppose that $b_1, b_2\in C_{\theta , \phi , \psi }(h(P))$. Then

$$\begin{aligned} \mu _{h(P)}(b_1)\geqslant \theta , \eta _{h(P)}(b_1)\geqslant \phi , v_{h(P)}(b_1)\leqslant \psi \end{aligned}$$

and $\mu _{h(P)}(b_2)\geqslant \theta , \eta _{h(P)}(b_2)\geqslant \phi , v_{h(P)}(b_2)\leqslant \psi$.

That is,

$$\begin{aligned} \mu _{h(P)}(h(a_1))\geqslant \theta , \eta _{h(P)}(h(a_1))\geqslant \phi , v_{h(P)}(h(a_1))\leqslant \psi \end{aligned}$$

and $\mu _{h(P)}(h(a_2))\geqslant \theta , \eta _{h(P)}(h(a_2))\geqslant \phi , v_{h(P)}(h(a_2))\leqslant \psi$ [where $b_1=h(a_1)$ and $b_2=h(a_2)$ for unique $a_1, a_2\in R_1$].

That is,

$\mu _{P}(a_1)\geqslant \theta , \eta _{P}(a_1)\geqslant \phi , v_{P}(a_1)\leqslant \psi$ and $\mu _{P}(a_2)\geqslant \theta , \eta _{P}(a_2)\geqslant \phi , v_{P}(a_2)\leqslant \psi$.

This gives, $a_1\in C_{\theta , \phi , \psi }(P)$ and $a_2\in C_{\theta , \phi , \psi }(P)$.

This implies, $a_1-a_2\in C_{\theta , \phi , \psi }(P)$ and $a_1\cdot a_2\in C_{\theta , \phi , \psi }(P)$ [as $C_{\theta , \phi , \psi }(P)$ is a crisp subring of $R_1$].

This implies, $h(a_1-a_2)\in h(C_{\theta , \phi , \psi }(P))=C_{\theta , \phi , \psi }(h(P))$ and $h(a_1\cdot a_2)\in h(C_{\theta , \phi , \psi }(P))=C_{\theta , \phi , \psi }(h(P))$.

This implies, $h(a_1)-h(a_2)\in C_{\theta , \phi , \psi }(h(P))$ and $h(a_1)\cdot h(a_2)\in C_{\theta , \phi , \psi }(h(P))$ [as h is a ring homomorphism].

This gives, $b_1-b_2$ and $b_1\cdot b_2\in C_{\theta , \phi , \psi }(h(P))$.

Thus, $C_{\theta , \phi , \psi }(h(P))$ is a crisp subring of $R_2$. Consequently, by Proposition 4, h(P) is a PFSR of $R_2$. $\square$

5 Conclusion

We notice that exploration of the theory of subring in context of PFS plays a vital role in the field of algebra. In this paper, we established the notion of PFSR of a crisp ring and investigated some basic results related to it. We studied some basic properties of PFSR in the environment of classical ring homomorphism. It is our hope that these works will help the researchers to develop the theory of subring in context of some other types of sets.