Fuzzy congruence relation on pseudo-UP algebra

ABSTRACT

In this paper, we investigated some related properties and applied the concept of fuzzy congruence relation to a pseudo-UP algebra X. Additionally, we used their level set to define the fuzzy congruence relation on X. The Cartesian product of two fuzzy congruences and the intersection of two fuzzy congruences are discussed. We also provided the connection between the collection of fuzzy pseudo-UP ideals and the collection of fuzzy congruence relations in pseudo-UP algebras. We then demonstrated that there exists an associated algebra X that is a pseudo-UP algebra for each fuzzy ideal ζ and fuzzy congruence ϕ_ζ generated by ζ. Additionally, we created a congruence relation on a pseudo-UP algebra using the fuzzy pseudo-UP ideal and fuzzy congruence relation ϕ_ζ produced by ζ.

KEYWORDS:

• pseudo-UP algebra
• congruence relation
• fuzzy pseudo-UP ideal
• and fuzzy congruence relation

1. Introduction

Georgescu and Iorgulescu (2001) introduced the concept of pseudo-BCK algebra as an extension of BCK algebra. Y. B. Jun (2003) introduced the pseudo-ideal of pseudo-BCK algebra. Dudek and Jun (2008) introduced and analyzed the concept of pseudo-BCI algebra as a generalization of BCI-algebra. Y. Jun et al. (2006) introduced the concept of pseudo-BCI ideal in pseudo-BCI algebra. Kim and Kim (2007) introduced the notion of BE-algebra which is another generalization of BCK-algebra. Borzooei et al. (2013) generalized the notion of BE-algebra and introduced the notion of pseudo BE algebra. Rezaei et al. (2014) introduced the notion of congruence relation on pseudo BE-algebra and constructed the quotient pseudo BE algebra via this congruence relation. Prabpayak and Leerawat (2009) introduced a new algebraic structure which is called KU-algebra. As a generalization of that Iampan (2017) introduced a new algebraic structure which is called UP-algebra, and he studied ideal, congruence, and homomorphism of UP-algebra and investigated some related properties. Moreover Iampan derived some properties of the relation between quotient UP-algebra and isomorphism. Romano (2020a) introduced the concept of pseudo-UP algebra and derived basic properties. Romano (2020b) in a forthcoming article introduced the concepts of pseudo-UP ideal and pseudo-UP filter in pseudo-UP algebra. As a continuation of these papers, Romano (2021) introduced the concept of homomorphism between pseudo-UP algebra. The notion of homomorphism between pseudo-UP algebra is designed in the same way as it was done in Y. B. Jun (2003), Y. Jun et al. (2006), and Lee and Park (2009) when analyzing pseudo-BCK and pseudo-BCI algebras. Furthermore, Iampan and Romano (2022) introduced the concept of congruence relation on a pseudo-UP algebra.

Zadeh (1965) introduced the notion of fuzzy sets and fuzzy relation. Then many authors have studied about it. Lee (2009) defined the notion of ideal in pseudo BCI-algebra. Dymek and Walendziak (2012) investigated fuzzy ideals of pseudo BCK-algebra. Recently, Rezaei et al. (2015) discussed on (fuzzy) congruence relation in (pseudo) CI/BE-algebras and studied some of their properties.

The notion of congruence relations are one of the important concept in algebraic structure. Motivated by all the above results, we applied, in this paper, the notion of fuzzy congruence relation on pseudo-UP algebra. Additionally, we used their level set to define the fuzzy congruence relation on X. The Cartesian product of two fuzzy congruences and the intersection of two fuzzy congruences are discussed. We also provided the connection between the collection of fuzzy pseudo-UP ideal and the collection of fuzzy congruence relation in pseudo-UP algebra. We then demonstrated that there exists an associated algebra X that is a pseudo-UP algebra for each fuzzy ideal ζ and fuzzy congruence ϕ_ζ generated by ζ. Additionally, we created a congruence relation on a pseudo-UP algebra using the fuzzy pseudo-UP ideal and fuzzy congruence relation ϕ_ζ produced by ζ.

2. Preliminaries

In this section, we discussed some fundamental concepts and important basic results associated with our study, like pseudo-UP algebras, pseudo-UP ideal and congruence relation of pseudo-UP algebra. Furthermore, fuzzy set, fuzzy pseudo-UP ideals are discussed. These concepts are taken from Iampan and Romano (2022), Romano (2020a, 2020b), and Zadeh (1965).

Definition 2.1.

A pseudo-UP algebra is an algebra $(X,\cdot ,\star ,0)$ of type $(2,2,0)$ which satisfies the following axioms: for any $x,y,z\in X$

1)	$$\begin{gathered} (y\cdot z)\cdot ((x\cdot y)\star (x\cdot z))=0and \\ (y\star z)\star ((x\star y)\cdot (x\star z))=0 \end{gathered}$$
2)	$x\cdot y=0=y\cdot xandx\star y=0=y\star x⟹x=y,$
3)	$(y\cdot 0)\star x=xand(y\star 0)\cdot x=x$
4)	$x⩽y⟺x\cdot y=0andx⩽y⟺x\star y=0.$

Lemma 2.2.

In a pseudo-UP algebra X the following holds, for each $x\in X$,

1)	$x\cdot 0=0andx\star 0=0$
2)	$0\cdot x=xand0\star x=x$ and
3)	$x\cdot x=0andx\star x=0.$

Proposition 2.3.

In a pseudo-UP algebra X the following holds: for any $x,y\in X$

1)	$x⩽y\cdot x$ and
2)	$x⩽y\star x$.

Theorem 2.4.

In a pseudo-UP algebra X the following holds: for any $x,y,z\in X$

1)	$x⩽yandy⩽z⟹x⩽z$
2)	$x⩽y⟹z\cdot x⩽z\cdot y$
3)	$x⩽y⟹z\star x⩽z\star y$
4)	$x⩽y⟹y\cdot z⩽x\cdot z$ and
5)	$x⩽y⟹y\star z⩽x\star z.$

Definition 2.5.

A pseudo-UP ideal of X is a nonempty subset J of a pseudo-UP algebra X that has the following: for each $x,y,z\in X$.

1)	$0\in J$
2)	$x\cdot (y\star z)\in Jandy\in J⟹x\cdot z\in J$ and
3)	$x\star (y\cdot z)\in Jandy\in J⟹x\star z\in J.$

The following theorem describes the distinguishing characteristics of these substructures.

Theorem 2.6.

Let J be a pseudo-UP ideal in a pseudo-UP algebra, $\mathrm{\forall }x,y,z\in X$. Then:

1)	$y\star z\in Jandy\in J⟹z\in J$
2)	$y\in J⟹x\star y\in J$
3)	$y\cdot z\in Jandy\in J⟹z\in J$ and
4)	$y\in J⟹x\cdot y\in J.$

Definition 2.7.

Let $(X,\cdot ,\star ,0)$ be a pseudo-UP algebra and ϕ be an equivalence relation on the set X.

i)	ϕ is a congruence of type 1 on X if the following holds: for all$x,y,z\in X$ $(x,y)\in \varphi ⟹(x\cdot z,y\cdot z)\in \varphi \wedge (z\cdot x,z\cdot y)\in \varphi$.
ii)	ϕ is a congruence of type 2 on X if the following holds $(x,y)\in \varphi ⟹(x\star z,y\star z)\in \varphi \wedge (z\star x,z\star y)\text{break}\in \varphi$.

Definition 2.8.

A fuzzy subset ζ of a pseudo-UP algebra of X is called fuzzy pseudo-UP ideal of X if and only if it fulfill the following axioms: for any $x,y,z\in X$

1)	$\zeta (0)\ge \zeta (x)$
2)	$\zeta (x\cdot z)\ge min\{\zeta (x\cdot (y\star z)),\zeta (y)\}$ and
3)	$\zeta (x\star z)\ge min\{\zeta (x\star (y\cdot z)),\zeta (y)\}.$

Theorem 2.9.

Let ζ be a fuzzy pseudo-UP ideal of a pseudo-UP algebra of X, then it satisfies the following assertions: for each $x,y,z\in X$

1)	$\zeta (z)\ge min\{\zeta (y\star z),\zeta (y)\}$
2)	$\zeta (x\star y)\ge \zeta (y)$
3)	$\zeta (z)\ge min\{\zeta (y\cdot z),\zeta (y)\}$
4)	$\zeta (x\cdot y)\ge \zeta (y).$

Definition 2.10.

Let ζ and µ be fuzzy subsets of X. Then

1)

$\zeta \cap \mu$ (intersection) is defined as $\begin{array}{l}(\zeta \cap \mu )(x)=min\{\zeta (x),\mu (x)\},forallx\in X.\end{array}$ More generally, if ${\left\{{\zeta }_i\right\}}_{i\in \mathrm{\Delta }}$ is a family of fuzzy subsets of X, then ${\bigcap }_{i\in \mathrm{\Delta }}{\zeta }_i$ (the arbitrary intersection) is a fuzzy subset of X defined by $\begin{array}{l}\bigcap _{i\in \mathrm{\Delta }}{\zeta }_i(x)=\underset{i\in \mathrm{\Delta }}{inf}{\zeta }_i(x),forallx\in X.\end{array}$

2)

$\zeta \cup \mu$ (union) is defined as $\begin{array}{l}(\zeta \cup \mu )(x)=max\{\zeta (x),\mu (x)\},forallx\in X.\end{array}$ More generally, if ${\left\{{\zeta }_i\right\}}_{i\in \mathrm{\Delta }}$ is a family of fuzzy subsets of X, then ${\bigcup }_{i\in \mathrm{\Delta }}{\zeta }_i$ (the arbitrary union) is a fuzzy subset of X defined by $\begin{array}{l}\bigcup _{i\in \mathrm{\Delta }}{\zeta }_i(x)=\underset{i\in \mathrm{\Delta }}{sup}{\zeta }_i(x),forallx\in X.\end{array}$

3. Fuzzy congruence relations on pseudo-up algebra

In this section, we study the basic properties of a fuzzy congruence relation on a pseudo-UP algebra. Let X be a pseudo-UP algebra. A fuzzy relation ϕ on X is a mapping $\varphi :X\times X\to [0,1]$.

Definition 3.1.

A fuzzy relation ϕ on X is called a fuzzy congruence relation on a pseudo-UP algebra of X, if it satisfies the following axioms: for any $x,y,z\in X$

1)	$\varphi (x,x)=\varphi (0,0)$
2)	$\varphi (x,y)=\varphi (y,x)$
3)	$\varphi (x,z)\ge min\{\varphi (x,y),\varphi (y,z)\}$
4)	$\varphi (x\cdot z,y\cdot z)\ge \varphi (x,y)$ and $\varphi (x\star z,y\star z)\ge \varphi (x,y)$ and
5)	$\varphi (z\cdot x,z\cdot y)\ge \varphi (x,y)$ and $\varphi (z\star x,z\star y)\ge \varphi (x,y).$

Example 3.2.

Let $X=\{0,a,b,c\}$ with two binary operations $"\cdot "$ and $"\star "$ defined by the following Cayley table.

$\begin{array}{lll}\begin{array}{lllll}\cdot & 0& \mathrm{a}& \mathrm{b}& \mathrm{c}\\ 0& 0& \mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{a}& 0& 0& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& 0& \mathrm{a}& 0& \mathrm{c}\\ \mathrm{c}& 0& \mathrm{a}& 0& 0\end{array}& & \begin{array}{lllll}\star & 0& \mathrm{a}& \mathrm{b}& \mathrm{c}\\ 0& 0& \mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{a}& 0& 0& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& 0& \mathrm{a}& 0& \mathrm{c}\\ \mathrm{c}& 0& \mathrm{a}& 0& 0\end{array}\end{array}$

Then $(X,\cdot ,\star ,0)$ is a pseudo-UP algebra. Consider a fuzzy relation ϕ from X × X to $[0,1]$ with $\varphi (0,0)=\varphi (a,a)=\varphi (b,b)=\varphi (c,c)=1$, $\varphi (a,c)=\varphi (b,0)=\varphi (c,a)=\varphi (0,b)=0.6$ and $\varphi (0,c)=\varphi (c,0)=\varphi (a,b)=\varphi (b,a)=\varphi (a,0)=\varphi (0,a)=\varphi (b,c)=\varphi (c,b)=0.4$. It is easily checked that ϕ a fuzzy congruence relation on X.

Proposition 3.3.

If ϕ is a fuzzy congruence relation on a pseudo-UP algebra of X, then, for all $x,y\in X$

i)	$\varphi (0,0)\ge \varphi (x,y)$
ii)	$\varphi (x,y)\le \varphi (x\cdot y,0)$ and $\varphi (x,y)\le \varphi (x\star y,0)$
iii)	if ϕ satisfies Definition 2.1 $(2),(3)$ (4) and (5), then $\varphi (x,x)=\varphi (0,0)$ is equivalent to $\varphi (0,0)\ge \varphi (x,y).$

Proof.

i)	We have $\varphi (0,0)=\varphi (x,x)\ge min\{\varphi (x,y),\varphi (y,x)\}\text{break}=\varphi (x,y)$, since $\varphi (x,y)=\varphi (y,x).$
ii)	By Definition 2.1 (5), we have $\varphi (x,y)\le \varphi (x\cdot y,y\cdot y)=\varphi (x\cdot y,0)$ and $\varphi (x,y)\le \varphi (x\star y,y\star y)=\varphi (x\star y,0)$.
iii)	Suppose $\varphi (0,0)=\varphi (x,x)$, ϕ satisfies Definition 2.1 (2) and (3), we have $\varphi (0,0)=\varphi (x,x)\ge min\{\varphi (x,y),\varphi (y,x)\}=\varphi (x,y)$. Conversely, assume $\varphi (0,0)\ge \varphi (x,y)$. Using (4), we have $\varphi (x,x)=\varphi (0\cdot x,0\cdot x)\ge \varphi (0,0)$. Again using (4) we have $\varphi (0,0)=\varphi (x\cdot 0,x\cdot 0)\ge \varphi (x,x).$ Similarly, by Definition 2.1 (4), we have $\varphi (x,x)=\varphi (0\star x,0\star x)\ge \varphi (0,0)$. Again using (4) we have $\varphi (0,0)=\varphi (x\star 0,x\star 0)\ge \varphi (x,x).$

Theorem 3.4.

A fuzzy equivalence relation ϕ of X is a fuzzy congruence relation pseudo-UP algebra of X if and only if $\varphi (x\cdot u,y\cdot v)\ge min\{\varphi (x,y),\varphi (u,v)\}$ and $\varphi (x\star u,y\star v)\ge min\{\varphi (x,y),\varphi (u,v)\}$.

Proof.

Suppose that a fuzzy equivalence relation ϕ is a fuzzy congruence on X. Then for all $x,y,u,v\in X$.

$min\{\varphi (x,y),\varphi (u,v)\}=min\{min\{\varphi (x,y),\varphi (u,u)\},\text{break}min\{\varphi (y,y),\varphi (u,v)\}\},\text{since}\varphi (0,0)\ge \varphi (x,y)$

$\begin{array}{ll}& \le min\{\varphi (x\cdot u,y\cdot u),\varphi (y\cdot u,y\cdot v)\},\\ & (\text{by~ Definition~3.1,(4) })\\ & \le \varphi (x\cdot u,y\cdot v).(\text{by~transitive})\end{array}$

And,

$\begin{array}{ll}min\{\varphi (x,y),\varphi (u,v)\}& =min\{min\{\varphi (x,y),\varphi (u,u)\},\\ & min\{\varphi (y,y),\varphi (u,v)\}\}\\ & \le min\{\varphi (x\star u,y\star u),\\ & \varphi (y\star u,y\star v)\}& \\ & \le \varphi (x\star u,y\star v).& \end{array}$

Therefore,  $\varphi (x\cdot u,y\cdot v)\ge min\{\varphi (x,y),\varphi (u,v)\}$ and $\varphi (x\star u,y\star v)\ge min\{\varphi (x,y),\varphi (u,v)\}$.

Conversely, assume ϕ is a fuzzy equivalence relation on X that satisfies $\varphi (x\cdot u,y\cdot v)\ge min\{\varphi (x,y),\varphi (u,v)\}$ and $\varphi (x\star u,y\star v)\ge min\{\varphi (x,y),\varphi (u,v)\}$.

Now, $\varphi (x,y)=min\{\varphi (x,y),\varphi (z,z)\}\le \varphi (x\cdot z,y\cdot z)$ and $\varphi (x,y)=min\{\varphi (x,y),\varphi (z,z)\}\le \varphi (x\star z,y\star z).$ Similarly, $\varphi (x,y)=min\{\varphi (z,z),\varphi (x,y)\}\le \varphi (z\cdot x,z\cdot y)$ and $\varphi (x,y)=min\{\varphi (z,z),\varphi (x,y)\}\le \varphi (z\star x,z\star z).$ Therefore, ϕ is a fuzzy congruence relation on a pseudo-UP algebra of X.

Theorem 3.5.

If ϕ and θ are fuzzy congruence relations on X. Then $\varphi \times \theta$ is a fuzzy congruence relation on X × X.

Proof.

1)	Let $x\in X$. Then, $\begin{array}{ll}(\varphi \times \theta )((x,x),(x,x))& =min\{\varphi (x,x),\theta (x,x)\}\\ & =min\{\varphi (0,0),\theta (0,0)\}\\ & =(\varphi \times \theta )((0,0),(0,0)).\end{array}$
2)	Let $x_i,y_i\in X$, for any $i=1,2$. $\begin{array}{ll}(\varphi \times \theta )((x_1,x_2),(y_1,y_2))& =min\{\varphi (x_1,y_1),\theta (x_2,y_2)\}\\ & =min\{\varphi (y_1,x_1),\theta (y_2,x_2)\}\\ & =(\varphi \times \theta )((y_1,y_2),(x_1,x_2)).\end{array}$
3)	Let $x_i,y_i,z_i\in X$, for any $i=1,2.$ $\begin{array}{ll}& (\varphi \times \theta )((x_1,x_2),(y_1,y_2))\\ & =min\{\varphi (x_1,y_1),\theta (x_2,y_2)\}\\ & \ge min\{min\{\varphi (x_1,z_1),\varphi (z_1,y_1)\},\\ & min\{\theta (x_2,z_2),\theta (z_2,y_2)\}\}\\ & =min\{min\{\varphi (x_1,z_1),\theta (x_2,z_2)\},\\ & min\{\varphi (z_1,y_1),\theta (z_2,y_2)\}\}\\ & =min\{(\varphi \times \theta )((x_1,x_2),(z_1,z_2)),\\ & (\varphi \times \theta )((z_1,z_2),(y_1,y_2))\}.\end{array}$
4)	Let $x_i,y_i,u_i\in X$ for any $i=1,2$. $\begin{array}{ll}& (\varphi \times \theta )((x_1,x_2)\cdot (u_1,u_2),(y_1,y_2)\cdot (u_1,u_2))\\ & =(\varphi \times \theta )((x_1\cdot u_1,x_2\cdot u_2),(y_1\cdot u_1,y_2\cdot u_2))\\ & =min\{\varphi (x_1\cdot u_1,y_1\cdot u_1),\theta (x_2\cdot u_2,y_2\cdot u_2)\}\\ & \ge min\{\varphi (x_1,y_1),\theta (x_2,y_2)\}\\ & =(\varphi \times \theta )((x_1,x_2),(y_1,y_2)).\end{array}$ Similarly, $(\varphi \times \theta )((x_1,x_2)\star (u_1,u_2),(y_1,y_2)\star (u_1,u_2))\ge (\varphi \times \theta )((x_1,x_2),(y_1,y_2)).$
5)	By similarly way, we have $(\varphi \times \theta )\text{break}((u_1,u_2)\cdot (x_1,x_2),(u_1,u_2)\cdot (y_1,y_2))\ge (\varphi \times \theta )\text{break}((x_1,x_2),(y_1,y_2))$. Therefore, $\varphi \times \theta$ is a fuzzy congruence relation on X × X.

Definition 3.6.

Let ${\varphi }_1,{\varphi }_2$ be fuzzy congruence relations of X. Define the composition ${\varphi }_1\circ {\varphi }_2$ by:

$\begin{array}{l}({\varphi }_1\circ {\varphi }_2)(x,y)=\underset{z\in X}{sup}\{min\{{\varphi }_1(x,z),{\varphi }_2(z,y)\}\}.\end{array}$

Definition 3.7.

Let ϕ be a fuzzy congruence relation on X and $t\in [0,1]$. Then, the level congruence relation $U(\varphi :t)$ of ϕ and strong level congruence $U^{+}(\varphi :t)$ of ϕ are defined as the follows:

$\begin{array}{ll}U(\varphi :t):& =\{(x,y)\in X\times X:\varphi (x,y)\ge t\}\text{and }{U}^{+}(\varphi :t):\\ & =\{(x,y)\in X\times X:\varphi (x,y)>t\}.\end{array}$

Example 3.8.

Consider the pseudo-UP algebra given in Example 3.2. Then

i)	if t = 0.4, then $U(\varphi :t)=X\times X,$
ii)	if t = 0.6, then $$\begin{gathered} U(\varphi :t)=\{(0,0),(a,a),(b,b), \\ (c,c),(a,c),(c,a),(b,0),(0,b)\}, \end{gathered}$$
iii)	if t = 1, then $U(\varphi :t)=△$, where $△=\{(x,x):\text{break}x\in X\},$
iv)	if t = 0.4, then $$\begin{gathered} U^{+}(\varphi :t)=\{(0,0),(a,a),(b,b), \\ (c,c),(a,c),(c,a),(b,0),(0,b)\}, \end{gathered}$$
v)	if t = 0.6, then $U^{+}(\varphi :t)=△$, where $△=\{(x,x):\text{break}x\in X\},$
vi)	if t = 1, then $U^{+}(\varphi :t)=\mathrm{\varnothing }.$

Theorem 3.9.

Let $\varphi ,{\varphi }_1$ and ϕ₂ be fuzzy congruence relations of X. Then

i)	$U(\varphi :t)={\bigcap }_{0\le s<t}{U}^{+}(\varphi :s)$ and $U^{+}(\varphi :t)\text{break}={\bigcup }_{t<s\le 1}U(\varphi :s)$,
ii)	For any $t\in [0,1],{\varphi }_1={\varphi }_2$ if and only if $U^{+}({\varphi }_1:t)\text{break}=U^{+}({\varphi }_2:t)$ and $U^{+}({\varphi }_1\circ {\varphi }_2:t)=U^{+}({\varphi }_1:t)\circ U^{+}({\varphi }_2:t).$

Proof.

i)

Let $t\in [0,1]$ and s < t. Then $U(\varphi :t)\subseteq U^{+}(\varphi :s)$ and so, $\begin{array}{l}U(\varphi :t)\subseteq \bigcap _{0\le s<t}{U}^{+}(\varphi :s)\cdots \cdots \cdots (\star )\end{array}$ Conversely, let ɛ > 0 be given and let $(x,y)\in {\bigcap }_{0\le s<t}{U}^{+}(\varphi :s)$. Then $\varphi (x,y)\ge t-\epsilon$ implies $\varphi (x,y)\ge t$, since ɛ be arbitrary. From this $(x,y)\in U(\varphi :t)$. Which implies that $\begin{array}{l}\bigcap _{0\le s<t}{U}^{+}(\varphi :s)\subseteq U(\varphi :t)\cdots \cdots \cdots (\star \star )\end{array}$ . From $(\star )$ and $(\star \star )$, we get $$\begin{gathered} {\bigcap }_{0\le s<t}{U}^{+}(\varphi :s) \\ =U(\varphi :t). \end{gathered}$$ And $\begin{array}{ll}{U}^{+}(\varphi :t)& =\{(x,y)\in X\times X|\varphi (x,y)>t\}\\ & =\bigcup _{t<s\le 1}\{(x,y)\in X\times X|\varphi (x,y)\ge s\}\\ & =\bigcup _{t<s\le 1}U(\varphi :s).\end{array}$

ii)

Suppose ${\varphi }_1={\varphi }_2$ and $(x,y)\in U^{+}({\varphi }_1:t).$ Then ${\varphi }_2(x,y)={\varphi }_1(x,y)>t$ and so $(x,y)\in U^{+}({\varphi }_2:t)$. Hence, $U^{+}({\varphi }_1:t)\subseteq U^{+}({\varphi }_2:t)$. In the same way, we get $U^{+}({\varphi }_2:t)\subseteq U^{+}({\varphi }_1:t)$. Therefore, $U^{+}({\varphi }_1:t)=U^{+}({\varphi }_2:t)$. Conversely, suppose $U^{+}({\varphi }_1:t)=U^{+}({\varphi }_2:t)$. We need to show that ${\varphi }_1={\varphi }_2.$ Assume ${\varphi }_1\ne {\varphi }_2$. Then, there exists $(x,y)\in X\times X$ such that $t_1={\varphi }_1(x,y)\ne {\varphi }_2(x,y)=t_2$. Without loss of generality , let $t_1>t_2$ . Then ${\varphi }_1(x,y)=t_1>t_2$ and so $(x,y)\in U^{+}({\varphi }_1:t_1)=U^{+}({\varphi }_2:t_1)$. Which implies that ${\varphi }_2(x,y)>t_1$ implies $t_2>t_1$ which is a contradiction. Therefore, ${\varphi }_1={\varphi }_2$. Also for any $(x,y)\in X\times X$ and $t\in [0,1]$. Then, $\begin{array}{ll}& (x,y)\in U^{+}({\varphi }_1\circ {\varphi }_2:t)\\ & ⟺{\varphi }_1\circ {\varphi }_2(x,y)>t\\ & ⟺su{p}_{z\in X}\{min\{{\varphi }_1(x,z),{\varphi }_2(z,y)\}\}>t\\ & ⟺\mathrm{\exists }{z}_0\in X,min\{{\varphi }_1(x,z_0),{\varphi }_2(z_0,y)\}>t\\ & ⟺{\varphi }_1(x,z_0)\ge tand{\varphi }_2(z_0,y)>t\\ & ⟺(x,z_0)\in U^{+}({\varphi }_1:t)and\\ & (z_0,y)\in U^{+}({\varphi }_2:t)\\ & ⟺(x,y)\in U^{+}({\varphi }_1:t)\circ U^{+}({\varphi }_2:t)& \end{array}$

Therefore, $U^{+}({\varphi }_1\circ {\varphi }_2)=U^{+}({\varphi }_1:t)\circ U^{+}({\varphi }_2:t)$.

Theorem 3.10.

Let $U(\varphi :t)$ be non empty and for any $t\in [0,1]$. A fuzzy relation ϕ on X is a fuzzy congruence relation on X if and only if $U(\varphi :t)$ is a congruence relation on X.

Proof.

Suppose that ϕ is fuzzy congruence relation on X. We need to prove that $U(\varphi :t)$ is a congruence relation on X. Let $t\in [0,1]$ be such that $U(\varphi :t)\ne \mathrm{\varnothing }$. Let $u,v\in X$ such that $(u,v)\in U(\varphi :t)$. Then $\varphi (u,v)\ge t$. Since ϕ is fuzzy congruence relation, then we have $t\le \varphi (u,v)\le \varphi (0,0)=\varphi (x,x)$. Which implies that $(x,x)\in U(\varphi :t),\mathrm{\forall }t\in [0,1]$. Thus $U(\varphi :t)$ is reflexive. Let $(x,y)\in U(\varphi :t)$. Then $\varphi (y,x)=\varphi (x,y)\ge t$. This implies that $(y,x)\in U(\varphi :t)$. Hence, $U(\varphi :t)$ is symmetric. Let $(x,y),(y,z)\in U(\varphi :t)$. Then $\varphi (x,y)\ge t$ and $\varphi (y,z)\ge t$. Now, $\varphi (x,z)\ge min\{\varphi (x,y),\varphi (y,z)\}\ge min\{t,t\}=t$ implies that $(x,z)\in U(\varphi :t)$. Hence $U(\varphi :t)$ is transitive. Let $(x,y)\in U(\varphi :t),$ then $t\le \varphi (x,y)\le \varphi (x\cdot z,y\cdot z)$ and $t\le \varphi (x,y)\le \varphi (x\star z,y\star z)$ implies that $(x\cdot z,y\cdot z)$ and $(x\star z,y\star z)$ belongs to $U(\varphi :t)$. Similarly, Let $(x,y)\in U(\varphi :t),$ then $t\le \varphi (x,y)\le \varphi (z\cdot x,z\cdot y)$ and $t\le \varphi (x,y)\le \varphi (z\star x,z\star y)$ implies that $(z\cdot x,z\cdot y)$ and $(z\star x,z\star y)$ belongs to $U(\varphi :t)$. Therefore, $U(\varphi :t)$ is a congruence relation on pseudo-UP algebra of X.

Conversely, assume a level subset $U(\varphi :t)$ is a congruence relation on X. We need to show that ϕ is a fuzzy congruence relation on X. Let $t\in [0,1]$ be such that $U(\varphi :t)\ne \mathrm{\varnothing }$. Let $x\in X$ such that $(x,x)\in U(\varphi :t)$. Then $\varphi (x,x)\ge t$ and take $t=\varphi (x,x)$. Since $U(\varphi :t)$ is a congruence relation on X, we have $(0,0)\in U(\varphi :t)$ such that $\varphi (0,0)\ge t=\varphi (x,x)$. In the same way, we have $\varphi (x,x)\ge \varphi (0,0)$. Hence, $\varphi (x,x)=\varphi (0,0)$. Let $\varphi (x,y)=t,$ then $(x,y)\in U(\varphi :t)$ and $(y,x)\in U(\varphi :t)$. Because $U(\varphi :t)$ is a congruence relation on X. Thus $\varphi (y,x)\ge t=\varphi (x,y)$. In the same way, we have $\varphi (x,y)\ge \varphi (y,x)$. Hence, $\varphi (x,y)=\varphi (y,x)$. Let $(x,y),(y,z)\in U(\varphi :t)$, then $\varphi (x,y)=t_1$ and $\varphi (y,z)=t_2$ and take $t=min\{t_1,t_2\}$. Since $U(\varphi :t)$ is a congruence relation, we have $(x,z)\in U(\varphi :t)$. Then $\varphi (x,z)\ge t=min\{t_1,t_2\}=min\{\varphi (x,y),\varphi (y,z)\}$. Let $\varphi (x,y)=t,$ then $(x,y)\in U(\varphi :t)$. Since, $U(\varphi :t)$ is a congruence relation on X, we have $(x\cdot z,y\cdot z)$ and $(x\star z,y\star z)\in U(\varphi :t)$. Then $\varphi (x\cdot z,y\cdot z)\ge t=\varphi (x,y)$ implies that $\varphi (x\cdot z,y\cdot z)\ge \varphi (x,y)$ and $\varphi (x\star z,y\star z)\ge t=\varphi (x,y)$ implies that $\varphi (x\star z,y\star z)\ge \varphi (x,y).$ Similarly, $\varphi (z\cdot x,z\cdot y)\ge t=\varphi (x,y)$ implies that $\varphi (z\cdot x,z\cdot y)\ge \varphi (x,y)$ and $\varphi (z\star x,z\star y)\ge t=\varphi (x,y)$ implies that $\varphi (z\star x,z\star y)\ge \varphi (x,y)$. Therefore, ϕ is a fuzzy congruence relation on pseudo-UP algebra of X.

Proposition 3.11.

A relation ϕ of X is a congruence relation of X if and only if its characteristic function χ_ϕ is a fuzzy congruence relation on X.

Proof.

Suppose a relation ϕ of X is a congruence relation on X. We need to show that χ_ϕ is a fuzzy congruence relation on X. For $x\in X$, $(x,x)\in \varphi$, then ${\chi }_{\varphi }(x,x)=1={\chi }_{\varphi }(0,0)$, since $(0,0)\in \varphi$ because ϕ is a congruence relation on X. Let $(x,y)\in \varphi$, then ${\chi }_{\varphi }(x,y)=1$, since ϕ is a congruence relation, we have $(y,x)\in \varphi$, then ${\chi }_{\varphi }(y,x)=1$. So ${\chi }_{\varphi }(x,y)={\chi }_{\varphi }(y,x).$ If ${\chi }_{\varphi }(x,y)=0,$ then ${\chi }_{\varphi }(y,x)=0,$ otherwise ${\chi }_{\varphi }(y,x)=1$ which is a contradiction. This implies that $(y,x)\in \varphi$. Therefore, in any case ${\chi }_{\varphi }(x,y)={\chi }_{\varphi }(y,x)$. Let $(x,y),(y,z)\in \varphi$. Since ϕ is a congruence relation X, then $(x,z)\in \varphi$. Now, ${\chi }_{\varphi }(x,z)=1\ge min\{1,1\}=min\{{\chi }_{\varphi }(x,y),{\chi }_{\varphi }(y,z)\}$. Suppose $(x,y),(y,z)\notin \varphi$, then ${\chi }_{\varphi }(x,y)=0={\chi }_{\varphi }(y,z)$. Now, ${\chi }_{\varphi }(x,z)\ge 0=min\{0,0\}=min\{{\chi }_{\varphi }(x,y),{\chi }_{\varphi }(y,z)\}$. Finally, let $(x\cdot z,y\cdot z)$ and $(x\star z,y\star z)\in \varphi$, then ${\chi }_{\varphi }(x\cdot z,y\cdot z)=1={\chi }_{\varphi }(x\star z,y\star z)$. Since ϕ is a congruence relation such that $(x,y)\in \varphi$. Then ${\chi }_{\varphi }(x\cdot z,y\cdot z)=1\ge 1={\chi }_{\varphi }(x,y)$ and ${\chi }_{\varphi }(x\star z,y\star z)=1\ge 1={\chi }_{\varphi }(x,y)$. Similarly, ${\chi }_{\varphi }(z\cdot x,z\cdot y)=1\ge 1={\chi }_{\varphi }(x,y)$ and ${\chi }_{\varphi }(z\star x,z\star y)=1\ge 1={\chi }_{\varphi }(x,y)$. Therefore, χ_ϕ is a fuzzy congruence relation on X.

Conversely, assume χ_ϕ is a fuzzy congruence relation on X. We want to prove that ϕ is a congruence relation on X. Let $(0,0)\in \varphi$, then ${\chi }_{\varphi }(x,x)={\chi }_{\varphi }(0,0)=1$. Thus $(x,x)\in \varphi$. Let $(x,y)\in \varphi$, then $1={\chi }_{\varphi }(x,y)={\chi }_{\varphi }(y,x).$ Thus $(y,x)\in \varphi .$ For any $(x,y),(y,z)\in \varphi$ then ${\chi }_{\varphi }(x,z)\ge min\{{\chi }_{\varphi }(x,y),{\chi }_{\varphi }(y,z)\}=min\{1,1\}=1$. Thus $(x,z)\in \varphi .$ Let $(x,y)\in \varphi$, then $1={\chi }_{\varphi }(x,y)\le {\chi }_{\varphi }(x\cdot z,y\cdot z)$ and $1={\chi }_{\varphi }(x,y)\le {\chi }_{\varphi }(x\star z,y\star z)$. Hence, $(x\cdot z,y\cdot z)$ and $(x\star z,y\star z)$ belongs to ϕ. Similarly, $1={\chi }_{\varphi }(x,y)\le {\chi }_{\varphi }(z\cdot x,z\cdot y)$ and $1={\chi }_{\varphi }(x,y)\le {\chi }_{\varphi }(z\star x,z\star y)$. Hence, $(z\cdot x,z\cdot y)$ and $(z\star x,z\star y)$ belongs to ϕ. Therefore, ϕ is a congruence relation on X.

Theorem 3.12.

If $\{{\varphi }_i:i\in \mathrm{\Lambda }\}$ is a family of fuzzy congruence relation on X. Then ${\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i$ is a fuzzy congruence relation of X.

Proof.

Suppose ${\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(x,x)=in{f}_{i\in \mathrm{\Lambda }}({\varphi }_i(x,x))$. Then $in{f}_{i\in \mathrm{\Lambda }}({\varphi }_i(x,x))=in{f}_{i\in \mathrm{\Lambda }}({\varphi }_i(0,0))={\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(0,0)$. For $x,y\in X$, ${\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)=in{f}_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)=in{f}_{i\in \mathrm{\Lambda }}{\varphi }_i(y,x)\text{break}={\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(y,x)$. For any $x,y,z\in X$, ${\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(x,z)=in{f}_{i\in \mathrm{\Lambda }}\text{break}{\varphi }_i(x,z)\ge in{f}_{i\in \mathrm{\Lambda }}min\{{\varphi }_i(x,y),{\varphi }_i(y,z)\}=min\{in{f}_{i\in \mathrm{\Lambda }}\text{break}{\varphi }_i(x,y),in{f}_{i\in \mathrm{\Lambda }}{\varphi }_i(y,z)\}$ $=min\{{\cap }_{\in \mathrm{\Lambda }}{\varphi }_i(x,y),{\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i\text{break}(y,z)\}.$ Finally, ${\cap }_{i\in \mathrm{\Lambda }}\varphi (x\cdot z,y\cdot z=in{f}_{i\in \mathrm{\Lambda }}({\varphi }_i(x\cdot z,y\cdot z))\ge in{f}_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)={\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)$ and ${\cap }_{i\in \mathrm{\Lambda }}\varphi (x\star z,y\star z)=in{f}_{i\in \mathrm{\Lambda }}({\varphi }_i(x\star z,y\star z))\ge in{f}_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)={\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)$. Similarly, ${\cap }_{i\in \mathrm{\Lambda }}\varphi (z\cdot x,z\cdot y\text{break}=in{f}_{i\in \mathrm{\Lambda }}({\varphi }_i(z\cdot x,z\cdot y))\ge in{f}_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)={\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)$ and ${\cap }_{i\in \mathrm{\Lambda }}\varphi (z\star x,z\star y)=in{f}_{i\in \mathrm{\Lambda }}({\varphi }_i(z\star x,z\star y))\ge in{f}_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y)={\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i(x,y).$ Therefore, ${\cap }_{i\in \mathrm{\Lambda }}{\varphi }_i$ is a fuzzy congruence relation on X.

Remark 3.13.

The union of any two fuzzy congruence relation on X is not necessarily a fuzzy congruence relation on X.

Example 3.14.

Let $X=\{0,a,b\}$ be a set with binary operations $"\cdot "$ and $"\star "$ by the following Cayley table.

$\begin{array}{lll}\begin{array}{llll}\cdot & 0& \mathrm{a}& \mathrm{b}\\ 0& 0& \mathrm{a}& \mathrm{b}\\ \mathrm{a}& 0& 0& \mathrm{a}\\ \mathrm{b}& 0& 0& 0\end{array}& & \begin{array}{llll}\star & 0& \mathrm{a}& \mathrm{b}\\ 0& 0& \mathrm{a}& \mathrm{b}\\ \mathrm{a}& 0& 0& \mathrm{a}\\ \mathrm{b}& 0& 0& 0\end{array}\end{array}$

Then $(X;\cdot ,\star ,0)$ is a pseudo-UP algebra. Define the fuzzy relations ϕ and θ as follows:

$\begin{array}{lll}\begin{array}{llll}\varphi & 0& a& b\\ 0& 0.7& 0.1& 0.1\\ a& 0.1& 0.7& 0.3\\ b& 0.1& 0.3& 0.7\end{array}& & \begin{array}{llll}\theta & 0& a& b\\ 0& 0.5& 0.2& 0.2\\ a& 0.2& 0.5& 0.4\\ b& 0.2& 0.4& 0.5\end{array}\end{array}$

Then ϕ and θ are fuzzy congruence relation on X. It is easily checked that $(\varphi \cup \theta )(x,y)=max\{\varphi (x,y),\theta (x,y)\}$ is not a fuzzy congruence relation on X. Because Definition 2.1 (3) is not valid.

Theorem 3.15.

Let f be an endomorphism of X. If ϕ is fuzzy congruence relation of X, ϕ_f is defined by ${\varphi }_f(x,y)\text{break}=\varphi (f(x),f(y))$ is also a fuzzy congruence relation on X.

Proof.

For any $x,y,u,v\in X$. If $(x,y)=(u,v)$. Then,

$\begin{array}{lll}& ⟹f(x,y)=f(u,v)& \\ & ⟹\varphi (f(x,y))=\varphi (f(u,v))& \\ & ⟹\varphi (f(x),f(y))=\varphi (f(u),f(v))& \\ & ⟹{\varphi }_f(x,y)={\varphi }_f(u,v)& \end{array}$

Therefore, ϕ_f is well defined. Let $x,y,z\in X.$

1)	${\varphi }_f(x,x)=\varphi (f(x),f(x))=\varphi (0,0)={\varphi }_f(0,0).$
2)	${\varphi }_f(x,y)=\varphi (f(x),f(y))=\varphi (f(y),f(x))={\varphi }_f(y,x).$
3)	${\varphi }_f(x,z)=\varphi (f(x,z))=\varphi (f(x),f(z))\ge min\{\varphi (f(x),\text{break}f(y)),\varphi (f(y),f(z))\}=min\{{\varphi }_f(x,y),{\varphi }_f(y,z)\}.$
4)	${\varphi }_f(x\cdot z,y\cdot z)=\varphi (f(x\cdot z,y\cdot z))=\varphi (f(x)\cdot f(z),\text{break}f(y)\cdot f(z))\ge \varphi (f(x),f(y))={\varphi }_f(x,y)$. Similarly, ${\varphi }_f(x\star z,y\star z)=\varphi (f(x\star z,y\star z))=\varphi (f(x)\star f(z),\text{break}f(y)\star f(z))\ge \varphi (f(x),f(y))={\varphi }_f(x,y).$
5)	${\varphi }_f(z\cdot x,z\cdot y)=\varphi (f(z\cdot x,z\cdot y))=\varphi (f(z)\cdot f(x),\text{break}f(z)\cdot f(y))\ge \varphi (f(x),f(y))={\varphi }_f(x,y)$. Similarly, ${\varphi }_f(z\star x,z\star y)=\varphi (f(z\star x,z\star y))=\varphi (f(z)\star f(x),\text{break}f(z)\star f(y))\ge \varphi (f(x),f(y))={\varphi }_f(x,y)$. Thus by$(1-5)$ϕfis a fuzzy congruence relation on a pseudo-UP algebra of X.

Definition 3.16.

Let ϕ be a fuzzy congruence relation in X and $x\in X$. Define the fuzzy set ${\varphi }^x:X\to [0,1]$ which is defined by ${\varphi }^x(y)=\varphi (x,y)$ for any $y\in X$ is called a fuzzy congruence relation class containing x.

Theorem 3.17.

If ϕ is a fuzzy congruence relation on X. Then ϕ⁰ is a fuzzy pseudo-UP ideal of X.

Proof.

Suppose ϕ is a fuzzy congruence relation on X. We need to show that ϕ⁰ is a fuzzy pseudo-UP ideals of X. From Definition 3.16 ${\varphi }^0(0)=\varphi (0,0)$ and by Proposition  3.3(i), we have $\varphi (0,0)\ge \varphi (0,x)={\varphi }^0(x)$ which implies that ${\varphi }^0(0)\ge {\varphi }^0(x)$. Next for any $x,y,z\in X$ be such that ${\varphi }^0(y)=\varphi (0,y)$ and ${\varphi }^0(x\cdot (y\star z))=\varphi (0,x\cdot (y\star z))$. From ${\varphi }^0(y)=\varphi (0,y)=\varphi (y,0)\le \varphi (y\star z,0\star z)=\varphi (y\star z,z)\le \varphi (x\cdot (y\star z),x\cdot z).$ Which implies that ${\varphi }^0(y)\le \varphi (x\cdot (y\star z),x\cdot z)\dots \dots \dots ..(\star ).$

Also, since ϕ is a fuzzy congruence relation on X, we have ${\varphi }^0(x\cdot z)=\varphi (0,x\cdot z)\ge min\{\varphi (0,x\cdot (y\star z),\varphi (x\cdot (y\star z),x\cdot z))\}$ and from $(\star )$, we have $\varphi (x\cdot (y\star z),x\cdot z)\ge \varphi (0,y)$ implies that $\varphi (0,x\cdot z)\ge \varphi (0,y)$, by transitive. Which implies that $\varphi (0,x\cdot z)\ge min\{\varphi (0,x\cdot (y\star z)),\varphi (0,y)\}$. Hence, ${\varphi }^0(x\cdot z)\ge min\{{\varphi }^0(x\cdot (y\star z)),{\varphi }^0(y)\}$.

Similarly, for any $x,y,z\in X$ be such that ${\varphi }^0(y)=\varphi (0,y)$ and ${\varphi }^0(x\star (y\cdot z))=\varphi (0,x\star (y\cdot z))$. From ${\varphi }^0(y)=\varphi (0,y)=\varphi (y,0)\le \varphi (y\cdot z,0\cdot z)=\varphi (y\cdot z,z)\le \varphi (x\star (y\cdot z),x\star z).$ Which implies that ${\varphi }^0(y)\le \varphi (x\star (y\cdot z),x\star z)\dots \dots \dots ..(\star ).$

Also, since ϕ is a fuzzy congruence relation on X, we have ${\varphi }^0(x\star z)=\varphi (0,x\star z)\ge min\{\varphi (0,x\star (y\cdot z),\varphi (x\star (y\cdot z)))\}$ and from $(\star )$, we have $\varphi (x\star (y\cdot z),x\star z)\ge \varphi (0,y)$ implies that $\varphi (0,x\star z)\ge \varphi (0,y)$, by transitive. Which implies that $\varphi (0,x\star z)\ge min\{\varphi (0,x\star (y\cdot z)),\varphi (0,y)\}$. Thus, ${\varphi }^0(x\star z)\ge min\{{\varphi }^0(x\star (y\cdot z)),{\varphi }^0(y)\}$.

Therefore, ϕ⁰ is a fuzzy pseudo-UP ideal of X.

4. Fuzzy congruence relations induced by fuzzy pseudo-UP ideal

Let ζ be a fuzzy pseudo-UP ideals of X, let us define the fuzzy relation ϕ_ζ on X define as follows:

${\varphi }_{\zeta }(x,y)=min\{\zeta (x\cdot y),\zeta (y\cdot x)\}$ =min$\{\zeta (x\star y),\zeta (y\star x)\}.$ Then we have the following results.

Theorem 4.1.

The fuzzy relation ϕ_ζ is a fuzzy congruence relation on X.

Proof.

Assume that ζ is a fuzzy pseudo-UP ideals of X. Define a fuzzy relation on X by:

${\varphi }_{\zeta }(x,y)=min\{\zeta (x\cdot y),\zeta (y\cdot x)\}$. Then for all $x,y\in X$, we have,

1)	$\begin{array}{ll}{\varphi }_{\zeta }(x,x)& =min\{\zeta (x\cdot x),\zeta (x\cdot x)\}\\ & =min\{\zeta (0),\zeta (0)\}\\ & =min\{\zeta (0\cdot 0),\zeta (0\cdot 0)\}& & \\ & ={\varphi }_{\zeta }(0,0).& \end{array}$ Similarly, $\begin{array}{ll}{\varphi }_{\zeta }(x,x)& =min\{\zeta (x\star x),\zeta (x\star x)\}\\ & =min\{\zeta (0),\zeta (0)\}\\ & =min\{\zeta (0\star 0),\zeta (0\star 0)\}& & \\ & ={\varphi }_{\zeta }(0,0).& \end{array}$
2)	$\begin{array}{ll}{\varphi }_{\zeta }(x,y)& =min\{\zeta (x\cdot y),\zeta (y\cdot x)\}\\ & =min\{\zeta (y\cdot x),\zeta (x\cdot y)\}\\ & ={\varphi }_{\zeta }(y,x).& \end{array}$ Similarly, $\begin{array}{ll}{\varphi }_{\zeta }(x,y)& =min\{\zeta (x\star y),\zeta (y\star x)\}\\ & =min\{\zeta (y\star x),\zeta (x\star y)\}\\ & ={\varphi }_{\zeta }(y,x).& \end{array}$
3)	For $x,y,z\in X.$ Then by Theorem 2.4 (1), we have, $\begin{array}{ll}{\varphi }_{\zeta }(x,z)& =min\{\zeta (x\cdot z),\zeta (z\cdot x)\}\\ & \ge min\{min\{\zeta (x\cdot y),\zeta (y\cdot z)\},\\ & min\{\zeta (y\cdot x),\zeta (z\cdot y)\}\}\\ & =min\{min\{\zeta (x\cdot y),\zeta (y\cdot x)\},\\ & min\{\zeta (y\cdot z),\zeta (z\cdot y)\}\}& \\ & =min\{{\varphi }_{\zeta }(x,y),{\varphi }_{\zeta }(y,z)\}.& \end{array}$ Similarly, $\begin{array}{ll}{\varphi }_{\zeta }(x,z)& =min\{\zeta (x\star z),\zeta (z\star x)\}\\ & \ge min\{min\{\zeta (x\star y),\zeta (y\star z)\},\\ & min\{\zeta (y\star x),\zeta (z\star y)\}\}\\ & =min\{min\{\zeta (x\star y),\zeta (y\star x)\},\\ & min\{\zeta (y\star z),\zeta (z\star y)\}\}& \\ & =min\{{\varphi }_{\zeta }(x,y),{\varphi }_{\zeta }(y,z)\}.\end{array}$
4)	Let $x,y,z\in X$ such that $x\cdot y\le (z\cdot x)\star (z\cdot y)$, $y\cdot x\le (z\cdot y)\star (z\cdot x)$ and similarly, $x\star y\le (z\star x)\cdot (z\star y)$, $y\star x\le (z\star y)\cdot (z\star x).$ From Theorem 2.4, we have  $x\cdot y\star (z\cdot x)\star (z\cdot y)=0$, $(y\cdot x)\star (z\cdot y)\star (z\cdot x)=0$ and similarly, $(x\star y)\cdot (z\star x)\cdot (z\star y)=0$, $(y\star x)\cdot (z\star y)\cdot (z\star x)=0.$ So, $\zeta (z\cdot x)\cdot (z\cdot y)\ge \zeta (x\cdot y),\zeta (z\cdot y)\cdot \zeta (z\cdot x)\ge \zeta (y\cdot x)$. Then, ${\varphi }_{\zeta }(z\cdot x,z\cdot y)=min\{\zeta (z\cdot x)\cdot (z\cdot y),\zeta (z\cdot y)\cdot (z\cdot x)\}$ $\begin{array}{lll}& \ge min\{\zeta (x\cdot y),\zeta (y\cdot x)\}& \\ & ={\varphi }_{\zeta }(x,y).& \end{array}$ By similar argument, ${\varphi }_{\zeta }(z\star x),(z\star y)\ge {\varphi }_{\zeta }(x,y)$.
5)	${\varphi }_{\zeta }(x\cdot z,y\cdot z)\ge {\varphi }_{\zeta }(x,y)$ and ${\varphi }_{\zeta }(x\star z,y\star z)\ge {\varphi }_{\zeta }(x,y).$

Therefore, by $(1)-(5)$ ϕ_ζ is a fuzzy congruence relation on pseudo-UP algebra of X.

From this, a fuzzy congruence relation ϕ_ζ we say that it is generated by fuzzy pseudo-UP ideal ζ of X.

Theorem 4.2.

Let ζ be a fuzzy pseudo-UP ideals of X. Then, there is a fuzzy congruence relation ϕ in X such that $({\varphi }_{\zeta }{)}^0=\zeta$.

Proof.

By Theorem 3.17 and 4.1 it is easily checked that $({\varphi }_{\zeta }{)}^0=\zeta$ for each fuzzy pseudo-UP ideal ζ of X and fuzzy congruence relation ϕ of X.

Let ζ be a fuzzy pseudo-UP ideal of $X,$  ζ^x denote the fuzzy congruence class of x induced by ζ in X for every $x\in X$.

Proposition 4.3.

Let ζ be a fuzzy pseudo-UP ideal of X. Then ${\zeta }^x={\zeta }^y$ if and only if $\zeta (x\cdot y)=\zeta (y\cdot x)=\zeta (0)=\zeta (x\star y)=\zeta (y\star x)$.

Proof.

Let ${\zeta }^x={\zeta }^y$, for $x,y\in X$. We have ${\zeta }^u(v)={\varphi }_{\zeta }(u,v)\}=min\{\zeta (u\cdot v),\zeta (v\cdot u)\}=min\{\zeta (u\star v),\zeta (v\star u)\}$, for any $u,v\in X$. Since ${\zeta }^x={\zeta }^y,\text{then}{\zeta }^x(x)={\zeta }^y(x),$ for any $x,y\in X$. It follows that $min\{\zeta (x\cdot x),\zeta (x\cdot x)\}=min\{\zeta (y\cdot x),\zeta (x\cdot y)\}$. Since X is pseudo-UP algebra such that $x\cdot x=0$.

$\begin{array}{ll}& ⟹min\{\zeta (0),\zeta (0)\}=min\{\zeta (y\cdot x),\zeta (x\cdot y)\}\\ & ⟹\zeta (0)=min\{\zeta (y\cdot x),\zeta (x\cdot y)\}\\ & ⟹\zeta (0)=\zeta (y\cdot x)=\zeta (x\cdot y),\\ & \text{by~Definitions~of~fuzzy~pseudo-UP~ideal~of~X}.\end{array}$

Similarly, $min\{\zeta (x\star x),\zeta (x\star x)\}=min\{\zeta (y\star x),\zeta (x\star y)\}$. Since X is pseudo-UP algebra such that $x\star x=0$.

$\begin{array}{ll}& ⟹min\{\zeta (0),\zeta (0)\}=min\{\zeta (y\star x),\zeta (x\star y)\}\\ & ⟹\zeta (0)=min\{\zeta (y\star x),\zeta (x\star y)\}\\ & ⟹\zeta (0)=\zeta (y\star x)=\zeta (x\star y),\\ & \text{by~Definitions~of~fuzzy~pseudo-UP~ideal~of~X}.\end{array}$

Conversely, let $\zeta (x\cdot y)=\zeta (y\cdot x)=\zeta (x\star y)=\zeta (y\star x)=\zeta (0).$ By Theorem 2.9, we have $\zeta (x\cdot z)\ge min\{\zeta (x\cdot y),\zeta ((x\cdot y)\star (x\cdot z)\}$, since X is a pseudo-UP algebra, then we have $(y\cdot z)\star ((x\cdot y)\star (x\cdot z))=0$. Which implies that $\zeta (x\cdot z)\ge min\{\zeta (x\cdot y),\zeta (y\cdot z)\}$ implies $\zeta (x\cdot z)\ge min\{\zeta (0),\zeta (y\cdot z)\}$, since $\zeta (x\cdot y)=\zeta (0)$. So, $\zeta (x\cdot z)\ge \zeta (y\cdot z)$, since ζ is fuzzy pseudo-UP ideal of X, i.e $\zeta (0)\ge \zeta (x),\mathrm{\forall }x\in X$. Thus $\zeta (x\cdot z)\ge \zeta (y\cdot z)$…….$(\star )$. Again by Theorem 2.9 $\zeta (y\cdot z)\ge min\{\zeta (y\cdot x),\zeta ((y\cdot x)\star (y\cdot z))\}$, since X is a pseudo-UP algebra then we have $(x\cdot z)\star ((y\cdot x)\star (x\cdot z))=0$. Which implies that $\zeta (y\cdot z)\ge min\{\zeta (y\cdot x),\zeta (x\cdot z)\}$ implies $\zeta (y\cdot z)\ge min\{\zeta (0),\zeta (x\cdot z)\}=\zeta (x\cdot z)$, since $\zeta (y\cdot x)=\zeta (0)$. Thus $\zeta (y\cdot z)\ge \zeta (x\cdot z)$……..$(\star \star )$. From $(\star )$ and $(\star \star )$ we have $\zeta (y\cdot z)=\zeta (x\cdot z)$. Similarly, we have $\zeta (y\star z)=\zeta (x\star z)$. Now ${\zeta }^x(z)=min\{\zeta (x\cdot z),\zeta (z\cdot x)\}=min\{\zeta (y\cdot z),\zeta (z\star y)\}={\zeta }^y(z),\mathrm{\forall }z\in X$. Therefore, ${\zeta }^x={\zeta }^y$.

Theorem 4.4.

Let ζ be a fuzzy pseudo-UP ideal of X and ϕ_ζ be a fuzzy congruence relation on X. Then $(X/{\varphi }_{\zeta },\cdot ,\star ,0)$ is a pseudo-UP algebra of X.

Proof.

For every ${\zeta }^x,{\zeta }^y\in X/{\varphi }_{\zeta }$, we define ${\zeta }^x\cdot {\zeta }^y={\zeta }^{x\cdot y}$ and ${\zeta }^x\star {\zeta }^y={\zeta }^{x\star y}$. Let ${\zeta }^x,{\zeta }^y,{\zeta }^z\in X/{\varphi }_{\zeta }$, we have $({\zeta }^y\cdot {\zeta }^z)\star (({\zeta }^x\cdot {\zeta }^y)\star ({\zeta }^x\cdot {\zeta }^z))={\zeta }^{(y\cdot z)\star ((x\cdot y)\star (x\cdot z))}={\zeta }^0$, since X is pseudo-UP algebra. Similarly, $({\zeta }^y\star {\zeta }^z)\cdot (({\zeta }^x\star {\zeta }^y)\cdot ({\zeta }^x\star {\zeta }^z))={\zeta }^{(y\star z)\cdot ((x\star y)\cdot (x\star z))}={\zeta }^0$, since X is pseudo-UP algebra. Let ${\zeta }^x\cdot {\zeta }^y={\zeta }^0$ and ${\zeta }^y\cdot {\zeta }^x={\zeta }^0$.

$\begin{array}{ll}& ⟹\zeta (x\cdot y)=\zeta (y\cdot x)=\zeta (0)\\ & ⟹min\{\zeta (x\cdot y),\zeta (0)\}andmin\{\zeta (y\cdot x),\zeta (0)\}& \\ & ⟹min\{\zeta (0\cdot (x\cdot y)),\zeta ((x\cdot y)\cdot 0)\}and\\ & min\{\zeta (0\cdot (y\cdot x)),\zeta ((y\cdot x)\cdot 0)\}.& \end{array}$

It follows that ${\varphi }_{\zeta }((x\cdot y),0)$ and ${\varphi }_{\zeta }((y\cdot x),0)$. This means, $\zeta (x\cdot y)=\zeta (0\cdot (x\cdot y))$ and $\zeta (0)=\zeta ((x\cdot y)\cdot 0)$ and $\zeta (y\cdot x)=\zeta (0\cdot (y\cdot x))$ and $\zeta (0)=\zeta ((y\cdot x)\cdot 0)$. That is ${\varphi }_{\zeta }(x,y)$. Thus ${\zeta }^x={\zeta }^y$. Similarly, Let ${\zeta }^x\star {\zeta }^y={\zeta }^0$ and ${\zeta }^y\star {\zeta }^x={\zeta }^0$.

$\begin{array}{lll}& ⟹\zeta (x\star y)=\zeta (y\star x)=\zeta (0)& \\ & ⟹min\{\zeta (x\star y),\zeta (0)\}and\\ & min\{\zeta (y\star x),\zeta (0)\}& \\ & ⟹min\{\zeta (0\star (x\star y)),\\ & \zeta ((x\star y)\star 0)\}and\\ & min\{\zeta (0\star (y\star x)),\zeta ((y\star x)\star 0)\}.& \end{array}$

It follows that ${\varphi }_{\zeta }((x\star y),0)$ and ${\varphi }_{\zeta }((y\star x),0)$. This means, $\zeta (x\star y)=\zeta (0\star (x\star y))$ and $\zeta (0)=\zeta ((x\star y)\star 0)$ and $\zeta (y\star x)=\zeta (0\star (y\star x))$ and $\zeta (0)=\zeta ((y\star x)\star 0)$. That is ${\varphi }_{\zeta }(x,y)$. Thus ${\zeta }^x={\zeta }^y$. Let ${\zeta }^x,{\zeta }^y,{\zeta }^0\in X/\zeta$, then $({\zeta }^y\cdot {\zeta }^0)\star {\zeta }^x=\zeta {(}^{y\cdot 0)\star x}={\zeta }^x$ and $({\zeta }^y\star {\zeta }^0)\cdot {\zeta }^x=\zeta {(}^{y\star 0)\cdot x}={\zeta }^x$. Therefore, $(X/{\varphi }_{\zeta ,}\cdot ,\star ,0)$ is a pseudo-UP algebra of X.

5. Conclusion

In this paper, we introduced the notion fuzzy congruence relation on pseudo-UP algebra, considering the concept of fuzzy congruence relation in certain algebraic structures and investigated some associated properties. We characterized the fuzzy congruence relation on X by their level set. The intersection of two fuzzy congruence and the Cartesian product of two fuzzy congruence are discussed. We provided the relationship between the set of fuzzy pseudo-UP ideal and the set of fuzzy congruence relation in pseudo-UP algebra. Then, we show that for each fuzzy ideal ζ and fuzzy congruence ϕ_ζ generated by ζ, there is an associated algebra $X/{\varphi }_{\zeta }$ that is a pseudo-UP algebra. Moreover, using fuzzy pseudo-UP ideal and fuzzy congruence relation ϕ_ζ generated by ζ, we constructed a congruence relation on a pseudo-UP algebra. Since congruence relations are interesting and important subjects in fuzzy logic, we hope that we helped to open new fields to anyone that is interested to studying of these concepts in pseudo-UP algebra.

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No potential conflict of interest was reported by the authors.

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Funding

The authors received no direct funding from public or non-public organizations for this research work.

Notes on contributors

Alachew Amaneh Mechdesro

Alachew Amaneh Mechderso received the MSc degree in mathematics from Ambo University, in 2018. He is currently a lecturer at Department of Mathematics, Kebri Dehar University, Ethiopia, and also a PhD student at Department of Mathematics, Bahir Dar University, Ethiopia. His research interests include Boolean algebra, fuzzy set theory and soft set theory. Alachew Amaneh Mechderso is a PhD scholar at the Department of Mathematics, Science College, Bahir Dar University, Ethiopia. His research interest includes the development of a Fuzzy pseudo-UP Algebra.