Fuzzy pseudo-UP ideal of pseudo-UP algebra

ABSTRACT

Keeping in view the importance of fuzzy set theory and pseudo-UP algebra, in this article the notion of fuzzy pseudo-UP subalgebra (respectively, fuzzy pseudo-UP filter and fuzzy pseudo-UP ideal) of pseudo-UP algebra is acquainted, and some basic properties of the defined notion are discussed. Then, we proved its generalizations and characterizations of pseudo-UP subalgebra (respectively, pseudo-UP filter and pseudo-UP ideal) and also we prove that every fuzzy pseudo-UP subalgebra is not fuzzy pseudo-UP filter and every fuzzy pseudo-UP filter is not fuzzy pseudo-UP ideal. Furthermore, we discussed the relations between fuzzy pseudo-UP subalgebra (respectively, fuzzy pseudo-UP filter, and fuzzy pseudo-UP ideal) and their level sets. Wherever necessary, the concepts for the defined notion are elaborated by examples.

KEYWORDS:

• pseudo-UP algebra
• fuzzy pseudo-UP subalgebra
• fuzzy pseudo-UP filter
• fuzzy pseudo-UP ideal

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-algebras. Y. Jun et al. (2006) introduced the concept of pseudo-BCI ideal in pseudo-BCI algebra. As a generalization of KU-algebra, Iampan (2017) introduced a new algebraic structure called UP-algebra. (The reader can learn more about these algebra in article Chanwit and Utsanee (2009).) In UP-algebra, he studied ideal and congruence. He also investigated the concept of UP-algebra homomorphism and some related properties. Romano (2020a) introduced pseudo-UP algebra and derived basic properties. Romano (2020a) introduced the notions of pseudo-UP ideal and pseudo-UP filter of pseudo-UP algebra. Oner et al. (2021) introduced Sheffer Stroke UP-algebra (in short, SUP-algebra) and studied its properties. And also Oner demonstrated the Cartesian product of two SUP-algebras is a SUP-algebra. After presenting SUP-subalgebra, Oner et al. defined SUP-homomorphisms between SUP-algebra.

A fuzzy subset of a set X is a function that connects X to the closed interval [0, 1]. Zadeh (1965) was the first to consider the concept of a fuzzy subset of a set. Zadeh was the first to develop the fuzzy set theories, and others have found numerous applications in mathematics and other fields. Several researchers investigated the generalization of the concept of fuzzy sets in algebraic classes such as BCC/BCK/BCI/KU-algebras (see for more information Dudek and Zhang (1999), Lele et al. (2001), Y. B. Jun et al. (2004), Y. B. Jun (2005), Kordi and Moussavi (2007), and Mostafa et al. (2011)). Iampan (2017) defined a new algebraic structure known as a UP-algebra. The concepts of fuzzy subalgebras and fuzzy ideal are crucial in studying the various logical algebras. Somjanta et al. (2016) introduced and investigated some of the properties of fuzzy UP-subalgebra and fuzzy UP-ideal of UP-algebra. Prabpayak et al. (2018) investigated some results on fuzzy pseudo KU-algebra. Oner et al. (2022) applied the concepts of (sup-hesitant) fuzzy SUP-subalgebras and fuzzy duplex SUP-sets on Sheffer stroke UP-algebra (in short, SUP-algebra). After defining (sup- hesitant) fuzzy SUP-subalgebra and fuzzy duplex SUP-sets of a SUP-algebra, Oner et al. studied some of their properties and analysed whether the intersection or union of these subalgebra is a (sup-hesitant) fuzzy SUP-subalgebra of a SUP-algebra. Oner et al. proofed that the level sets of (sup-hesitant) fuzzy SUP-subalgebra of a SUP-algebra are its SUP-subalgebras

In this present paper, the notions of fuzzy pseudo-UP subalgebra, fuzzy pseudo-UP filter and fuzzy pseudo ideal of pseudo-UP algebra are introduced, and some algebraic properties of fuzzy pseudo-UP subalgebra (respectively, filter, ideal) of pseudo-UP algebra are studied. We proved generalizations and characterizations of fuzzy pseudo-UP subalgebra (respectively, filter, ideal) of pseudo-UP algebra. Finally, the relationships between fuzzy pseudo-UP subalgebra (respectively, fuzzy pseudo-UP filter and fuzzy pseudo-UP ideal) and their level subsets are discussed.

2. Preliminaries

In this section, we discussed some fundamental concepts and important basic results associated with our study, like pseudo-UP algebra, pseudo-UP ideal, pseudo-UP filter, and subalgebra of pseudo-UP algebra. Furthermore, fuzzy set, level subsets of ζ such as the upper t-level, the upper t-strong level, the lower t-level and the lower t-strong level are also discussed. These concepts are taken from Zadeh (1965), Rosenfeld (1971), Somjanta (Somjanta et al., 2016), Romano (2020a, 2020b), and Yousef and Khalaf (2022).

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)	$(y\cdot z)\star ((x\cdot y)\star (x\cdot z))=0$  and  $(y\star z)\cdot ((x\star y)\cdot (x\star z))=0$,
2)	$x\cdot y=0=y\cdot x\text{and}x\star y=0=y\star x⟹x=y,$
3)	$(y\cdot 0)\star x=x\text{and}(y\star 0)\cdot x=x$,
4)	$x⩽y⟺x\cdot y=0\text{and}x⩽y⟺x\star y=0$.

Lemma 2.2.

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

1)	$x\cdot 0=0\text{and}x\star 0=0$
2)	$0\cdot x=x\text{and}0\star x=x$ and
3)	$x\cdot x=0\text{and}x\star x=0$

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)	$x⩽y\star x$.

Definition 2.4.

A pseudo-UP subalgebra of X is a non-empty subset S of a pseudo-UP algebra if it satisfies the following conditions:

1)	$0\in \mathrm{S},$
2)	S is closed under two binary operations $"\cdot "$ and $"\star "$

Definition 2.5.

A pseudo-UP filter of X is a nonempty subset F of a pseudo-UP algebra X that has the following conditions : $\mathrm{\forall }x,y,z\in X$

1)	$0\in F$
2)	$x\in F\text{and}x\cdot y\in F⟹y\in F$ and
3)	$x\in F\text{and}x\star y\in F⟹y\in F$

Definition 2.6.

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)	$x\cdot (y\star z)\in J\text{and}y\in J⟹x\cdot z\in J$ and
3)	$x\star (y\cdot z)\in J\text{and}y\in J⟹x\star z\in J$

The following theorem describes the distinguishing characteristics of these substructures.

Theorem 2.7.

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

1)	$y\star z\in J\text{and}y\in J⟹z\in J$
2)	$y\in J⟹x\star y\in J$
3)	$y\cdot z\in J\text{and}y\in J⟹z\in J$ and
4)	$y\in J⟹x\cdot y\in J$

Definition 2.8.

Let X be any nonempty set. A mapping ζ : $X\to [0,1]$ is called a fuzzy subset of X.

Definition 2.9.

Suppose that ζ is a fuzzy set in $X,\mathrm{\forall }t\in [0,1]$. The set

(1) $\begin{array}{ll}U(\zeta ;t)& =\{x\in X\mid \zeta (x)\ge t\}\text{and}{U}^{+}(\zeta ;t)\\ & =\{x\in X\mid \zeta (x)>t\}\end{array}$(1)

are called an upper t-level subset and an upper t-strong level subset of ζ, respectively. The set

(2) $\begin{array}{ll}L(\zeta ;t)& =\{x\in X\mid \zeta (x)\le t\}\text{ and }{L}^{-}(\zeta ;t)\\ & =\{x\in X\mid \zeta (x)<t\}\end{array}$(2)

are called a lower t-level subset and a lower t-strong level subset of ζ, respectively.

Lemma 2.10.

Consider ζ is a fuzzy set in X. Then, $\mathrm{\forall }x,y\in X$,

1)	$1-max\{\zeta (x),\zeta (y)\}=min\{1-\zeta (x),1-\zeta (y)\}$
2)	$1-min\{\zeta (x),\zeta (y)\}=max\{1-\zeta (x),1-\zeta (y)\}$

Definition 2.11.

A fuzzy set ζ in X has sup property if for any nonempty subset T of X, there exists $t_0\in T$ such that $\zeta \left(t_0\right)=sup\{\zeta (t){\}}_{t\in T}$.

Definition 2.12.

Let f be any function from a set X to a set $T;\zeta$ be any fuzzy subset of X, and η be any fuzzy subset of T. The image of ζ under f, denoted by $f(\zeta )$, is a fuzzy subset of T defined by

(3) $f(\zeta )(y)=\{\begin{array}{ll}sup\{\zeta (x)|x\in f^{-1}(y)\}& \text{ if }{f}^{-1}(y)\ne \mathrm{\varnothing }\\ 0& \text{ otherwise },\end{array}$(3)

where $y\in T$. The pre-image of η under f, symbolized by $f^{-1}(\eta )$, is a fuzzy subset of T defined by

(4) $f^{-1}(\eta )(x)=\eta (f(x)),\text{ for all }x\in X.$(4)

3. Fuzzy pseudo-UP subalgebra

In what follows, let X denote a pseudo-UP algebra unless otherwise specified. In this section, we study fuzzy pseudo-UP subalgebra of pseudo-UP algebra and study some basic properties of fuzzy pseudo-UP subalgebra of pseudo-UP algebra.

Definition 3.1.

A fuzzy subset ζ of a pseudo-UP algebra X is called fuzzy pseudo-UP subalgebra of X, if it fulfills the following axioms: $\mathrm{\forall }x,y\in X,$

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

Example 3.2.

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

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

Clearly, $(X,\cdot ,\star ,0)$ is a pseudo-UP subalgebra of X. We define a fuzzy subset ζ in X as follow

$\zeta (0)=1,\zeta (b)=0.9,\zeta (a)=0.3=\zeta (c)$. Obviously, ζ is a fuzzy pseudo-UP subalgebra of X.

Example 3.3.

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

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

Clearly,$(X,\cdot ,\star ,0)$ is a pseudo-UP algebra. We define a fuzzy subset ζ in X as follows:

$\zeta (0)=0.6,\zeta (a)=0.2,\zeta (b)=0.2$. Clearly, ζ is fuzzy pseudo-UP subalgebra of X.

Lemma 3.4.

If ζ is fuzzy pseudo-UP subalgebra of X. Then $\zeta (0)\ge \zeta (x),\mathrm{\forall }x\in X$.

Proof.

Suppose that ζ is a fuzzy pseudo-UP subalgebra of X. Then

$\zeta (0)=\zeta (x\cdot x)$, by Lemma 2.2.

$\ge min\{\zeta (x),\zeta (x)\}$

$=\zeta (x)$, by Definition 3.1.

Similarly, $\zeta (0)=\zeta (x\star x)$

$\ge min\{\zeta (x),\zeta (x)\}$

$=\zeta (x)$.

Therefore, $\zeta (0)\ge \zeta (x).$

Theorem 3.5.

If ζ and η are fuzzy pseudo-UP subalgebra of X. Then $\zeta \cap \eta$ is also fuzzy pseudo-UP subalgebra of X.

Proof.

Suppose ζ and η are fuzzy pseudo-UP subalgebra of X and $\mathrm{\forall }x,y\in X$ .

$\begin{array}{ll}(\zeta \cap \eta )(x\cdot y)=& min\{\zeta (x\cdot y),\eta (x\cdot y)\}\\ & \ge min\{min\{\zeta (x),\zeta (y)\},\\ & min\{\eta (x),\eta (y)\}\}& \\ & =min\{min\{\zeta (x),\eta (x)\},\\ & min\{\zeta (y),\eta (y)\}\}& \\ & =min\{\zeta \cap \eta (x),\zeta \cap \eta (y)\}.& \\ & ⟹(\zeta \cap \eta )(x\cdot y)\ge min\{(\zeta \cap \eta )(x),\\ & (\zeta \cap \eta )(y)\}.\end{array}$

$\begin{array}{ll}(\zeta \cap \eta )(x\star y)=& min\{\zeta (x\star y),\eta (x\star y)\}\\ & \ge min\{min\{\zeta (x),\zeta (y)\},\\ & min(\eta (x),\eta (y)\}\}& \\ & =min\{min(\zeta (x),\eta (x)\},\\ & min\{\zeta (y),\eta (y)\}\}& \\ & =min\{\zeta \cap \eta (x),\zeta \cap \eta (y)\}.& \\ & ⟹(\zeta \cap \eta )(x\star y)\ge min\{(\zeta \cap \eta )(x),\\ & (\zeta \cap \eta )(y)\}.\end{array}$

Therefore, $\zeta \cap \eta$ is fuzzy pseudo-UP subalgebra of X.

Corollary 3.6.

Let $\{{\zeta }_i/i\in \mathrm{\Lambda }\}$ be a family of fuzzy pseudo-UP subalgebra of X. Then $\bigcap _{i\in \mathrm{\Lambda }}{\zeta }_i$ is also fuzzy pseudo-UP subalgebra of X.

Remark 3.7.

The union of any two fuzzy pseudo-UP subalgebra of a pseudo-UP algebra X is not necessarily a fuzzy pseudo-UP subalgebra of X.

Example 3.8.

Let $X=\{0,1,2,3\}$ be a set with a binary operation $"\cdot "$ and $"\star "$ defined by the following cayley table:

$\begin{array}{lll}\begin{array}{lllll}\cdot & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 3& 0\\ 2& 0& 1& 0& 0\\ 3& 0& 1& 3& 0\end{array}& & \begin{array}{lllll}\star & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 3& 0\\ 2& 0& 1& 0& 0\\ 3& 0& 1& 3& 0\end{array}\end{array}$

Clearly, $(X,\cdot ,\star ,0)$ is a pseudo-UP algebra of X. We define $\zeta :X\to [0,1]$ as follows, $\zeta (0)=0.9,\zeta (1)=0.7,\zeta (2)=0.6,\zeta (3)=0.2$ and define $\eta :X\to [0,1]$ as follows, $\eta (0)=0.8,\eta (1)=0.8,\eta (2)=0.7,\eta (3)=0.1$

$(\zeta \cup \eta )(1\cdot 2)=max\{\zeta (1\cdot 2),\eta (1\cdot 2)\}\text{break}=max\{\zeta (3),\eta (3)\}=max\{0.2,0.1\}=0.2$.

$(\zeta \cup \eta )(1\star 2)=max\{\zeta (1\star 2),\eta (1\star 2)\}\text{break}=max\{\zeta (3),\eta (3)\}=max\{0.2,0.1\}=0.2$.

Hence, $(\zeta \cup \eta )(1\cdot 2)=0.2$ and $(\zeta \cup \eta )(1\star 2)=0.2$ ……………………….$(\star )$

$\begin{array}{ll}(\zeta \cup \eta )(1\cdot 2)& =max\{\zeta (1\cdot 2),\eta (1\cdot 2)\}\\ & \ge max\{min\{\zeta (1),\zeta (2)\},\\ & min\{\eta (1),\eta (2)\}\}\\ & =min\{max\{\zeta (1),\eta (1)\},max\{\zeta (2),\eta (2)\}\}\\ & =min\{max\{0.7,0.8\},max\{0.6,0.7\}\}\\ & =min\{0.8,0.7\}=0.7.\end{array}$

$\begin{array}{ll}(\zeta \cup \eta )(1\star 2)& =max\{\zeta (1\star 2),\eta (1\star 2)\}\\ & \ge max\{min\{\zeta (1),\zeta (2)\},min\{\eta (1),\eta (2)\}\}\\ & =min\{max\{\zeta (1),\eta (1)\},max\{\zeta (2),\eta (2)\}\}\\ & =min\{max\{0.7,0.8\},max\{0.6,0.7\}\}\\ & =min\{0.8,0.7\}=0.7.\end{array}$

Thus $(\zeta \cup \eta )(1\cdot 2)=0.7$ and $(\zeta \cup \eta )(1\star 2)=0.7$……………………..$(\star \star )$

From $(\star )\text{and}(\star \star )$ we get $=0.2\le 0.7$ which is contradict to Definition 3.1. This implies that the union of any two fuzzy pseudo-UP subalgebra of X is may not be a fuzzy pseudo-UP subalgebra of X.

Proposition 3.9.

Let X be a pseudo-UP algebra. Then the union of two fuzzy pseudo-UP subalgebra of X is a fuzzy pseudo-UP subalgebra of X if it satisfies the following conditions $\mathrm{\forall }x,y\in X$.

1)	$x\cdot y=y$ or $x\cdot y=x$ or $x\cdot y=0$,
2)	$x\star y=y$ or $x\star y=x$ or $x\star y=0$.

Proof.

Let ${\zeta }_1,{\zeta }_2$ be fuzzy pseudo-UP subalgebra of X. If $x\cdot y=y$ and $x\star y=y$, $\mathrm{\forall }x,y\in X$. Then,

$\begin{array}{ll}({\zeta }_1\cup {\zeta }_2)(x\cdot y)& =max\{{\zeta }_1(x\cdot y),{\zeta }_2(x\cdot y)\}\\ & \ge max\{min\{{\zeta }_1(x),{\zeta }_1(y)\},\\ & min\{{\zeta }_2(x),{\zeta }_2(y)\}\}\\ & =min\{max\{{\zeta }_1(x),{\zeta }_2(x)\},\\ & max\{{\zeta }_1(y),{\zeta }_2(y)\}\}\\ & =min\{({\zeta }_1\cup {\zeta }_2)(x),({\zeta }_1\cup {\zeta }_2)(y)\}.\\ & ⟹({\zeta }_1\cup {\zeta }_2)(y)\ge min\{({\zeta }_1\cup {\zeta }_2)(x),\\ & ({\zeta }_1\cup {\zeta }_2)(y)\}.\end{array}$

$\begin{array}{ll}({\zeta }_1\cup {\zeta }_2)(x\star y)& =max\{{\zeta }_1(x\star y),{\zeta }_2(x\star y)\}\\ & \ge max\{min\{{\zeta }_1(x),{\zeta }_1(y)\},\\ & min\{{\zeta }_2(x),{\zeta }_2(y)\}\}\\ & =min\{max\{{\zeta }_1(x),{\zeta }_2(x)\},\\ & max\{{\zeta }_1(y),{\zeta }_2(y)\}\}\\ & =min\{({\zeta }_1\cup {\zeta }_2)(x),({\zeta }_1\cup {\zeta }_2)(y)\}.\\ & ⟹({\zeta }_1\cup {\zeta }_2)(y)\ge min\{({\zeta }_1\cup {\zeta }_2)(x),\\ & ({\zeta }_1\cup {\zeta }_2(y)\}.\end{array}$

Similarly, if $x\cdot y=x=x\star y=$ and $x\cdot y=0=x\star y$. Then $({\zeta }_1\cup {\zeta }_2)(x\cdot y)\ge min\{({\zeta }_1\cup {\zeta }_2)(x),({\zeta }_1\cup {\zeta }_2)(y)\}$ and $({\zeta }_1\cup {\zeta }_2)(x\star y)\ge min\{({\zeta }_1\cup {\zeta }_2)(x),({\zeta }_1\cup {\zeta }_2)(y)\}.$

Theorem 3.10

Let S be a non-empty subset of a pseudo-UP algebra X and a fuzzy subset ζ of X is defined by

$\begin{array}{l}\zeta (x)=\{\begin{array}{ll}p& \text{if }x\in S,\\ q& \text{if}x\notin S,\mathrm{\forall }p,q\in [0,1],\text{with}p\ge q.\end{array}\end{array}$

Then ζ is a fuzzy pseudo-UP subalgebra of X if and only if S is a pseudo-UP subalgebra of X.

Proof.

Suppose that ζ is a fuzzy pseudo-UP subalgebra of X. Then, we want to show that S is a pseudo-UP subalgebra of X. Since X is a pseudo-UP algebra then $0\in X$ and S is non empty subset of X such that $x\in S$. Now, $\zeta (0)\ge \zeta (x)=p$. Hence, $0\in S$. Next, for any $x,y\in X$ such that $x,y\in S$. Since ζ is a fuzzy pseudo-UP subalgebra of X, we have $\zeta (x\cdot y)\ge min\{\zeta (x),\zeta (y)\}=min\{p,p\}=p$. Similarly, $\zeta (x\star y)\ge min\{\zeta (x),\zeta (y)\}=min\{p,p\}=p.$ Which implies that $0,x\cdot y,x\star y\in S$. Therefore, S is a pseudo-UP subalgebra of pseudo-UP algebra X.

Conversely, assume that S is a pseudo-UP subalgebra of X. We need to show that ζ is a fuzzy pseudo-UP subalgebra of X. Consider the following cases:

Case 1: If $x,y\in S$. Then $x\cdot y,x\star y\in S$, since S is pseudo-UP subalgebra of X. Thus $\zeta (x\cdot y)=p\ge min\{\zeta (x),\zeta (y)\}$ and $\zeta (x\star y)=p\ge min\{\zeta (x),\zeta (y)\}.$

Case 2: If $x\in S$ and $y\notin S$, then $\zeta (x)=p$ and $\zeta (y)=q$. Thus $\zeta (x\cdot y)\ge min\{p,q\}=min\{\zeta (x),\zeta (y)\}$. Similary, $\zeta (x\star y)\ge min\{p,q\}=min\{\zeta (x),\zeta (y)\}.$

Case 3: If $x\notin S$ and $y\in S$, then interchanging the role of x and y in case 2 yields similar results. $\zeta (x\cdot y)\ge min\{\zeta (x),\zeta (y)\}$ and $\zeta (x\star y)\ge min\{\zeta (x),\zeta (y)\}.$

Case 4: If $x,y\notin S$, then $\zeta (x)=q=\zeta (y)$ then, $\zeta (x\cdot y)\ge q=min\{\zeta (x),\zeta (y)\}$ and $\zeta (x\star y)\ge q=min\{\zeta (x),\zeta (y)\}$. Therefore, ζ is a fuzzy pseudo-UP subalgebra of pseudo-UP algebra X.

Theorem 3.11.

Assume S is a non-empty subset of X. Then χ_S is fuzzy pseudo-UP subalgebra of X if and only if S is pseudo-UP subalgebra of X.

Theorem 3.12.

Let $U(\zeta :t)$ is non-empty and $\mathrm{\forall }t\in [0,1]$. A fuzzy subset ζ of pseudo-UP algebras X is fuzzy pseudo-UP subalgebra of X, if and only if $U(\zeta :t)$ is pseudo-UP subalgebra of X.

Proof.

Suppose that ζ is fuzzy pseudo-UP subalgebra of X. We show that $U(\zeta :t)$ is a pseudo-UP subalgebra of X,$\mathrm{\forall }t\in [0,1]$. Since X is pseudo-UP algebra, then $0\in X$ implies that $\zeta (0)\ge t.$

$⟹0\in U(\zeta :t)$

$⟹U(\zeta :t)\ne \mathrm{\varnothing },t\in [0,1]$.

Let $x,y\in U(\zeta :t)$. Then, $\zeta (x)\ge t$ and $\zeta (y)\ge t$. We have

$\zeta (x\cdot y)\ge min\{\zeta (x),\zeta (y)\}\ge t$*                          $(\star )$

$\zeta (x\star y)\ge min\{\zeta (x),\zeta (y)\}\ge t$.*                          $(\star \star )$

From $(\star )$ and $(\star \star )$ we get $x\cdot y,x\star y\in U(\zeta :t).$ Therefore, $U(\zeta :t)$ is a pseudo-UP subalgebra of X.

Conversely, assume $U(\zeta :t)$ is a pseudo-UP subalgebra of X and $\cup (\zeta :t)$ is non-empty. We prove that ζ is a fuzzy pseudo-UP subalgebra of X. For $x,y\in X$, take $\zeta (x)=t_1$ and $\zeta (y)=t_2$ and $t=min\{t_1,t_2\}$,$\mathrm{\forall }t,t_1,t_2\in [0,1]$.

Since $U(\zeta :t)$ is a pseudo-UP subalgebra of X, so that $x,y,x\cdot y,x\star y\in U(\zeta :t)$. Then, $\zeta (x\cdot y)\ge t=min\{t_1,t_2\}$=min$\{\zeta (x),\zeta (y)\}$. Thus $\zeta (x\cdot y)\ge min\{\zeta (x),\zeta (y)\}$. Similarly, $\zeta (x\star y)\ge$ min$\{\zeta (x),\zeta (y)\}.$

Therefore, ζ is fuzzy pseudo-UP subalgebra of X.

Corollary 3.13.

Let $U^{+}(\zeta :t)$ is non-empty and $\mathrm{\forall }t\in [0,1]$. A fuzzy subset of pseudo-UP algebra of X is a fuzzy pseudo-UP subalgebra of X if and only if $U^{+}(\zeta :t)$ is a pseudo-UP subalgebra of X.

Theorem 3.14.

Let $L(\zeta :t)$ is non-empty and $\mathrm{\forall }t\in [0,1]$. Let ζ be a fuzzy subset of X. Then, ζ^c is a fuzzy pseudo-UP subalgebra of X if and only if $L(\zeta :t)$ is a pseudo-UP subalgebra of X.

Proof.

Suppose ζ^c is a fuzzy pseudo-UP subalgebra of X. We need to prove that $L(\zeta :t)$ is a pseudo-UP subalgebra. Since $L(\zeta :t)\ne \mathrm{\varnothing }$, then $a\in L(\zeta :t)$ implies that $\zeta (a)\le t$. From our suppostion, we have ${\zeta }^c(0)\ge {\zeta }^c(a)$ implies that $1-\zeta (0)\ge 1-\zeta (a)$. Thus $\zeta (0)\le \zeta (a)\le t$. Hence $0\in L(\zeta :t).$ Let $x,y\in L(\zeta :t)$ then, $\zeta (x)\le t$ and $\zeta (y)\le t,\mathrm{\forall }t\in [0,1]$. Since ζ^c is a fuzzy pseudo-UP subalgebra of X. Then, we have ${\zeta }^c(x\cdot y)\ge min\{{\zeta }^c(x),{\zeta }^c(y)\}$ and ${\zeta }^c(x\star y)\ge min\{{\zeta }^c(x),{\zeta }^c(y)\}.$

$\begin{array}{ll}1-\zeta (x\cdot y)& \ge min\{1-\zeta (x),1-\zeta (y)\}\\ & =1-max\{\zeta (x),\zeta (y)\}.\\ & ⟹1-\zeta (x\cdot y)\ge 1-max\{\zeta (x),\zeta (y)\}\\ & ⟹\zeta (x\cdot y)\le max\{\zeta (x),\zeta (y)\}\\ & \le max\{t,t\}=t.& \\ & ⟹\zeta (x\cdot y)\le t.& \\ & ⟹x\cdot y\in L(\zeta :t).\end{array}$

$\begin{array}{ll}1-\zeta (x\star y)& \ge min\{1-\zeta (x),1-\zeta (y)\}\\ & =1-max\{\zeta (x),\zeta (y)\}.& \\ & ⟹1-\zeta (x\star y)\ge 1-max\{\zeta (x),\zeta (y)\}& \\ & ⟹\zeta (x\star y)\le max\{\zeta (x),\zeta (y)\}\\ & \le max\{t,t\}=t.& \\ & ⟹\zeta (x\star y)\le t.& \\ & ⟹x\star y\in L(\zeta :t).\end{array}$

Therefore, $L(\zeta :t)$ is a pseudo-UP subalgebra of X.

Conversely, assume $L(\zeta :t)$ is a pseudo-UP subalgebra of X and $L(\zeta :t)$ is non-empty. We need to prove that ζ is a fuzzy pseudo-UP subalgebra of X. For $x,y\in X$, choose $t=max\{\zeta (x),\zeta (y)\}$. Then $\zeta (x)\le t$, $\zeta (y)\le t$ implies that $x,y\in L(\zeta :t).$ Since $L(\zeta :t)$ is a pseudo-UP subalgebra of X, $x\cdot y,x\star y\in L(\zeta :t).$ Then, $\zeta (x\cdot y)\le t=max\{\zeta (x),\zeta (y)\}$. We have,

$\begin{array}{ll}{\zeta }^c(x\cdot y)& =1-\zeta (x\cdot y)\\ & \ge 1-max\{\zeta (x),\zeta (y)\}& \\ & =min\{1-\zeta (x),1-\zeta (y)\}& \\ & =min\{{\zeta }^c(x),{\zeta }^c(y)\}.& \\ & ⟹{\zeta }^c(x\cdot y)\ge min\{{\zeta }^c(x),{\zeta }^c(y)\}.\end{array}$

Similarly, $\zeta (x\star y)\le t=max\{\zeta (x),\zeta (y)\}$. We have,

$\begin{array}{ll}{\zeta }^c(x\star y)& =1-\zeta (x\star y)\\ & \ge 1-max\{\zeta (x),\zeta (y)\}& \\ & =min\{1-\zeta (x),1-\zeta (y)\}& \\ & =min\{{\zeta }^c(x),{\zeta }^c(y)\}.& \\ & ⟹{\zeta }^c(x\star y)\ge min\{{\zeta }^c(x),{\zeta }^c(y)\}.\end{array}$

Therefore, ζ^c is a fuzzy pseudo-UP subalgebra of X.

Corollary 3.15.

Let ζ be fuzzy subset of pseudo-UP algebra of X and let $L^{-}(\zeta :t)$ is non-empty. Then ζ^c is fuzzy pseudo-UP subalgebra of X if and only if $L^{-}(\zeta :t)$ is pseudo-UP subalgebra of X, $\mathrm{\forall }t\in [0,1]$.

Theorem 3.16.

Let $(A,{\cdot }_A,{\star }_A,0_A)$ and $(B,{\cdot }_B,{\star }_B,0_B)$ be pseudo-UP algebra. Let $f:A\to B$ be homomorphism and η be a fuzzy pseudo-UP subalgebra of B. Then $f^{-1}(\eta )$ is a fuzzy pseudo-UP subalgebra of A.

Proof.

Let η be a fuzzy pseudo-UP subalgebra of B and for any $x,y\in A$. Then,

$\begin{array}{ll}{f}^{-1}(\eta )(x{\cdot }_{A}y)& =\eta (f(x{\cdot }_{A}y))\\ & =\eta (f(x{\cdot }_{A}y))& \\ & =\eta (f(x){\cdot }_{B}f(y))& \\ & \ge min\{\eta (f(x)),\eta (f(y))\}& \\ & =min\{f^{-1}(\eta )(x),(f^{-1}(\eta )(y))\}.& \\ & ⟹f^{-1}(\eta )(x{\cdot }_{A}y)\\ & \ge min\{f^{-1}(\eta )(x),f^{-1}(\eta )(y)\}.\end{array}$

$\begin{array}{ll}{f}^{-1}(\eta )(x{\star }_{A}y)& =\eta (f(x{\star }_{A}y))\\ & =\eta (f(x{\star }_{A}y))\\ & =\eta (f(x){\star }_{B}f(y))& \\ & \ge min\{\eta (f(x)),\eta (f(y))\}& \\ & =min\{f^{-1}(\eta )(x),(f^{-1}(\eta )(y))\}.\\ & ⟹f^{-1}(\eta )(x{\star }_{A}y)\\ & \ge min\{f^{-1}(\eta )(x),f^{-1}(\eta )(y)\}.\end{array}$

Hence, $f^{-1}(\eta )$ is a fuzzy pseudo-UP subalgebra of A.

Theorem 3.17.

Let $(A,\cdot ,\star ,0)$ and $(B,\cdot ,\star ,0)$ be pseudo-UP algebra. A mapping $f:A\to B$ is surjective and ζ is a fuzzy pseudo-UP subalgebra of A. Then $f(\zeta )$ is a fuzzy pseudo-UP subalgebra of B, provided that the sup property holds.

Proof.

Let ζ be a fuzzy pseudo-UP subalgebra of A and let $x,y\in B$. Then, there exist $c,d\in A$ such that $f(c)=x,f(d)=y$ and ζ holds sup property that is $\zeta (a)=sup\{\zeta (t)|t\in f^{-1}(x)\}$ and $\zeta (b)=sup\{\zeta (t)|t\in f^{-1}(y)\}$ for some $a,b\in A$. Since f is surjective, $a\cdot b\in f^{-1}(x\cdot y)$ and $a\star b\in f^{-1}(x\star y).$ Hence,

$\begin{array}{ll}f(\zeta )(x\cdot y)& \ge \zeta (a\cdot b)\\ & \ge min\{\zeta (a),\zeta (b)\}\\ & =min\{sup\zeta (t)|t\in f^{-1}(x),\\ & sup\zeta (t)|t\in f^{-1}(y)\}& \\ & =min\{f(\zeta )(x),f(\zeta )(y)\}.\end{array}$

Similarly,

$\begin{array}{ll}f(\zeta )(x\star y)& \ge \zeta (a\star b)\\ & \ge min\{\zeta (a),\zeta (b)\}& \\ & =min\{sup\zeta (t)|t\in f^{-1}(x),\\ & sup\zeta (t)|t\in f^{-1}(y)\}\\ & =min\{f(\zeta )(x),f(\zeta )(y)\}.\end{array}$

Therefore, $f(\zeta )$ is a fuzzy pseudo-UP subalgebra of B.

Proposition 1.

Let $A,B,C$ be pseudo-UP algebra. Let $f:A\to B$ and $g:B\to C$ be pseudo-UP homomorphism. Then the following holds:

1)	If $f,g$ are surjective, then $(g\circ f)(\zeta )$ is a fuzzy pseudo-UP subalgebra of C for any fuzzy pseudo-UP algebra ζ of A, provided that sup property holds.
2)	$(g\circ f{)}^{-1}(\eta )$ is a fuzzy pseudo-UP subalgebra of A for any fuzzy pseudo-UP subalgebra η of C.

Proof.

1) Let ζ be a fuzzy pseudo-UP subalgebra of A. We prove that $(g\circ f)(\zeta )$ is a fuzzy pseudo-UP subalgebra of C. For any $x,y\in C$, there exist $c,d\in A$ such that $(g\circ f)(c)=x$ and $(g\circ f)(d)=y$. Then, we have $\zeta (a)=sup\{\zeta (t)\mid t\in$ $(g\circ f{)}^{-1}(x)$ and $\zeta (b)=sup\{\zeta (t)\mid t\in (g\circ f{)}^{-1}(y)$ for some $a,b\in A$. Since $f,g$ are surjective, so is $g\circ f$, and hence $a\cdot b\in f^{-1}(x\cdot y)$ and $a\star b\in f^{-1}(x\star y)$. So we obtain

$\begin{array}{ll}(g\circ f)(\zeta )(x\cdot y)& \ge \zeta (a\cdot b)\\ & \ge min\{\zeta (a),\zeta (b)\}\\ & =min\{sup\zeta (t)|t\in (g\circ f{)}^{-1}(x),\\ & sup\zeta (t)|t\in (g\circ f{)}^{-1}(y)\}\\ & =min\{(g\circ f)(\zeta )(x),(g\circ f)(\zeta )(y)\}.\end{array}$

$\begin{array}{ll}(g\circ f)(\zeta )(x\star y)& \ge \zeta (a\star b)\\ & \ge min\{\zeta (a),\zeta (b)\}\\ & =min\{sup\zeta (t)|t\in (g\circ f{)}^{-1}(x),\\ & msup\zeta (t)|t\in (g\circ f{)}^{-1}(y)\}\\ & =min\{(g\circ f)(\zeta )(x),(g\circ f)(\zeta )(y)\}.\end{array}$

Therefore, $(g\circ f)(\zeta )$ is a fuzzy pseudo-UP subalgebra of C.

2) Let η be a fuzzy pseudo-UP subalgebra of C. For $x,y\in A$. Then,

$\begin{array}{ll}(g\circ f{)}^{-1}(\eta )(x\cdot y)& =\eta ((g\circ f)(x\cdot y))\\ & =\eta ((g\circ f)(x)\cdot (g\circ f)(y))& \\ & \ge min\{\eta ((g\circ f)(x)),\eta ((g\circ f)(y))\}& \\ & =min\{(g\circ f{)}^{-1}(\eta )(x),(g\circ f{)}^{-1}(\eta )(y)\}.\end{array}$

$\begin{array}{ll}(g\circ f{)}^{-1}(\eta )(x\star y)& =\eta ((g\circ f)(x\star y))\\ & =\eta ((g\circ f)\star (x)(g\circ f)(y))& \\ & \ge min\{\eta ((g\circ f)(x)),\eta ((g\circ f)(y))\}& \\ & =min\{(g\circ f{)}^{-1}(\eta )(x),(g\circ f{)}^{-1}(\eta )(y)\}.\end{array}$

Therefore, $(g\circ f{)}^{-1}(\eta )$ is a fuzzy pseudo-UP subalgebra of A.

Corollary 3.18.

Let $X_1,X_2,\dots X_n$ be a pseudo- UP algebra and $f_i:X_i\to X_{i+1}$ be pseudo-UP homomorphism for $n\in N$. Then,

1)	If fi is surjective for all $i\in \{1,\dots ,n-1\}$, then $(f_{n-1}\circ f_{n-2}\circ \cdots \circ f_1)(\zeta )$ is a fuzzy pseudo-UP subalgebra of Xn for any fuzzy pseudo-UP subalgebra ζ of X1.
2)	If ζ is a fuzzy pseudo-UP subalgebra of Xn. Then ${(f_{n-1}\circ f_{n-2}\circ \cdots \circ f_1)}^{-1}(\zeta )$ is a fuzzy pseudo-UP subalgebra of X1.

Theorem 2.

Let ζ be any fuzzy subset of X. Then $\zeta (x)=sup\{k\in [0,1]|x\in U(\zeta :k)\}$, where $x\in X$.

Proof.

Suppose $\alpha =sup\{k\in [0,1]|x\in U(\zeta :k)\}$ and let ϵ > 0 be arbitrary. Then $\alpha -ϵ<sup\{k|x\in U(\zeta :k)\}$

$\begin{array}{lll}& ⟹\alpha -ϵ<k\text{for some k such that }x\in U(\zeta :k)& \\ & ⟹\alpha -ϵ<\zeta (x),\text{ since}\zeta (x)\ge k& \\ & ⟹\alpha \le \zeta (x),\text{since}ϵ>0\text{is arbitrary}..(\star ).\\ & \text{Next we need to show that}\zeta (x)\le \alpha .\end{array}$

Assume $\zeta (x)=t$, then $x\in U(\zeta :t)$ and $t\in \{k\in [0,1]|x\in U(\zeta :k)\}$. Hence $t\le sup\{k|x\in U(\zeta :k)\}$ implies that $t\le \alpha$, it give that $\zeta (x)\le \alpha \dots \dots .(\star \star )$. From $(\star )$ and $(\star \star )$, we get $\zeta (x)=\alpha =sup\{k|x\in U(\zeta :k)\}$.

Theorem 3.

Let ζ be any fuzzy subset of X. Then the fuzzy subset ${\zeta }^{\star }$ of X defined by

${\zeta }^{\star }(x)=sup\{k\in [0,1]|x\in <U(\zeta :k)>\}$

where $x\in X$ is the least fuzzy pseudo-UP subalgebra of X that contains ζ and $<U(\zeta :k)>$ is the least pseudo-UP subalgebra contain $U(\zeta :k)$.

Proof.

Suppose ζ is a fuzzy subset of X and ${\zeta }^{\star }(x)=sup\{k|x\in <U(\zeta :k)>\},\mathrm{\forall }x\in X$. We need to show that ${\zeta }^{\star }$ is a fuzzy pseudo-UP subalgebra of X. Let $t\in Im{\zeta }^{\star },ϵ>0$ be arbitrary and $\alpha =t-ϵ$, then $x\in U({\zeta }^{\star }:t)$

$\begin{array}{lll}& ⟹t\le {\zeta }^{\star }(x)=sup\{k|x\in <U(\zeta :k)>\}& \\ & ⟹\alpha \le k\text{for some}k\text{such that}x\in <U(\zeta :k)>& \\ & ⟹U(\zeta :k)\subseteq U(\zeta :\alpha ),\text{by level subset of}\zeta & \\ & ⟹<U(\zeta :k)>\subseteq <U(\zeta :\alpha )>& \\ & ⟹x\in <U(\zeta :\alpha )>.\end{array}$

Hence, $U({\zeta }^{\star }:t)\subseteq <U(\zeta :\alpha )>.$………….$(\star ).$

Next, *  let $x\in <U(\zeta :\alpha )>$

$\begin{array}{lll}& ⟹\alpha \in \{k|x\in <U(\zeta :k)>\}& \\ & ⟹\alpha \le sup\{k|x\in <U(\zeta :k)>\}& \\ & ⟹t-ϵ\le {\zeta }^{\star }(x),\text{since}\alpha =t-ϵ& \\ & ⟹t\le {\zeta }^{\star }(x),\text{since}ϵ>0\text{is arbitrary}& \\ & ⟹x\in U({\zeta }^{\star }:t)& \\ & ⟹<U(\zeta :\alpha )>\subseteq U({\zeta }^{\star }:t)..(\star \star ).\end{array}$

From $(\star )$ and $(\star \star )$, we get $<U(\zeta :\alpha )>=U({\zeta }^{\star }:t)$. Hence $U({\zeta }^{\star }:t)$ is a pseudo-UP subalgebra of X generated by $U(\zeta :\alpha )$. Therefore, ${\zeta }^{\star }$ is a fuzzy pseudo-UP subalgebra of X. Next, we want to show that ${\zeta }^{\star }$ is contains ζ. For any $x\in X$ be arbitrary. Then $\zeta (x)=sup\{k\in [0,1]|x\in U(\zeta :k)\}$, by Theorem 2 $\le sup\{k\in [0,1]|x\in <U(\zeta :k)>\}={\zeta }^{\star }(x)$ implies that $\zeta (x)\le {\zeta }^{\star }(x),\mathrm{\forall }x\in X$. Hence ${\zeta }^{\star }$ is contains ζ. Finally, let η be a fuzzy pseudo-UP subalgebra of X contains ζ. For any $x\in X$, if ${\zeta }^{\star }(x)=0$, then clearly ${\zeta }^{\star }(x)\le \eta (x)$. Assume ${\zeta }^{\star }(x)=t\ne 0$. Then $x\in U({\zeta }^{\star }:t)=<U(\zeta :\alpha )>$, so $x\in <U(\zeta :\alpha )>$. Which implies that $\eta (x)\ge \zeta (x)\ge \alpha =t-ϵ$. Thus $\eta (x)\ge t={\zeta }^{\star }(x)$. Hence ${\zeta }^{\star }\subseteq \eta .$

4. Fuzzy pseudo-UP filter

In this section, we defined a fuzzy pseudo-UP filter on X and we proved that every fuzzy pseudo-UP filter is a fuzzy pseudo-UP subalgebra, as well as we proved that the converse is not always true. Furthermore, we demonstrated the relationship between the fuzzy pseudo-UP filter and its level subset of ζ.

Definition 4.1.

A fuzzy subsets $\zeta :X\to [0,1]$ is called a fuzzy pseudo-UP filter of X if and only if it satisfies the following axioms: for any $,\mathrm{\forall }x,y\in X$.

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

Theorem 4.2.

Every fuzzy pseudo-UP filter is a fuzzy pseudo-UP subalgebra.

Proof.

Assume ζ is a fuzzy pseudo-UP filter on X and for any $x,y\in X$. We want to show that ζ is fuzzy pseudo-UP subalgebra of X.

$\begin{array}{ll}\zeta (x\cdot y)& \ge min\{\zeta (y\star (x\cdot y)),\zeta (y)\}\\ & =min\{\zeta (0),\zeta (y)\},\text{by Proposition}2.3& \\ & =\zeta (y).& \\ & \ge min\{\zeta (x),\zeta (y)\}.\end{array}$

Next,

$\begin{array}{ll}\zeta (x\star y)& \ge min\{\zeta (y\cdot (x\star y)),\zeta (y)\}\\ & =min\{\zeta (0),\zeta (y)\},\text{by Proposition}2.3& \\ & =\zeta (y).& \\ & \ge min\{\zeta (x),\zeta (y)\}.\end{array}$

Therefore, ζ is a fuzzy pseudo-UP subalgebra of X.

Remark 4.3.

The converse may not be true.

Example 4.4.

Let $X=\{0,1,2,3\}$ be a set with a binary operation $"\cdot "$ and $"\star "$ defined by the following cayley table:

$\begin{array}{lll}\begin{array}{lllll}\cdot & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 1& 2\\ 2& 0& 0& 0& 1\\ 3& 0& 0& 0& 0\end{array}& & \begin{array}{lllll}\star & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 1& 2\\ 2& 0& 0& 0& 1\\ 3& 0& 0& 0& 0\end{array}\end{array}$

Then, it is clear that $(X,\cdot ,\star ,0)$ is a pseudo-UP algebra. We define a mapping $\zeta :X\to [0,1]$ as follows:

$\zeta (0)=1,\zeta (1)=0.6,\zeta (2)=0.4,\zeta (3)=0.1$. Then ζ is a fuzzy pseudo-UP subalgebra of X, such that $\zeta (1\cdot 2)\ge min\{\zeta (1),\zeta (2)\}\Rightarrow \zeta (1)\ge min\{\zeta (1),\zeta (2)\}$ implies that $0.6\ge 0.4$. But $\zeta (y)\ge min\{\zeta (x\cdot y),\zeta (x)\}$ which implies that $\zeta (2)\ge min\{\zeta (1\cdot 2),\zeta (1)\}$. Thus $0.4\ge 0.6$ which is contradict to Definition 4.1.

Proposition 4.5.

Let ζ be a fuzzy pseudo-UP filter of X and $x\le y$, then $\zeta (x)\ge \zeta (y)$, for any $x,y\in X$.

Proof.

Assume ζ is a fuzzy pseudo-UP filter of X and $x\le y$. We need to show that $\zeta (x)\ge \zeta (y)$.

Now, $\zeta (x)\ge min\{\zeta (yx),\zeta (y)\}$, by Definition 4.1. Which implies that $\zeta (x)\ge min\{\zeta (0),\zeta (y)\}=\zeta (y)$, by Lemma 2.2. Similarly, $\zeta (x)\ge min\{\zeta (y\star x),\zeta (y)\}=min\{\zeta (0),\zeta (y)\}=\zeta (y)$. Hence, $\zeta (x)\ge \zeta (y)$.

Theorem 4.6.

A non-empty subset F of X is a pseudo-UP filter of X if and only if the characteristic fuzzy set χ_F is a fuzzy pseudo-UP filter of X.

Theorem 4.

Let $U(\zeta :t)$ is non-empty, $\mathrm{\forall }t\in [0,1]$. A fuzzy subset ζ of a pseudo-UP algebra of X is fuzzy pseudo-UP filter of X, if and only if $U(\zeta :t)$ is pseudo-UP filter of X.

Proof.

Suppose ζ is a fuzzy pseudo-UP filter of X. Let $\mathrm{\forall }t\in [0,1]$ be $U(\zeta :t)\ne \mathrm{\varnothing }$, then there exist $x\in U(\zeta :t)$ such that $\zeta (x)\ge t$. From Definition 4.1 $\zeta (0)\ge \zeta (x)\ge t$. Thus $0\in U(\zeta :t)$. Let $x\cdot y,x\star y\in U(\zeta :t)$ and $x\in U(\zeta :t)$, then $\zeta (x\cdot y)\ge t$, $\zeta (x\star y)\ge t$ and $\zeta (x)\ge t$. By Definition 4.1, we have $\zeta (y)\ge min\{\zeta (x),\zeta (x\cdot y)\}$ and $\zeta (y)\ge min\{\zeta (x),\zeta (x\star y)\}$ implies that $\zeta (y)\ge min\{t,t\}=t$ and $\zeta (y)\ge min\{t,t\}=t$. So $y\in U(\zeta :t)$. Therefore, $U(\zeta :t)$ is a pseudo-UP filter of X.

Conversely, suppose $U(\zeta :t)$ is a pseudo-UP filter of X, $\mathrm{\forall }t\in [0,1]$. We want to show that ζ is a fuzzy pseudo- UP filter of X. Since $U(\zeta :t)\ne \mathrm{\varnothing }$, then there exists $x\in U(\zeta :t)$ such that $\zeta (x)\ge t$ and take $t=\zeta (x)$. From our assumption $0\in U(\zeta :t)$. Hence $\zeta (0)\ge t=\zeta (x)$. Let $x\cdot y,x\in U(\zeta :t)$. Then $\zeta (x)=t_1$ and $\zeta (x\cdot y)=t_2,\mathrm{\forall }{t}_1,t_2\in [0,1]$ and take $t=min\{t_1,t_2\}$. Since $U(\zeta :t)$ is a Pseudo-UP filter of X, then $y\in U(\zeta :t)$ such that $\zeta (y)\ge t=min\{t_1,t_2\}=min\{\zeta (x),\zeta (x\cdot y)\}$. Similarly, $x\star y\in U(\zeta :t)$ such that $\zeta (x\star y)=t_3$ and take $t=min\{t_1,t_3\}$. Then $\zeta (y)\ge t=min\{t_1,t_3\}=min\{\zeta (x),\zeta (x\star y)\}$. Therefore, ζ is a fuzzy pseudo-UP filter of X

Corollary 4.7.

Let $U^{+}(\zeta :t)$ is non-empty and $\mathrm{\forall }t\in [0,1]$. A fuzzy subset of pseudo-UP algebras of X is fuzzy pseudo-UP filter of X if and only if $U^{+}(\zeta :t)$ pseudo-UP filter of X.

Theorem 5.

Let $(\zeta :t)$ is non-empty and $\mathrm{\forall }t\in [0,1]$. A fuzzy subset ζ^c of X is fuzzy pseudo-UP filter of X if and only if $L(\zeta :t)$ is pseudo-UP filter of X.

Proof.

Suppose ζ^c is a fuzzy pseudo-UP filter of X. Since $L(\zeta :t)$ be non-empty, then there exists $x\in L(\zeta :t)$ such that $\zeta (x)\le t$. From our assumption ζ^c is a fuzzy pseudo-UP filter of X, then ${\zeta }^c(0)\ge {\zeta }^c(x)$. Thus $1-\zeta (0)\ge 1-\zeta (x)$. So $\zeta (0)\le \zeta (x)\le t$. Hence $0\in L(\zeta :t)$. Let $x\cdot y,x\star y\in L(\zeta :t)$ and $x\in L(\zeta :t)$ such that $\zeta (x\cdot y)\le t$, $\zeta (x\star y)\le t$ and $\zeta (x)\le t$. Now, ${\zeta }^c(y)\ge min\{{\zeta }^c(x),{\zeta }^c(x\cdot y)\}$. By Lemma 2.10, we have $1-\zeta (y)\ge min\{1-\zeta (x),1-\zeta (x\cdot y)\}=1-max\{\zeta (x),\zeta (x\cdot y)\}$ which implies that $\zeta (y)\le max\{\zeta (x),\zeta (x\cdot y)\}\le max\{t,t\}=t$. So $y\in L(\zeta :t)$. Similarly, $\zeta (y)\le max\{\zeta (x),\zeta (x\star y)\}$ implies that $y\in L(\zeta :t)$. Hence, $L(\zeta :t)$ is a pseudo-UP filter of X.

Conversely, assume $L(\zeta :t)$ is a pseudo-UP filter of X. We need to show that ζ^c is a fuzzy pseudo-UP filter of X. Since $L(\zeta :t)\ne \mathrm{\varnothing }$ then there exist $x\in L(\zeta :t)$ such that $\zeta (x)\le t$ and choose $t=\zeta (x)$. Now $\zeta (0)\le t=\zeta (x)$. Hence, ${\zeta }^c(0)=1-\zeta (0)\ge 1-\zeta (x)={\zeta }^c(x)$. Let $x\cdot y,x\star y\in L(\zeta :t)$ and $x\in L(\zeta :t)$ such that $\zeta (x\cdot y)\le t,\zeta (x\star y)\le t$ and $\zeta (x)\le t$. Choose $t=max\{\zeta (x),\zeta (x\cdot y)\}$. Then $\zeta (y)\le t=max\{\zeta (x),\zeta (x\cdot y)\}$. By Lemma 2.10 we have ${\zeta }^c(y)=1-\zeta (y)\ge 1-max\{\zeta (x),\zeta (x\cdot y)\}$ $=min\{1-\zeta (x),1-\zeta (x\cdot y)\}$ $=min\{{\zeta }^c(x),{\zeta }^c(x\cdot y)\}$. Similarly, Choose $t=max\{\zeta (x),\zeta (x\star y)\}$. Then ${\zeta }^c(y)\ge min\{{\zeta }^c(x),{\zeta }^c(x\star y)\}$. Therefore, ζ^c is a fuzzy pseudo-UP filter.

Corollary 4.8.

Let ζ be fuzzy subset of pseudo-UP algebra of X and let $L^{-}(\zeta :t)$ is non-empty. Then, ζ^c is fuzzy pseudo-UP filter of X if and only if $L^{-}(\zeta :t)$ is pseudo-UP filter of X, for any $t\in [0,1]$.

Theorem 6.

Let $(A,\cdot ,\star ,0)$ and $(B,\cdot ,\star ,0)$ be pseudo-UP algebra. A mapping $f:A\to B$ is epimorphism and η is a fuzzy pseudo-UP filter of B. Then $f^{-1}(\eta )$ is a fuzzy pseudo-UP filter of A.

Proof.

Let η be a fuzzy pseudo-UP filter of B and let $0\in A$ then,

$\begin{array}{ll}{f}^{-1}(\eta )(0)& =\eta (f(0))\\ & \ge \eta (f(x)\\ & \Rightarrow f^{-1}(\eta )(0)\\ & \ge (f^{-1}(\eta )(x).\end{array}$

For $x,y\in A$. Then,

Next,

$\begin{array}{ll}{f}^{-1}(\eta )(y)& =\eta (f(y))\\ & \ge min\{\eta (f(x)),\eta (f(x\cdot y))\}& \\ & =min\{f^{-1}(\eta )(x),(f^{-1}(\eta )(x\cdot y)\}.& \\ & ⟹f^{-1}(\eta )(y)\ge min\{f^{-1}(\eta )(x),f^{-1}(\eta )(x\cdot y)\}.\end{array}$

Similarly,

$\begin{array}{ll}{f}^{-1}(\eta )(y)& =\eta (f(y))\\ & \ge min\{\eta (f(x)),\eta (f(x\star y))\}& \\ & =min\{f^{-1}(\eta )(x),(f^{-1}(\eta )(x\star y))\}.& \\ & ⟹f^{-1}(\eta )(y)\ge min\{f^{-1}(\eta )(x),f^{-1}(\eta )(x\star y)\}.\end{array}$

Hence, $f^{-1}(\eta )$ is a fuzzy pseudo-UP filter of A.

Theorem 7.

Let $(A,\cdot ,\star ,0)$ and $(B,\cdot ,\star ,0)$ be a pseudo-UP algebra and $f:A\to B$ be surjective and ζ be a fuzzy pseudo-UP filter of A. Then $f(\zeta )$ is a fuzzy pseudo-UP filter of B, provided that sup property holds.

Proof.

Let $0\in B$, then there exist $0\in A$ such that $f(0)=0$.

Now, $f(\zeta )(0)=\{sup\zeta (t)|t\in f^{-1}(0)\}=\zeta (0)\ge \zeta (a),\mathrm{\forall }a\in A$. Let $x\in B$. Since f is surjective, we have $f^{-1}(x)\ne \mathrm{\varnothing }$ and $\zeta (0)\ge \zeta (t)|t\in f^{-1}(x).$ Which implies that $f(\zeta )(0)\ge \{sup\zeta (t)|t\in f^{-1}(x)\}=f(\zeta )(x),\mathrm{\forall }x\in B$. Hence, $f(\zeta )(0)\ge f(\zeta )(x),\mathrm{\forall }x\in B$. For any $x,y\in B$, then there exist $a,b\in A$ such that $f(a)=x,f(b)=y$. Let $\zeta (a\cdot b)=sup\zeta (t)|t\in f^{-1}(x\cdot y)$ and $\zeta (a)=sup\zeta (t)|t\in f^{-1}(x)$. Then,

$\begin{array}{ll}f(\zeta )(y)& =\{sup\zeta (t)|t\in f^{-1}(y)\}\\ & =\zeta (b)& \\ & \ge min\{\zeta (a),\zeta (a\cdot b)\}& \\ & =min\{sup\zeta (t)|t\in f^{-1}(x),\\ & sup\zeta (t)|t\in f^{-1}(x\cdot y)\}& \\ & =min\{f(\zeta )(x),\zeta (y\cdot z)\}.\end{array}$

Similarly,

$\begin{array}{ll}f(\zeta )(y)& =\{sup\zeta (t)|t\in f^{-1}(y)\}\\ & =\zeta (b)& \\ & \ge min\{\zeta (a),\zeta (a\star b)\}& \\ & =min\{sup\zeta (t)|t\in f^{-1}(x),\\ & sup\zeta (t)|t\in f^{-1}(x\star y)\}& \\ & =min\{f(\zeta )(x),\zeta (y\star z)\}.\end{array}$

Therefore, $f(\zeta )$ is a fuzzy pseudo-UP filter of B.

5. Fuzzy pseudo-UP ideal

In this phase of the article, we initiated the basic definition fuzzy pseudo-UP ideal. Further, we discussed the properties of given notions and the relationship between fuzzy pseudo-UP ideal of X and their level set of ζ. And also we proved that every fuzzy pseudo-UP ideal is fuzzy pseudo-UP subalgebra and fuzzy pseudo-UP filter of X.

Definition 5.1.

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

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

Example 5.2.

Let $X=\{0,1\}$ be a set with binary operations $"\cdot "$ and $"\star "$ defined by the following cayley table:

$\begin{array}{lll}\begin{array}{lll}\cdot & 0& 1\\ 0& 0& 1\\ 1& 0& 0\end{array}& & \begin{array}{lll}\star & 0& 1\\ 0& 0& 1\\ 1& 0& 0\end{array}\end{array}$

Clearly, $(X,\cdot ,\star ,0)$ is pseudo-UP algebra, we define a fuzzy subset of X as follows:

Let $\zeta (0)=0.5,\zeta (1)=0.3$. Then, ζ is a fuzzy pseudo-UP ideal of X.

Theorem 5.3

Every fuzzy pseudo-UP ideal of X is fuzzy pseudo-UP subalgebra of X.

Proof.

Suppose ζ is fuzzy pseudo-UP ideal of X. Then,

$\zeta (x\cdot y)\ge min(\zeta (x\cdot (y\star y)),\zeta (y))$, by Definition 5.1

$\begin{array}{lll}& =min\{\zeta (x\cdot 0),\zeta (y)\},\text{by Lemma}2.2& \\ & =min\{\zeta (0),\zeta (y)\},\text{by Lemma}2.2& \\ & =\zeta (y),\text{by Definition}5.1& \\ & \ge min\{\zeta (x),\zeta (y)\}.& \\ & ⟹\zeta (x\cdot y)\ge min\{\zeta (x),\zeta (y)\}.\end{array}$

Similarly  $\zeta (x\star y)\ge min\{zeta(x\star (y\cdot y)),\zeta (y)\}$, by Definition 5.1

$\begin{array}{lll}& =min\{\zeta (x\star 0),\zeta (y)\},\text{by ~Proposition}2.3& \\ & =min\{\zeta (0),\zeta (y)\},\text{by~ Lemma}2.2& \\ & =\zeta (y),\text{by ~Definitions}5.1& \\ & \ge min\{\zeta (x),\zeta (y)\}.& \\ & ⟹\zeta (x\star y)\ge min\{\zeta (x),\zeta (y)\}.\end{array}$

Hence, ζ is fuzzy pseudo-UP subalgebra of X.

Remark 5.4.

The conversemay not be true. See Example 3.3

Theorem 5.5.

Every fuzzy pseudo-UP ideal of X is a fuzzy pseudo-UP filter of X.

Proof.

Assume that ζ is a fuzzy pseudo-UP ideal of X. We need to show that ζ is a fuzzy pseudo-UP filter of X. Let $x,y\in X$. Then, $\zeta (0)\ge \zeta (x),\mathrm{\forall }x\in X$

Next,

$\begin{array}{ll}\zeta (y)& =\zeta (0\cdot y),\text{by~Lemma}2.2\\ & \ge min\{\zeta (0\cdot (x\star y)),\zeta (x)\},\text{by~ Definitions}5.1& \\ & =min\{\zeta (x\star y),\zeta (x)\},\text{by~Lemma }2.2& \\ & ⟹\zeta (x\cdot y)\ge min\{\zeta (x\star y),\zeta (x)\}.\end{array}$

Similarly,

$\begin{array}{ll}\zeta (y)& =\zeta (0\star y),\text{by~Lemma}2.2\\ & \ge min\{\zeta (0\star (x\cdot y)),\zeta (x)\},\text{by Definitions }5.1& \\ & =min\{\zeta (x\cdot y),\zeta (x)\},\text{by ~Lemma}2.2& \\ & ⟹\zeta (x\star y)\ge min\{\zeta (x\cdot y),\zeta (x)\}.\end{array}$

Therefore, ζ is a fuzzy pseudo-UP filter of X.

Remark 5.6.

The converse may not be true.

Example 5.7.

Let $X=\{0,1,2,3\}$ be a set with a binary operation $"\cdot "$ and $"\star "$ defined by the following cayley table:

$\begin{array}{lll}\begin{array}{lllll}\cdot & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 2& 2\\ 2& 0& 1& 0& 2\\ 3& 0& 1& 0& 0\end{array}& & \begin{array}{lllll}\star & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 2& 2\\ 2& 0& 1& 0& 2\\ 3& 0& 1& 0& 0\end{array}\end{array}$

Then $(X,\cdot ,\star ,0)$ is a pseudo-UP algebra. We define a mapping $\zeta :X\to [0,1]$ as follows:

Let $\zeta (0)=1,\zeta (1)=0.2,\zeta (2)=0.1,\zeta (3)=0.1$ then ζ is a fuzzy pseudo-UP filter of X. Since $\zeta (2\cdot 3)\ge$ $min\{\zeta (2\cdot (1\star 3)),\zeta (1)\}$ implies that $\zeta (2)\ge min\{\zeta (0),\zeta (1)\}$. Thus $0.1\ge 0.2$, which is contradict to the definitions of a fuzzy pseudo-UP ideal of X.

Theorem 5.8.

If ζ is a fuzzy pseudo-UP ideal (respectively, fuzzy pseudo-UP filter and fuzzy pseudo-UP subalgebra) of X, then the set

$J:=\{x\in X\mid \zeta (x)=\zeta (0)\}$

is a pseudo-UP ideal (respectively, pseudo-UP filter and pseudo-UP subalgebra) of X.

Proof.

Suppose ζ is a fuzzy pseudo-UP ideal of X. We need to show that J is pseudo-UP ideal of X. Clearly $0\in J$. Let $x,y,z\in X$ suchthat $x\cdot (y\star z),x\star (y\cdot z)\in J$ and $y\in J$. Then $\zeta (x\cdot (y\star z))=\zeta (0)=\zeta (x\cdot z)$ and $\zeta (y)=\zeta (0)$. By Definition of 5.1, we have $\zeta (x\cdot z)\ge min\{\zeta (x\cdot (y\star z)),\zeta (y)\}=min\{\zeta (0),\zeta (0)\}=\zeta (0)$ and $\zeta (x\star z)\ge min\{\zeta (x\star (y\cdot z)),\zeta (y)\}=min\{\zeta (0),\zeta (0)\}=\zeta (0)$. Then $\zeta (x\cdot z)=\zeta (0)=\zeta (x\star z)$. Which implies that $x\cdot z,x\star z\in J$. As a result J is a pseudo-UP ideal of X. Similarly, we have other outcomes.

Theorem 5.9.

Let J be a non-empty subset of pseudo-UP algebra X and ζ be a fuzzy set in X defined by $\zeta (x)=\{\begin{array}{ll}p& \text{if }x\in J,\\ q& \text{if}x\notin J,\mathrm{\forall }p,q\in [0,1]\text{with},p\ge q\end{array}$

Then ζ is a fuzzy pseudo-UP ideal of X if and only if J is a pseudo-UP ideal of X.

Proof.

Suppose ζ is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X. We need to show that J is pseudo-UP ideal of X. Let $x\in X$ such that $x\in J$. Then, $\zeta (0)\ge \zeta (x)=p$. Hence $0\in P.$ Next, for $x\cdot (y\star z),y\in J$, then $\zeta (x\cdot z)\ge min\{\zeta (x\cdot (y\star z)),\zeta (y)\}=min\{p,p\}=p$ implies that $x\cdot z\in J$. Similarly, $x\star (y\cdot z),y\in J$, then $\zeta (x\star z)\ge min\{\zeta (x\star (y\cdot z)),\zeta (y)\}=min\{p,p\}=p$ implies that $x\star z\in J$. Hence, J is a pseudo-UP ideal of a pseudo-UP algebra X.

Conversely, assume J is a pseudo-UP ideal of X. We need to show that ζ is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X. Let $x\in X$. Since $0\in J$, then $\zeta (0)=p$. Clearly, $p\ge \zeta (x),\mathrm{\forall }x\in X$. Hence, $\zeta (0)\ge \zeta (x).$ Next, consider the following cases:

Case 1: If $x\cdot (y\star z)$ and $y\in J$. Then $x\cdot z\in J$. Thus $\zeta (x\cdot z)=p\ge min\{\zeta (x\cdot (y\star z)),\zeta (y)\}$ and if $x\star (y\cdot z)$ and $y\in J$. Then $x\star z\in J$. Thus $\zeta (x\star z)=p\ge min\{\zeta (x\star (y\star z)),\zeta (y)\}.$

Case 2: If $x\cdot (y\star z)\notin J$ or $y\notin J$. Then $$\begin{gathered} \zeta (x\cdot z)\ge \\ q=min\{\zeta (x\cdot (y\star z)),\zeta (y)\} \end{gathered}$$ and if $x\star (y\cdot z)\notin J$ or $y\notin J$. Then $$\begin{gathered} \zeta (x\star z)\ge \\ q=min\{\zeta (x\star (y\cdot z)),\zeta (y)\}. \end{gathered}$$

Case 3: If $x\cdot (y\star z)\notin J$ and $y\notin J$. Then $\zeta (x\cdot z)\ge q=min\{\zeta (x\cdot (y\star z)),\zeta (y)\}$ and if $x\star (y\cdot z)\notin J$ and $y\notin J$. Then $\zeta (x\star z)\ge q=min\{\zeta (x\star (y\cdot z)),\zeta (y)\}.$

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

Theorem 5.10.

Let J be a non-empty subsets of X. Then J is a pseudo-UP ideal of X if and only if the characteristic functions of χ_J is fuzzy pseudo-UP ideal of X.

Theorem 5.11.

Let ζ be fuzzy pseudo-UP ideals of X and if $y\le z$, then $\zeta (y)\ge \zeta (z)$.

Proof.

Let $x,y,z\in X$. Then,

$\begin{array}{l}\begin{array}{lll}\zeta (y)& =\zeta (0y),\text{by~Lemma}2.2& \\ & \ge min\{\zeta (0(z\star y)),\zeta (z)\},\text{by~ Definitions}5.1& \\ & =min\{\zeta (z\star y),\zeta (z)\}\text{by ~Lemma}2.2& \\ & =min\{\zeta (0),\zeta (z)\},\text{since}y\le z\Rightarrow y\star z=0& \\ & =\zeta (z),\text{by Definitions}5.1& \\ & ⟹\zeta (y)\ge \zeta (z).& \end{array}\end{array}$

Similarly,

$\begin{array}{l}\begin{array}{lll}\zeta (y)& =\zeta (0\star y)& \\ & \ge min\{\zeta (0\star (zy)),\zeta (z)\},\text{by ~Definitions of}5.1& \\ & =min\{\zeta (zy),\zeta (z)\},\text{by Lemma}2.2& \\ & =min\{\zeta (0),\zeta (z)\},\text{since}y\le z\Rightarrow yz=0& \\ & =\zeta (z),\text{by~ Definitions}5.1& \\ & ⟹\zeta (y)\ge \zeta (z).& \end{array}\end{array}$

Proposition 5.12.

If ζ is fuzzy pseudo-UP ideals of X, then $\zeta (x\cdot (x\cdot y))\ge \zeta (y)$ and $\zeta (x\star (x\star y))\ge \zeta (y)$

Proof.

Let $x,y\in X$.Then,

$\zeta (x\cdot (x\cdot y))\ge min(\zeta (x\cdot ((x\cdot y)\star (x\cdot y))),\zeta (x\cdot y))$,by Definition of 5.1.

$\begin{array}{lll}& =min\{\zeta (x\cdot 0),\zeta (x\cdot y)\}& \\ & =min\{\zeta (0),\zeta (x\cdot y)\},\text{by Lemma}2.2& \\ & =\zeta (x\cdot y).& \\ & \ge min\{\zeta (x\cdot (y\star y)),\zeta (y)\}& \\ & =min\{\zeta (x\cdot 0),\zeta (y)\}& \\ & =min\{\zeta (0),\zeta (y)\}& \\ & =\zeta (y).& \\ & ⟹\zeta (x\cdot (x\cdot y))\ge \zeta (y).\end{array}$

Finally, $\zeta (x\star (x\star y))\ge min\{\zeta (x\star ((x\star y)\cdot (x\star y)),\zeta (x\star y)\}$,by Definition of 5.1.

$\begin{array}{lll}& =min\{\zeta (x\star 0),\zeta (x\star y)\}& \\ & =min\{\zeta (0),\zeta (x\star y)\},\text{by Lemma~}2.2& \\ & =\zeta (x\star y).& \\ & \ge min\{\zeta (x\star (y\cdot y)),\zeta (y)\}& \\ & =min\{\zeta (x\star 0),\zeta (y)\}& \\ & =min\{\zeta (0),\zeta (y)\}& \\ & =\zeta (y).& \\ & ⟹\zeta (x\star (x\star y))\ge \zeta (y).\end{array}$

Theorem 5.13.

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)	$\zeta (x\cdot y)\ge \zeta (y).$

Proof.

1)	Suppose ζ is a fuzzy pseudo-UP ideal of X. Take x = 0 in Definition 5.1 (2) . Then, $\zeta (0\cdot z)\ge min\{\zeta (0\cdot (y\star z)),\zeta (y)\}$. Implies that $\zeta (z)\ge min\{\zeta (y\star z),\zeta (y)\}$, by Definition 5.1.
2)	Put z = y in Definition 5.1 (3). Then, $\zeta (x\star y)\ge \zeta (x\star (y\cdot y),\zeta (y))$, this implies that $min\{\zeta (0),\zeta (y)\}=\zeta (y)$. Hence, $\zeta (x\star y)\ge \zeta (y).$

Theorem 5.14.

If ζ and η are fuzzy pseudo-UP ideal of X. Then $\zeta \cap \eta$ is also fuzzy pseudo-UP ideal of X.

Proof.

Let $x,y,z\in X$.Then $(\zeta \cap \eta )(0)=min\{\zeta (0),\eta (0)\}$ $\ge min\{\zeta (x),\eta (x)\}$ $=(\zeta \cap \eta )(x)$ which implies that $(\zeta \cap \eta )(0)\ge (\zeta \cap \eta )(x).$

Next,* $(\zeta \cap \eta )(x\cdot z)=min\{\zeta (x\cdot z),\eta (x\cdot z)\}$

$\begin{array}{ll}& \ge min\{min\{\zeta (x\cdot (y\star z)),\zeta (y)\},\\ & min\{\eta (x\cdot (y\star z)),\eta (y)\}\}& \\ & =min\{min\{\zeta (x\cdot (y\star z))),\eta (x\cdot (y\star z)\},\\ & min\{\zeta (y),\eta (y)\}\}& \\ & =min\{(\zeta \cap \eta )(x\cdot (y\star z)),(\zeta \cap \eta )(y)\}.& \\ & ⟹(\zeta \cap \eta )(x\cdot z)\ge min\{(\zeta \cap \eta )(x\cdot (y\star z)),\\ & (\zeta \cap \eta )(y)\}.\end{array}$

$(\zeta \cap \eta )(x\star z)=min\{\zeta (x\star z),\eta (x\star z)\}$

$\begin{array}{ll}& \ge min\{min\{\zeta (x\star (y\cdot z)),\zeta (y)\},\\ & min\{\eta (x\star (y\cdot z)),\eta (y)\}\}& \\ & =min\{min\{\zeta (x\star (y\cdot z))),\eta (x\star (y\cdot z)\},\\ & min\{\zeta (y),\eta (y)\}\}& \\ & =min\{(\zeta \cap \eta )(x\star (y\cdot z)),(\zeta \cap \eta )(y)\}.& \\ & ⟹(\zeta \cap \eta )(x\star z)\ge min\{(\zeta \cap \eta )(x\star (y\cdot z)),\\ & (\zeta \cap \eta )(y)\}.\end{array}$

Therefore, $\zeta \cap \eta$ is a fuzzy pseudo-UP ideal of X.

Corollary 5.15.

Let $\{{\zeta }_i/i\in \mathrm{\Lambda }\}$ be a family of fuzzy pseudo-UP ideal of X. Then $\bigcap _{i\in \mathrm{\Lambda }}{\zeta }_i$ is also fuzzy pseudo-UP ideal of X.

Remark 5.16.

The union of any two fuzzy pseudo-UP ideal of a pseudo-UP algebra X is not necessarily a fuzzy pseudo-UP ideal of X.

Example 5.17.

Let $X=\{0,1,2,3\}$ be a set with a binary operation $"\cdot "$ and $"\star "$ defined by the following cayley table:

$\begin{array}{lll}\begin{array}{lllll}\cdot & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 3& 3\\ 2& 0& 1& 0& 0\\ 3& 0& 1& 3& 0\end{array}& & \begin{array}{lllll}\star & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 0& 0& 3& 3\\ 2& 0& 1& 0& 0\\ 3& 0& 1& 3& 0\end{array}\end{array}$

Clearly, $(X,\cdot ,\star ,0)$ is a pseudo-UP algebra. We define a fuzzy set $\zeta :X\to [0,1]$ as follows, $\zeta (0)=1,\zeta (1)=0.6,\zeta (2)=0.4,\zeta (3)=0.3$ and a fuzzy set $\eta :X\to [0,1]$ define as follows, $\eta (0)=1,\eta (1)=0.4,\eta (2)=0.5,\eta (3)=0.3.$ Now, $(\zeta \cup \eta )(1\cdot 3)=max\{\zeta (1\cdot 3),\eta (1\cdot 3)\}\text{break}=max\{\zeta (3),\eta (3)\}=max\{0.3,0.3\}=0.3$.

$(\zeta \cup \eta )(1\star 3)=max\{\zeta (1\star 3),\eta (1\star 3)\}\text{break}=max\{\zeta (3),\eta (3)\}=max\{0.3,0.3\}=0.3$ and

Hence, $(\zeta \cup \eta )(1\cdot 3)$ = 0.3=$(\zeta \cup \eta )(1\star 3)$ ……………….$(\star )$.

$(\zeta \cup \eta )(1\cdot 3)=max\{\zeta (1\cdot 3),\eta (1\cdot 3)\}$

$\begin{array}{ll}& \ge max\{min\{\zeta (1\cdot (2\star 3)),\zeta (2)\},\\ & min\{\eta (1\cdot (2\star 3)),\eta (2)\}\}& \\ & =min\{max\{\zeta (1\cdot (2\star 3)),\eta (1\cdot (2\star 3)\},\\ & max\{\zeta (2),\eta (2)& \\ & =min\{max\{1\cdot 0),\eta (1\cdot 0)\},max\{\zeta (2),\eta (2)\}\}& \\ & =min\{max\{\zeta (0),\eta (0)\},max\{\zeta (2),\eta (2)\}\}& \\ & =min\{max\{1,1\},max\{0.4,0.5\}\}=min\{1,0.5\}=0.5.\end{array}$

$(\zeta \cup \eta )(1\star 3)=max\{\zeta (1\star 3),\eta (1\star 3)\}$

$\begin{array}{ll}& \ge max\{min\{\zeta (1\star (2\cdot 3)),\zeta (2)\},\\ & min\{\eta (1\star (2\cdot 3)),\eta (2)\}\}& \\ & =min\{max\{\zeta (1\star (2\cdot 3)),\eta (1\star (2\cdot 3)\},\\ & max\{\zeta (2),\eta (2)\}\}& \\ & =min\{max\{1\star 0),\eta (1\star 0)\},max\{\zeta (2),\eta (2)\}\}& \\ & =min\{max\{\zeta (0),\eta (0)\},max\{\zeta (2),\eta (2)\}\}& \\ & =min\{max\{1,1\},max\{0.4,0.5\}\}=min\{1,0.5\}=0.5.\end{array}$

Hence, $(\zeta \cup \eta )(1\cdot 3)$ = 05= $(\zeta \cup \eta )(1\star 3)$……………….$(\star \star )$.

From $(\star )$ and $(\star \star )$ we see that $0.3\le 0.5$ which is contradict to Definition 5.1. This shows that the union of any two fuzzy pseudo-UP ideal of X may not be a fuzzy pseudo-UP ideal of X.

Theorem 5.18.

Let $U(\zeta :t)$ is non-empty and let ζ be a fuzzy subsets of X. Then ζ is a fuzzy pseudo-UP ideal of X if and only if $U(\zeta :t)$ is pseudo-UP ideal of X,$\mathrm{\forall }t\in [0,1]$.

Proof.

Suppose ζ is a fuzzy pseudo-UP ideal of X. Let $t\in [0,1]$ and $U(\zeta :t)$ is a non empty.

Let $a\in U(\zeta :t)$. Then $\zeta (a)\ge t$. Since ζ is fuzzy pseudo-UP ideal of X, we have $\zeta (0)\ge \zeta (a)$. Thus $0\in U(\zeta :t)$. Next for $x,y,z\in X$ and $x\cdot (y\star z),y\in U(\zeta :t)$. Then $\zeta (x\cdot (y\star z))\ge t$ and $\zeta (y)\ge t.$ Now, $\zeta (x\cdot z)\ge min\{\zeta (x\cdot (y\star z)),\zeta (y)\}\ge min\{t,t\}=t$ implies that $\zeta (x\cdot z)\ge t$. Thus $x\cdot z\in U(\zeta :t)$. Similarly, for $x,y,z\in X$ and $x\star (y\cdot z),y\in U(\zeta :t)$. Then $\zeta (x\star (y\cdot z))\ge t$ and $\zeta (y)\ge t.$ Now, $\zeta (x\star z)\ge min\{\zeta (x\star (y\cdot z)),\zeta (y)\}\ge min\{t,t\}=t$ implies that $\zeta (x\star z)\ge t$ which implies that $x\star z\in U(\zeta :t)$. Therefore, $U(\zeta :t)$ is a pseudo-UP ideal of X.

Conversely, assume that $U(\zeta :t)$ is pseudo-UP ideal of X and $U(\zeta :t)$ is non empty, $\mathrm{\forall }t\in [0,1]$. We need to show that ζ is fuzzy pseudo-UP ideal of X. Let $x\in U(\zeta :t)$. Then $\zeta (x)\in [0,1]$ and choose $t=\zeta (x)$. From our assumption, $U(\zeta :t)$ is pseudo-UP ideal of X. So, $0\in U(\zeta :t)$. Which implies that $\zeta (0)\ge t$. Hence, $\zeta (0)\ge \zeta (x).$ Next, for $x,y,z\in X$ and let $x\cdot (y\star z),y\in U(\zeta :t)$. Then $\zeta (x\cdot (y\star z))=t_1$ and $\zeta (y)=t_2$, where, $t_1,t_2\in [0,1]$ and take $t=min\{t_1,t_2\}$. Now, $\zeta (x\cdot z)\ge t=min\{t_1,t_2\}=min\{\zeta (x\cdot (y\star z)),\zeta (y)\}$. Thus $\zeta (x\cdot z)\ge min\{\zeta (x\cdot (y\star z)),\zeta (y)\}$. Similarly, let $x,y,z\in X$ and let $x\star (y\cdot z),y\in U(\zeta :t)$. Then $\zeta (x\star (y\cdot z))=t_1$ and $\zeta (y)=t_2$, where, $t_1,t_2\in [0,1]$ and choose $t=min\{t_1,t_2\}$. Then $\zeta (x\star z)\ge t=min\{t_1,t_2\}=min\{\zeta (x\star (y\cdot z)),\zeta (y)\}$. Thus $\zeta (x\star z)\ge min\{\zeta (x\star (y\cdot z)),\zeta (y)\}$. Therefore, ζ is fuzzy pseudo-UP ideal of X.

Corollary 5.19.

Let $U^{+}(\zeta :t)$ is non empty and et ζ be a fuzzy pseudo-UP ideal of X. Then ζ is a fuzzy pseudo-UP ideal of X if and only if $U^{+}(\zeta :t)$ is pseudo-UP ideal of X,$\mathrm{\forall }t\in [0,1]$.

Theorem 5.20.

Let $L(\zeta :t)$ is non empty and let ζ be a fuzzy subsets of X. Then ζ^c is a fuzzy pseudo-UP ideal of X if and only if $L(\zeta :t)$ is a pseudo-UP ideal, for all $t\in [0,1].$

Proof.

Suppose ζ^c is a fuzzy Pseudo-UP ideal of X. We need to show that $L(\zeta :t)$ is a pseudo-UP ideal of X.

Since $L(\zeta :t)\ne \mathrm{\varnothing }$, then there exist $x\in L(\zeta :t)$ such that $\zeta (x)\le t$. Thus

${\zeta }^c(0)={\zeta }^c(x\cdot 0)$, by Lemma 2.2

$\begin{array}{ll}& \ge min\{{\zeta }^c(x\cdot (x\star 0)),{\zeta }^c(x)\},\\ & \text{since}{\zeta }^c\text{fuzzy pseudo-UP ideal}& \\ & =min\{{\zeta }^c(x\cdot 0),{\zeta }^c(x)\}& \\ & =min\{{\zeta }^c(0),{\zeta }^c(x)\}& \\ & ={\zeta }^c(x).& \\ & ⟹{\zeta }^c(0)\ge {\zeta }^c(x)& \\ & ⟹1-\zeta (0)\ge 1-\zeta (x)& \\ & ⟹\zeta (0)\le \zeta (x)\le t& \\ & ⟹\zeta (x)\le t.& \\ & ⟹0\in L(\zeta :t).\end{array}$

Next for $x,y,z\in X$ be such that $x\cdot (y\star z),y\in L(\zeta :t)$. Then $\zeta (x\cdot (y\star z))\le t$ and $\zeta (y)\le t$. Now,

${\zeta }^c(x\cdot z)\ge min\{{\zeta }^c(x\cdot (y\star z)),{\zeta }^c(y)\}$

$\begin{array}{lll}& ⟹1-\zeta (x\cdot z)\ge min\{1-\zeta (x\cdot (y\star z)),1-\zeta (y)\}& \\ & =1-max\{\zeta (x\cdot (y\star z)),\zeta (y)\}.& \\ & ⟹\zeta (x\cdot z)\le max\{\zeta (x\cdot (y\star z)),\zeta (y)\}\le max\{t,t\}=t.& \\ & ⟹\zeta (x\cdot z)\le t.\text{Thus}x\cdot z\in L(\zeta :t).\end{array}$

Similarly, let $x,y,z\in X$ be such that $x\star (y\cdot z),y\in L(\zeta :t)$. Then $\zeta (x\star (y\cdot z))\le t$ and $\zeta (y)\le t$.

Now, ${\zeta }^c(x\star z)\ge min\{{\zeta }^c(x\star (y\cdot z)),{\zeta }^c(y)\}$

$\begin{array}{lll}& ⟹1-\zeta (x\star z)\ge min\{1-\zeta (x\star (y\cdot z)),1-\zeta (y)\}& \\ & =1-max\{\zeta (x\star (y\cdot z)),\zeta (y)\}.& \\ & ⟹\zeta (x\star z)\le max\{\zeta (x\star (y\cdot z)),\zeta (y)\}\le max\{t,t\}=t.& \\ & ⟹\zeta (x\star z)\le t.\text{Thus}x\star z\in L(\zeta :t).\end{array}$

Therefore, $L(\zeta :t)$ is pseudo-UP ideal of X.

Conversely, assume that $L(\zeta :t)$ is pseudo-UP ideal of X and $L(\zeta :t)\ne \mathrm{\varnothing }$, for every $t\in [0,1]$. We want to prove that ζ^c is fuzzy pseudo-UP ideal of X. Let $x\in L(\zeta :t)$, then $\zeta (x)\le t$ and choose $t=\zeta (x)$. Since, $L(\zeta :t)$ is a pseudo-UP ideal of X. Then $0\in L(\zeta :t)$ such that $\zeta (0)\le t=\zeta (x)$. Hence ${\zeta }^c(0)=1-\zeta (0)\ge 1-\zeta (x)={\zeta }^c(x)$. Next, for $x,y,z\in X$. Then $\zeta (x\cdot (y\star z)),\zeta (y)\in [0,1]$ and choose $t=max\{\zeta (x\cdot (y\star z)),\zeta (y)\}$. Then $\zeta (x\cdot (y\star z))\le t$ and $\zeta (y)\le t$. Thus $x\cdot (y\star z),y\in L(\zeta :t).$ Since $L(\zeta :t)$ is pseudo-UP ideal of X. Thus $x\cdot z\in L(\zeta :t)$. Which implies $\zeta (x\cdot z)\le t=max\{\zeta (x)\cdot (y\star z),\zeta (y)\}$. Now,

$\begin{array}{ll}{\zeta }^c(x\cdot z)& =1-\zeta (x\cdot z)\\ & \ge 1-max\{\zeta (x\cdot (y\star z)),\zeta (y)\}& \\ & =min\{1-\zeta (x\cdot (y\star z)),1-\zeta (y)\}& \\ & =min\{{\zeta }^c(x\cdot (y\star z)),{\zeta }^c(y)\}.\end{array}$

Similarly, let $\zeta (x\star (y\cdot z)),\zeta (y)\in [0,1]$ and take $t=max\{\zeta (x\star (y\cdot z)),\zeta (y)\}$. Then $\zeta (x\star (y\cdot z))\le t$ and $\zeta (y)\le t$. Thus $x\star (y\cdot z),y\in L(\zeta :t).$ Since $L(\zeta :t)$ is pseudo-UP ideal of X. Thus $x\star z\in L(\zeta :t).$ Which implies $\zeta (x\star z)\le t=max\{\zeta (x)\star (y\cdot z),\zeta (y)\}$. Now,

$\begin{array}{ll}{\zeta }^c(x\star z)& =1-\zeta (x\star z)\\ & \ge 1-max\{\zeta (x\star (y\cdot z)),\zeta (y)\}& \\ & =min\{1-\zeta (x\star (y\cdot z)),1-\zeta (y)\}& \\ & =min\{{\zeta }^c(x\star (y\cdot z)),{\zeta }^c(y)\}.\end{array}$

Therefore, ζ if fuzzy pseudo-UP ideal of X.

Corollary 5.21.

Let ζ be a fuzzy subsets of X and let $L^{-}(\zeta :t)$ is non-empty. Then ζ^c is a fuzzy pseudo-UP ideal of X if and only if $L^{-}(\zeta :t)$ is pseudo-UP ideal of X,$\mathrm{\forall }t\in [0,1]$.

Theorem 5.22.

Let ζ be any fuzzy subset of X. Then the fuzzy subset ${\zeta }^{\star }$ of X defined by

${\zeta }^{\star }=sup\{k\in [0,1]|x\in <U(\zeta :k)>\}$

where $x\in X$ is the least fuzzy pseudo-UP ideal of X that contains ζ and $<U(\zeta :k)>$ is the least pseudo-UP ideal contains $U(\zeta :k)$.

Proof.

Suppose ζ is a fuzzy subset of X and ${\zeta }^{\star }(x)=sup\{k\in [0,1]|x\in <U(\zeta :k)>\},\mathrm{\forall }x\in X$. We need to show that ${\zeta }^{\star }$ is fuzzy pseudo-UP ideal of X. Let $t\in Im{\zeta }^{\star },ϵ>0$ be arbitrary and $\alpha =t-ϵ$, then $x\in U({\zeta }^{\star }:t)$

$\begin{array}{lll}& ⟹t\le {\zeta }^{\star }(x)=sup\{k\in [0,1]|x\in <U(\zeta :k)>\}& \\ & ⟹\alpha \le k\text{for some}k\text{such that}x\in <U(\zeta :k)>& \\ & ⟹U(\zeta :k)\subseteq U(\zeta :\alpha ),\text{by the level subsets of}\zeta & \\ & ⟹<U(\zeta :k)>\subseteq <U(\zeta :\alpha )>& \\ & ⟹x\in <U(\zeta :\alpha )>.\end{array}$

Hence $U({\zeta }^{\star }(x))\subseteq <U(\zeta :\alpha )>$………………..$(\star )$

Next, let $x\in <U(\zeta :\alpha )>$

$\begin{array}{lll}& ⟹\alpha \in \{k\in [0,1]|x\in <U(\zeta :k)>\}& \\ & ⟹\alpha \le sup\{k|x\in <U(\zeta :k)>\}& \\ & ⟹t-ϵ\le {\zeta }^{\star }(x)& \\ & ⟹t\le {\zeta }^{\star }(x),\text{since}ϵ\text{is arbitrary}& \\ & ⟹x\in U({\zeta }^{\star }:t)& \\ & ⟹<U(\zeta :\alpha )>\subseteq U({\zeta }^{\star }:t).(\star \star )\end{array}$

From $(\star )$ and $(\star \star )$, we get $<U(\zeta :\alpha )>=U({\zeta }^{\star }:t)$. Hence $U({\zeta }^{\star }:t)$ is a pseudo-UP ideal of X generated by $U(\zeta :\alpha )$. Therefore, ${\zeta }^{\star }$ is a fuzzy pseudo-UP ideal of X. Next, we want to show that ${\zeta }^{\star }$ is contains ζ. For any $x\in X$ be arbitrary. Then $\zeta (x)=sup\{k|x\in U(\zeta :k)\}$, by Theorem 2

$\le sup\{k|x\in <U(\zeta :k)>\}={\zeta }^{\star }(x)$ implies that $\zeta (x)\le {\zeta }^{\star }(x),\mathrm{\forall }x\in X$. Hence ${\zeta }^{\star }$ is contains ζ. Finally, let η be a fuzzy pseudo-UP ideal of X contains ζ. For any $x\in X$, if ${\zeta }^{\star }(x)=0$, then clearly ${\zeta }^{\star }(x)\le \eta (x)$. Assume ${\zeta }^{\star }(x)=t\ne 0$. Then $x\in U({\zeta }^{\star }:t)=<U(\zeta :\alpha )>$, so $x\in <U(\zeta :\alpha )>$. Which implies that $\eta (x)\ge \zeta (x)\ge \alpha =t-ϵ$. Thus $\eta (x)\ge t={\zeta }^{\star }(x)$. Hence ${\zeta }^{\star }\subseteq \eta .$

Theorem 5.23.

Let $(A,\cdot ,\star ,0)$ and $(B,\cdot ,\star ,0)$ be pseudo-UP algebras and let $f:A\to B$ be surjective and η be a fuzzy pseudo-UP ideals of B. Then $f^{-1}(\eta )$ is a fuzzy pseudo-UP ideals of A

Proof.

Let η is a fuzzy pseudo-UP ideal of B. Let $x,y\in A$. Then $f^{-1}(\eta )(0)=\eta (f(0))$ $\ge \eta (f(x))$

Let $x,y,z\in A$. Then,

$\begin{array}{ll}{f}^{-1}(\eta (x\cdot z))& =\eta (f(x\cdot z))\\ & =\eta (f(x)\cdot f(z))& \\ & \ge min\{\eta (f(x)\cdot (f(y)\star f(z))),\eta (f(y))\}& \\ & =min\{\eta (f(x\cdot (y\star z))),\eta (f(y))\}& \\ & =min\{f^{-1}(\eta )(x\cdot (y\star z)),f^{-1}(\eta )(y)\}.& \end{array}$

Finally,

$\begin{array}{ll}{f}^{-1}(\eta (x\star z)))& =\eta (f(x\star z))\\ & =\eta (f(x)\star f(z))& \\ & \ge min\{\eta (f(x)\star (f(y)\cdot f(z))),\eta (f(y))\}& \\ & =min\{\eta (f(x\star (y\cdot z))),\eta (f(y))\}& \\ & =min\{f^{-1}(\eta (x\star (y\cdot z))),f^{-1}(\eta (y))\}.& \end{array}$

Therefore, $f^{-1}(\eta )$ is a fuzzy pseudo-UP ideal of A.

Theorem 5.24.

Let $(A,\cdot ,\star ,0)$ and $(B,\cdot ,\star ,0)$ be a pseudo-UP algebra. $f:A\to B$ is surjective and ζ is a fuzzy pseudo-UP ideal of A. Then $f(\zeta )$ is a fuzzy pseudo-UP ideal of B, provided that the sup property holds.

Proof.

Let $0\in B$, then there exist $0\in A$ such that $f(0)=0$. Now, $f(\zeta )(0)=\{sup\zeta (t)|t\in f^{-1}(0)\}=\zeta (0)\ge \zeta (a),\mathrm{\forall }a\in A$. Let $x\in B$. Since f is surjective, we have $f^{-1}(x)\ne \mathrm{\varnothing }$ and $\zeta (0)\ge \zeta (t)|t\in f^{-1}(x).$ Which implies that $f(\zeta )(0)\ge \{sup\zeta (t)|t\in f^{-1}(x)\}=f(\zeta )(x),\mathrm{\forall }x\in B$. Hence, $f(\zeta )(0)\ge f(\zeta )(x),\mathrm{\forall }x\in B$. For every $x,y,z\in B$, then there exist $a,b,c\in A$ such that $f(a)=x,f(b)=y,f(c)=z$. Let $\zeta (a\cdot c)=sup\zeta (t)|t\in f^{-1}(x\cdot z)$ and $\zeta (b)=sup\zeta (t)|t\in f^{-1}(y)$.

Then,

$\begin{array}{ll}f(\zeta )(x\cdot z)& =\{sup\zeta (t)|t\in f^{-1}(x\cdot z)\}\\ & =\zeta (a\cdot c)& \\ & \ge min\{\zeta (a\cdot (b\star c)),\zeta (b)\}& \\ & =min\{sup\zeta (t)|t\in f^{-1}(x\cdot (y\star z)),\\ & sup\zeta (t)|t\in f^{-1}(y)\}& \\ & =min\{f(\zeta )(x\cdot (y\star z)),\zeta (y)\}.\end{array}$

$\begin{array}{ll}f(\zeta )(x\star z)& =\{sup\zeta (t)|t\in f^{-1}(x\star z)\}\\ & =\zeta (a\star c)& \\ & \ge min\{\zeta (a\star (b\cdot c)),\zeta (b)\}& \\ & =min\{sup\zeta (t)|t\in f^{-1}(x\star (y\cdot z)),\\ & sup\zeta (t)|t\in f^{-1}(y)\}& \\ & =min\{f(\zeta )(x\star (y\cdot z)),\zeta (y)\}.\end{array}$

Therefore, $f(\zeta )$ is a fuzzy pseudo-UP ideal of B.

6. Conclusion

In this paper, we introduced the concepts of fuzzy pseudo-UP subalgebra (respectively, fuzzy pseudo-UP filter and fuzzy pseudo-UP ideal) and obtained some results. The relationship between fuzzy pseudo-UP subalgebra (respectively, fuzzy pseudo-UP filter and fuzzy pseudo-UP ideal) and their level sets is discussed. Furthermore, we demonstrated that every fuzzy pseudo-UP ideal implies fuzzy pseudo-UP filter which implies fuzzy pseudo-UP subalgebra of pseudo-UP algebra, but the converse is not always true.

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No potential conflict of interest was reported by the author(s).

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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. His 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.