Full article: Fuzzy ideals and fuzzy filters of bounded semihoops

Formulae display: $MathJax Logo$ ?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.

ABSTRACT

In this research paper, we present the concept of fuzzy ideals and fuzzy filters within bounded semihoops, examining their properties. We derive several characterizations for the smallest fuzzy ideals and filters containing a given fuzzy set. Furthermore, we analyze the properties and characterizations of prime fuzzy ideals, prime fuzzy filters, fuzzy primary ideals, and fuzzy primary filters. Lastly, we explore the fuzzy congruence relation associated with fuzzy (primary) ideals and fuzzy (primary) filters.

KEYWORDS:

MATHEMATICS SUBJECT CLASSIFICATION:

1. Introduction

Hoops, introduced by Bosbach (Citation1969, Citation1970), are naturally ordered commutative residuated integral monoids. The class of basic hoop algebras, as established by Aglianó et al. (Citation2007), is a significant development within the realm of algebra. It has been proven that any variety of BL-algebras encompasses the variety of basic hoop algebras. Semihoops, a more general form of hoops, were introduced by Esteva et al. (Citation2003). A semihoop is a type of hoop that does not meet the divisibility requirement. It encompasses all algebraic structures that are induced by left-continuous t-norms, making it significant in fuzzy logics and related algebraic structures. The study of these algebras and the completeness of the correspondence logic system involves the important role of filter and ideal theory. The examination of ideals and filters on semihoops has been extensive, leading to interesting findings in (NiuIdeal et al., Citation2020) and (Borzooei & Aaly Kologani, Citation2015), respectively. Moreover, in these works, the definitions and characterizations of prime, primary, and perfect ideals (and filters) of semihoops were introduced.

The concept of fuzzy subsets, introduced by L. A. Zadeh as an extension of classical set theory in (Zadeh, Citation1965), revolutionized the way we perceive sets. A fuzzy subset of a set is mathematically defined by assigning a value within the unit interval $[0, 1]$ to each element in the universe of discourse, indicating its degree of membership. Fuzzy set theory offers methods to represent imprecision and uncertainty in situations lacking clear boundaries. This mathematical framework enables us to model imprecision and uncertainty inherent in phenomena with vague boundaries. Subsequently, extensive research on fuzzy sets has emerged, with numerous applications across different domains (see (Lu et al., Citation2023)).

Rosenfeld, in his seminal paper (Rosenfeld, Citation1971), has introduced the concept of fuzziness into group theory. This research has inspired scholars to reconsider various abstract algebra concepts and findings within the broader context of a fuzzy environment, providing motivation for further exploration. Recent studies have focused on exploring topics such as fuzzy ideals and fuzzy prime ideals in universal algebras (Alaba & Addis, Citation2019a, Citation2019b), expanding the application of fuzzy concepts. Furthermore, the extension of fuzzy ideals and filters in MS-algebras and distributive lattices has been the subject of investigation in a series of papers (Alaba & Alemayehu, Citation2019a, Citation2019b; Norahun, Citation2020), highlighting the growing interest in this area. Moreover, a fuzzy congruence relation within an algebraic structure is defined as a fuzzy equivalence relation compatible with all basic operations of the algebra. The study of fuzzy congruence relations has been extended to various algebraic structures, demonstrating the broad applicability of this concept. Notably, recent research has focused on this topic in different contexts: in Ockham algebras (see (Alemayehu et al., Citation2021, Citation2022)) and more broadly, in universal algebras (see (Addis, Citation2022)), showing the diverse applications of fuzzy congruence relations.

The study of fuzzification in algebraic structures remains an active field of mathematical research. In the existing literature, researchers have explored different aspects of fuzzy ideals and filters in universal algebras, distributive lattices, and MS-algebras, contributing to the advancement of understanding. The research conducted by these scholars has significantly advanced the understanding and application of fuzzy concepts in different algebraic structures and settings. It has been noted that many fundamental results in conventional algebraic structures also apply to fuzzified algebraic concepts, highlighting the relevance and significance of this research direction. The exploration of fuzzy ideals and fuzzy filters within bounded semihoops is motivated by the desire to extend traditional algebraic concepts into the realm of fuzzy algebra. By investigating these fuzzy structures in the context of bounded semihoops, we aim to enhance our understanding of how imprecision and uncertainty can be effectively modeled within algebraic systems. The concept of fuzzy ideals and fuzzy filters within bounded semihoops is a fascinating area of study in algebra.

Building upon the aforementioned studies, this paper delves into exploring fuzzy ideals and fuzzy filters within a bounded semihoop, examining their properties. Furthermore, it delves into characterizing fuzzy ideals and filters generated by a fuzzy subset. Additionally, it investigates prime fuzzy ideals and filters, fuzzy primary ideals and filters, along with their respective characterizations. Lastly, it explores the concept of fuzzy congruence relations associated with fuzzy (primary) ideals and fuzzy (primary) filters, contributing to the ongoing discourse in this evolving field.

The unit interval [0,1] is recognized for its numerous lattice theoretic and topological properties. Due to this, we opt to use the unit interval [0,1] as the set of truth degrees for our fuzzy statements throughout the paper. The paper is organized as follows. In Section 2, we gather the basic notions and results in fuzzy sets and bounded (semihoops). In Section 3 and Section 4, we introduce the notion of fuzzy ideals (respectively, filters) in bounded semihoop and investigate their properties. Also, characterizations of fuzzy ideals and fuzzy filters generated by a fuzzy subset are discussed. In Section 5, we explore the concept of fuzzy congruence relations associated with fuzzy (primary) ideals and fuzzy (primary) filters.

2. Preliminaries

In this topic, we start with basic definitions and important results in fuzzy set and semihoops which we need in the sequel.

Definition 2.1.

(Esteva et al., Citation2003)

An algebra $(H, ⊙, $ \break $ \to, \land, 1)$ of type $(2, 2, 2, 0)$ is called a semihoop if it satisfies the following conditions:

$(H, \land, 1)$ is a $\land$ semilattice with greatest element 1,
$(H, ⊙, 1)$ is a commutative monoid,
$(s ⊙ t) \to u = s \to (t \to u)$ , for all $s, t, u \in H$ .

By a fuzzy subset ξ of a semihoop H, we mean a mapping $ξ : H \to [0, 1]$ . The set ${ξ (s) : s \in H}$ is called the image of ξ, and is denoted by $Im (ξ)$ . For each $α \in [0, 1]$ , the set

\begin{array}{l} ξ^{- 1} ([α, 1]) = {s \in H : α \leq ξ (h) \leq 1} . \end{array}

is called the α-level subset of $ξ .$ For any fuzzy subset ξ of a semihoop H and each $t \in H$ , we have

\begin{array}{l} ξ (s) = ⋁ {α \in [0, 1] : α \in ξ^{- 1} ([α, 1])} . \end{array}

For any fuzzy subsets ξ and σ of a semihoop H, we write

\begin{array}{l} ξ \leq σ to\ mean ξ (s) \leq σ (s), for\ all s \in H \end{array}

in the ordering of $[0, 1]$ . It can be easily verified that $" \leq "$ is a partial order on the set $[0, 1]^{H}$ and is called the point wise ordering.

Given a semihoop $(H, ⊙, \to, \land, 1)$ , the binary relation ≤ on H defined by: $s \leq t if and only if s \to t = 1, ∀s, t \in H$ is a partial ordering on H which we call the natural ordering on H and $s \leq 1,$ for all $s \in H .$ For any $s \in H$ and $n \in N \cup {0},$ where $N$ is a set of natural number, define a power of s by:

\begin{array}{l} s^{n} = {\begin{cases} 1 & if n = 0 \\ s^{n - 1} ⊙ s & otherwise \end{cases} \end{array}

Theorem 2.2.

(Esteva et al., Citation2003)

Let $(H, ⊙, \to, \land, 1)$ be a semihoop. Then, the following properties are true for any $s, t, u \in H$ :

$s ⊙ t \leq u \Leftrightarrow$ $s \leq t \to u;$
$s ⊙ t \leq s, t;$
$1 \to s = s, s \to 1 = 1;$
$s^{n} \leq s$ ,
$t \leq s \to t;$
if $s \leq t$ , then $t \to u \leq s \to u, u \to s \leq u \to t$ and $s ⊙ u \leq t ⊙ u;$
$s \leq (s \to t) \to t;$
$s \to t \leq (u \to s) \to (u \to t), s \to t \leq (t \to u) \to (s \to u);$
$s \to (t \to u) = t \to (s \to u) .$

A semihoop H is said to be bounded if there exists $0 \in H$ such that $0 \leq s \forall s \in H$ . In a bounded semihoop H one can define the negation of each $s \in H$ by $s^{*} = s \to 0$ . An element s in a bounded hoop H is defined to be closed if $s^{* *} = s$ , and H is said to have the double negation property (or DNP for short) provided that each of its element is closed; i.e. $s^{* *} = s,$ for all $s \in H$ .

Theorem 2.3.

(Borzooei & Aaly Kologani, Citation2015)

If H is a bounded semihoop, then the following properties hold, for any $s, t \in H$ :

$1^{*} = 0, 0^{*} = 1;$
$s \leq s^{* *};$
$s ⊙ s^{*} = 0;$
$t^{*} \leq t \to s;$
$s \leq t$ implies $t^{*} \leq s^{*};$
$s \to t \leq t^{*} \to s^{*};$
If H has a DNP, then $s \to t = t^{*} \to s^{*}$ .

Lemma 2.4.

(NiuIdeal et al., Citation2020)

Let H be a bounded semihoop . Then define a binary operation $\oplus$ on H by:

\begin{array}{l} s \oplus t = s^{*} \to t, for any s, t \in H . \end{array}

Then the following properties hold for any $s, t, u \in H$ :

$s \leq t$ , then $s \oplus u \leq t \oplus u;$
$s \leq s \oplus t;$
$s \oplus s^{*} = 1;$
$0 \oplus s = s, s \oplus 0 = s^{* *};$
$s \oplus t = 1 \Leftrightarrow s^{*} \leq t^{*};$
$(s \oplus t) \oplus u \leq s \oplus (t \oplus u);$
if H has a DNP, then $s^{*} ⊙ t^{*} = (s \oplus t)^{*};$
if H has a DNP, then $s^{*} \oplus t^{*} = (s ⊙ t)^{*}$ .

Note that if H is a bounded semihoop with DNP, then the binary operation $\oplus$ is commutative and associative.

Theorem 2.5.

(Borzooei & Aaly Kologani, Citation2015)

Define a binary operation $^{'} \lor^{'}$ on a semihoop H by:

\begin{array}{l} s \lor t = ((s \to t) \to t) \land ((t \to s) \to s), for any s, t \in H . \end{array}

Then the following conditions are equivalent:

$\lor$ is associative;
for all $s, t, u \in H;$ $s \leq t \Rightarrow s \lor u \leq t \lor u;$
for all $s, t, u \in H;$ $s \lor (u \land t) \leq (s \lor t) \land (s \lor u);$
$s \lor t$ is the supremum of s and t with respect to the natural ordering on H.

A semihoop H is called a $\lor$ -semi hoop, if it satisfies one of the equivalent conditions of the above theorem.

Definition 2.6.

(NiuIdeal et al., Citation2020)

Let H be a bounded semihoop. A non-empty subset I of H said to be an ideal of H, if it satisfies:

$s \leq t$ and $t \in I$ imply $s \in I,$ for any $s, t \in H;$
$s \oplus t \in I$ , for any $s, t \in I .$

The set of all ideals of H will be denoted by $I (H)$ . If $\emptyset \neq S \subseteq H,$ then the intersection of all ideals of H containing S is denoted by $(S]$ and is characterized by:

\begin{array}{l} (S] = & {t \in H : ∃n \in N and s_{1}, \dots, s_{n} \in S, t \\ \leq s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots)} . \end{array}

In particular, for any element $s \in H$ we have

\begin{array}{l} (s] & = {t \in H : t \leq ns for some n \in N}, where \\ ns & = s \oplus (\dots \oplus (s \oplus (s \oplus s) \dots) \end{array}

Definition 2.7.

(NiuIdeal et al., Citation2020)

A proper ideal I of a bounded semihoop H is called a primary ideal, if for all $s, t \in H$ , $s ⊙ t \in I$ implies either $s^{n} \in I$ or $t^{n} \in I$ , for some positive integer n.

Definition 2.8.

(NiuIdeal et al., Citation2020)

Let H be a semihoop. A nonempty subset F is called a filter of H, if it satisfies the following conditions:

$s ⊙ t \in F$ , for all $s, t \in F;$
for all $s, t \in H,$ $s \leq t$ and $s \in F$ imply $t \in F$

Theorem 2.9.

(NiuIdeal et al., Citation2020)

A subset F of a semihoop H is said to be a filter of H if and only if the following conditions hold:

$1 \in F$ ;
$s, s \to t \in F \Rightarrow t \in F$ , for all $s, t \in H$ .

The set of all filters of H will be denoted by $F (H)$ . If $\emptyset \neq S \subseteq H,$ then the intersection of all filters of H containing S is denoted by $[S)$ and is characterized by:

\begin{array}{l} [S) = {t \in H : ∃n \in N and s_{1}, \dots, s_{n} \in S, s_{1} ⊙ \dots ⊙ s_{n} \leq t} \end{array}

In particular, for any element $s \in H$ we have $[s) = {t \in H : s^{n} \leq t, for\ some n \in N}$ .

Theorem 2.10.

(NiuIdeal et al., Citation2020)

A proper filter of K of a bounded semihoop H is called a primary filter, if for all $s, t \in H,$ $s ⊙ t \in K$ implies $(s^{n})^{*} \in K$ or $(t^{n})^{*} \in K$ , for some positive integer n .

3. Fuzzy ideals of on a bounded semihoops

In this section, we define fuzzy ideals in a bounded semihoop and investigate some of its properties. Throughout this paper, H stands for a bounded semihoop $(H, ⊙, \to, \land, 1)$ unless otherwise stated. We begin with the following.

Definition 3.1.

Let H be a bounded semihoop. Then, a fuzzy subset ξ of H is called the fuzzy ideal of ξ, if for all $s, t \in H$ the following holds:

$ξ (0) = 1;$
$ξ (s) \land ξ (t) \leq ξ (s \oplus t);$
$s \leq t$ implies $ξ (t) \leq ξ (s) .$

A fuzzy ideal ξ of H is called proper if $\exists s \in H$ such that $ξ (s) \neq 1.$

$FI (H)$ will represent the class of all fuzzy ideals of H. It is evident that the characteristic functions $χ_{{0}}$ and χ_H of ${0}$ and H respectively, belongs to $FI (H)$ and hence it is nonempty.

Example 3.2.

Let $H = {0, w, x, y, z, 1}$ with $0 < x, z < w < 1, 0 < y < c < 1$ , where w and y are incomparable, x and z are incomparable. Define $⊙$ and $\to$ on H as follows:

\begin{array}{l} \begin{array}{l} \begin{array}{l} ⊙ & 0 & w & x & y & z & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ w & 0 & x & x & x & 0 & w \\ x & 0 & x & x & 0 & 0 & x \\ y & 0 & z & 0 & x & z & y \\ z & 0 & 0 & 0 & z & 0 & z \\ 1 & 0 & w & x & y & z & 1 \end{array} & \begin{array}{l} \to & 0 & w & x & y & z & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ w & 0 & x & x & x & 0 & w \\ x & 0 & x & x & 0 & 0 & x \\ y & 0 & z & 0 & x & z & y \\ z & 0 & 0 & 0 & z & 0 & z \\ 1 & 0 & w & x & y & z & 1 \end{array} \end{array} \end{array}

Then

(H, ⊙, \to, \land, 1)

is a bounded semihoop, where

s \land t = s ⊙ (s \to t),

for any

s, t \in H

. Define a fuzzy subsets ξ and λ on H by:

\begin{array}{l} ξ (0) & = 1 = ξ (x), ξ (w) = ξ (y) = ξ (z) = 0.6, ξ (1) = 0.2 and \\ λ (0) & = 1, λ (y) = λ (z) = 0.7, λ (w) = λ (x) = λ (1) = 0.6 . \end{array}

Then it is routine to verify that ξ and λ are fuzzy ideals of H.

Theorem 3.3.

A fuzzy subset ξ of H is a fuzzy ideal of H if and only if $ξ^{- 1} ([α, 1])$ is an ideal of $H,$ for all $α \in [0, 1]$ .

Proof.

Suppose that ξ is a fuzzy ideal of H and $α \in [0, 1]$ . Then $α \leq 1 = ξ (0)$ and so $0 \in φ^{- 1} ([α, 1])$ . Again let $s, t \in φ^{- 1} ([α, 1]) .$ Then $α \leq ξ (s)$ and $α \leq ξ (t)$ This implies that $α \leq ξ (s) \land ξ (t) \leq ξ (s \oplus t)$ and hence $s \oplus t \in ξ^{- 1} ([α, 1])$ . Again let $s \leq t$ and $t \in φ^{- 1} ([α, 1]) .$ Then $α \leq ξ (t) \leq ξ (s)$ and so $s \in ξ^{- 1} ([α, 1]) .$ So $ξ^{- 1} ([α, 1])$ is an ideal of H.

Conversely, suppose that $ξ^{- 1} ([α, 1])$ is an ideal of $H,$ for all $α \in [0, 1]$ . In particular, $ξ^{- 1} ({1})$ is an ideal. Since $0 \in ξ^{- 1} ({1})$ , we have $ξ (0) = 1$ . Again let $s, t \in H$ and put $α = ξ (s) \land ξ (t)$ . This implies that $s, t \in ξ^{- 1} ([α, 1])$ so we have $s \oplus t \in φ^{- 1} ([α, 1])$ . Therefore $ξ (s) \land ξ (t) = α \leq ξ (s \oplus t) .$ Again let $s, t \in H$ such that $s \leq t .$ Put $β = ξ (t) .$ Then $t \in φ^{- 1} ([β, 1]) .$ Since $φ^{- 1} ([β, 1])$ is an ideal of H and $s \leq t,$ we have $s \in ξ^{- 1} ([β, 1]) .$ Therefore $ξ (t) = β \leq ξ (s)$ and so ξ is a fuzzy ideal of $H .$

Corollary 3.3.1.

I is an ideal of H if and only if the characteristic function χ_I of I is a fuzzy ideal of H.

Theorem 3.4.

Let H be a bounded semihoop and ξ be a fuzzy subset of H. Then the following conditions are equivalent:

ξ is a fuzzy ideal of $H;$
$ξ (0) = 1$ , $ξ (s) \land ξ (t) \leq ξ (s \oplus t)$ and $ξ (s^{*} ⊙ t) \land ξ (s) \leq ξ (t),$ for any $s, t \in H;$
$ξ (0) = 1$ , $ξ (s) \land ξ (t) \leq ξ (s \oplus t)$ and $ξ ((s^{*} \to t^{*})^{*}) \land ξ (s) \leq ξ (t),$ for any $s, t \in H;$
ξ is normalized, $ξ (s) \land ξ (t) \leq ξ (s \oplus t)$ and $ξ (s) \leq ξ (s \land t),$ for any $s, t \in H$ .

Proof.

$(1) \Rightarrow (2) .$ Suppose that ξ is a fuzzy ideal of H. Then, by definition, $ξ (0) = 1.$ Let $s, t \in H .$ Then $ξ (s) \land ξ (t) \leq ξ (s \oplus t)$ and since $t \to (s \oplus (s^{*} ⊙ t)) = t \to (s^{*} \to (s^{*} ⊙ t))) = (t ⊙ s^{*}) \Rightarrow (t ⊙ s^{*}) = 1$ we have $t \leq (s \oplus (s^{*} ⊙ t))$ and hence

\begin{array}{l} ξ (s^{*} ⊙ t) \land ξ (s) & \leq ξ (s^{*} ⊙ t) \land ξ (s) \\ \leq ξ (s \oplus (s^{*} ⊙ t)) \\ \leq ξ (t) . \dots (since t \leq (s \oplus (s^{*} ⊙ t))) \end{array}

Hence (2) holds.

$(2) \Rightarrow (3) .$ Suppose that (2) holds. Let $s, t \in H .$ Then since $s^{*} ⊙ t^{* *} \leq (s^{*} \to t^{* * *})^{*} = (s^{*} \to t^{*})^{*},$ we have

\begin{array}{l} ξ ((s^{*} \to t^{*})^{*}) \land ξ (s) & \leq ξ (s^{*} ⊙ t^{* *}) \land ξ (s) \\ \leq ξ (t^{* *}) \dots (by (2)) \\ \leq ξ (t) \dots (since t \leq t^{* *}) \end{array}

Hence (3) holds.

$(3) \Rightarrow (1) .$ Suppose that (3) holds. Then $ξ (0) = 1$ Again let $s, t \in H$ such that $s \leq t$ . Then $t^{*} \leq s^{*} .$ This implies that $(t^{*} \to s^{*})^{*} = (1)^{*} = 0$ and hence $ξ ((t^{*} \to s^{*})^{*} = 1.$ Thus $ξ (t) = ξ (t) \land 1 = ξ (t) \land ξ ((t^{*} \to s^{*})^{*}) \leq ξ (s),$ by $(3) .$ Hence $(3) \Rightarrow (1)$ holds true.

$(1) \Rightarrow (4) .$ It is clear.

$(4) \Rightarrow (1) .$ Suppose that (4) holds. Since ξ is normalized, there exists $u \in H$ such that $ξ (u) = 1.$ Now since $1 = ξ (u) \leq ξ (u \land 0) = ξ (u ⊙ (u \to 0)) = ξ (u ⊙ u^{*}) = ξ (0),$ we have $ξ (0) = 1.$ Again let $s, t \in H$ such that $s \leq t .$ Then $s \to t = 1$ . Now and $ξ (t) \leq ξ (t \land s) = ξ (s \land t) = ξ (s ⊙ (s \to t)) = ξ (s ⊙ 1) = ξ (s) .$ Hence (1) holds.

Corollary 3.4.1.

Let ξ is a fuzzy ideal of H and $s \in H .$ Then $ξ (s) = 1$ if and only if $ξ (s^{* *}) = 1.$

Proof.

Suppose that $ξ (s) = 1.$ Since $s^{*} \leq s^{* * *}$ we have $ξ ((s^{*} \to s^{* * *})^{*}) = ξ (1^{*}) = ξ (0) = 1.$ Now, by Theorem 3.4(3), $1 = ξ ((s^{*} \to s^{* * *})^{*}) \land ξ (s) \leq ξ (s^{* *}),$ we have $ξ (s^{* *}) = 1.$ Conversely, suppose $ξ (s^{* *}) = 1.$ Since $s \leq s^{* *},$ we have $1 = ξ (s^{* *}) \leq ξ (s)$ and hence $ξ (s) = 1.$

Theorem 3.5.

If ${ξ_{i} : i \in Ω}$ is a class of fuzzy ideals of H, then $\cap_{i \in Ω} ξ_{i}$ is a fuzzy ideal of H.

From Theorem 3.5, it can be easily observed that the class $FI (H)$ of all fuzzy ideals of H is closed under arbitrary intersection. It follows from this fact that for any fuzzy subset φ in H, we can find the smallest fuzzy ideal of H containing φ which we call the fuzzy ideal of H generated by φ and will be denoted by $(φ]$ . The following theorems give characterization for $(φ]$ in different context.

In the following theorem, we characterize fuzzy ideals of H generated by a fuzzy subset in terms of ideals generated by its level subset.

Theorem 3.6.

For any fuzzy subset φ in H and any $s \in H$ :

\begin{array}{l} (φ] (s) = ⋁ {α \in [0, 1] : s \in (φ^{- 1} ([α, 1])]} \end{array}

Proof.

Let us put ξ to be a fuzzy subset of H given by:

\begin{array}{l} ξ (s) = ⋁ {α \in [0, 1] : s \in (φ^{- 1} ([α, 1])]} \end{array}

We claim that ξ is the smallest fuzzy ideal of H containing φ. Let us first show that $\hat{ξ}$ is an $L$ -filter. Clearly $ξ (0) = 1$ . Let $s, t \in H .$ Then

\begin{array}{l} ξ (s) \land ξ (t) & = ⋁ {α \in [0, 1] : s \in (φ^{- 1} ([α, 1])]} \\ \land ⋁ {β \in [0, 1] : t \in (φ^{- 1} ([β, 1])]} \\ = ⋁ {α \land β : s \in (φ^{- 1} ([α, 1])], t \in (φ^{- 1} ([β, 1])]} \end{array}

If we put $γ = α \land β$ , then we get $(φ^{- 1} ([α, 1])] \subseteq (φ^{- 1} ([γ, 1])]$ and $(φ^{- 1} ([β, 1])] \subseteq (φ^{- 1} ([γ, 1])]$ So that $s, t \in (φ^{- 1} ([γ, 1])]$ . Now it follows from the above equality that;

\begin{array}{l} ξ (s) \land ξ (t) & = ⋁ {α \land β : s \in (φ^{- 1} ([α, 1])], t \in (φ^{- 1} ([β, 1])]} \\ \leq ⋁ {γ \in [0, 1] : s \oplus t \in (φ^{- 1} ([γ, 1])] \\ = ξ (s \oplus t) \end{array}

Let $s, t \in H$ such that $s \leq t .$ Now

\begin{array}{l} ξ (t) & = ⋁ {α \in [0, 1] : t \in (φ^{- 1} ([α, 1])]} \\ \leq ⋁ {α \in [0, 1] : s \in (φ^{- 1} ([γ, 1])] \\ = ξ (s) \end{array}

Therefore ξ is a fuzzy ideal of H. It is also clear to see that $φ \subseteq ξ$ . Suppose that σ is any other fuzzy ideal of H such that $φ \subseteq σ$ . Then it is clear that $(φ^{- 1} ([α, 1]] \subseteq σ^{- 1} ([α, 1]),$ for all $α \in [0, 1]$ . Now for any $s \in H$ consider:

\begin{array}{l} ξ (s) & = ⋁ {α \in [0, 1] : s \in (φ^{- 1} ([α, 1]]} \\ \leq ⋁ {α \in [0, 1] : s \in (σ^{- 1} ([α, 1]]} \\ \leq ⋁ {α \in [0, 1] : s \in σ^{- 1} ([α, 1)} \\ = σ (s) . \end{array}

Therefore ξ is the smallest fuzzy ideal containing φ, that is, $ξ = (φ]$ .

Theorem 3.7.

Let S be any subset of H. Then the $(χ_{S}] = χ_{(S]}$ .

Proof.

Since $(S]$ is an ideal of H, by Corollary 3.3.1, we have $χ_{(S]}$ is a fuzzy ideal of H. Again since $S \subseteq (S]$ , we have $χ_{S} \subseteq χ_{(S]}$ . Let ξ be any fuzzy ideal of H such that $χ_{S} \subseteq ξ$ . Now we claim that $χ_{(S]} \subseteq ξ$ . Let $s \in H$ . If $h \notin (S]$ , then $χ_{(S]} (s) = 0 \leq ξ (s)$ . Let $s \in (S]$ . Since $χ_{S} \subseteq ξ$ , we have $S \subseteq ξ^{- 1} ({1})$ (1-level subset of ξ). This implies that $(S] \subseteq (ξ^{- 1} ({1}] = ξ^{- 1} ({1})$ . Thus $s \in ξ_{1}$ and hence $ξ (s) = 1$ . Therefore $χ_{(S]} (s) = 1 = ξ (s)$ . So that $χ_{(S]} (s) \leq ξ (s),$ for all $s \in H .$ Hence the claim holds. Therefore $χ_{(S]}$ is the smallest fuzzy ideal of H containing χ_S and hence $(χ_{S}] = χ_{(S]} .$

The following is also another algebraic characterization of fuzzy ideals generated by a fuzzy subset of H.

Theorem 3.8.

Let φ be a fuzzy subset of H. Then, the fuzzy subset $\overset{―}{φ}$ defined by: $\overset{―}{φ} (0) = 1$ and for t ≠ 0,

\begin{array}{l} \overset{―}{φ} (t) & = ⋁ {⋀_{i = 1}^{n} φ (s_{i}) : t \leq s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots)), \\ s_{1}, \dots, s_{n} \in H} \end{array}

is the fuzzy ideal of H generated by φ.

Proof.

Let $0 \neq t \in H$ and put

\begin{array}{l} P_{t} & = {⋀_{i = 1}^{n} φ (s_{i}) : t \leq s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots)), s_{i} \in H} \\ and Q_{t} = {α \in [0, 1] : s \in (φ^{- 1} ([α, 1])]} . \end{array}

Then, by Theorem 3.6, it is enough to show that $⋁ P_{t} = ⋁ Q_{t}$ . Let $α \in P_{t}$ . Then $α = ⋀_{i = 1}^{n} φ (s_{i})$ for some $s_{1}, \dots, s_{n} \in H$ such that $t \leq s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots))$ . This implies that $α \leq φ (s_{i}),$ for all $1 \leq i \leq n$ and so $s_{i} \in φ^{- 1} ([α, 1]) \subseteq (φ^{- 1} ([α, 1])]$ . Since $(φ^{- 1} ([α, 1])]$ is an ideal, we have $s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots)) \in (φ^{- 1} ([α, 1])]$ and as $t \leq s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots)) \in (φ^{- 1} ([α, 1])]$ we have $t \in (φ^{- 1} ([α, 1])]$ , i.e. $α \in Q_{t}$ . Hence $P_{t} \subseteq Q_{t} .$ and so $⋁ P_{t} \leq ⋁ Q_{t}$ . Again let $β \in Q_{t}$ . Then $t \in ((φ^{- 1} ([β, 1])]$ . Then $t \leq s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots)$ for some $s_{1}, \dots, s_{n} \in φ^{- 1} ([β, 1]) .$ This implies that $β \leq ⋀_{i = 1}^{n} φ (s_{i})$ . Put $α = ⋀_{i = 1}^{n} φ (s_{i})$ . Then since $t \leq s_{n} \oplus (\dots \oplus (s_{3} \oplus (s_{2} \oplus s_{1}) \dots)$ , we have $α \in P_{t}$ . Thus for each $β \in Q_{t},$ we get $α \in P_{t}$ such that $β \leq α .$ So $⋁ Q_{t} \leq ⋁ P_{t}$ . Hence $⋁ P_{t} = ⋁ Q_{t}$ . Therefore $\overset{―}{φ} = (φ]$ .

Definition 3.9.

Let H be a bounded semihoop with DNP. The proper fuzzy ideal λ of H is called a prime fuzzy ideal of H, if for any fuzzy ideals ξ and ν of H,

\begin{array}{l} ξ \cap ν \subseteq λ implies\ either ξ \subseteq λ or ν \subseteq λ . \end{array}

Lemma 3.10.

Let I be an ideal of H and $α \in [0, 1)$ . Then the fuzzy subset $I_{α}^{1}$ of H defined by:

\begin{array}{l} I_{α}^{1} (s) = {\begin{cases} 1 & if s \in I \\ α & if s \notin I \end{cases}, \end{array}

for all $s \in H$ is a fuzzy ideal of H.

In the following we characterize prime fuzzy ideal of H in terms of prime ideals of H and an element in $[0, 1) .$

Theorem 3.11.

Let H be a bounded $\lor$ -semihoop with DNP, P be an ideal of H and $1 \neq α \in [0, 1]$ . Then $P_{α}^{1}$ is a prime fuzzy ideal of H if and only if P is a prime ideal of $H .$

Proof.

Suppose that $P_{α}^{1}$ is a prime fuzzy ideal of H. Now we show that P is a prime ideal of $H .$ Since $P_{α}^{1}$ is proper, we have $P \neq H .$ . Let R and S be ideals of H such that $R \cap S \subseteq P$ . Then $(S \cap R)_{α}^{1} \subseteq P_{α}^{1} .$ Now since $R_{α}^{1} \cap S_{α}^{1} = (R \cap S)_{α}^{1} \subseteq P_{α}^{1}$ and $P_{α}^{1}$ is a prime fuzzy ideal of H, we have either $R_{α}^{1} \subseteq P_{α}^{1}$ or $S_{α}^{1} \subseteq P_{α}^{1} .$ This implies that either $R \subseteq P$ or $S \subseteq P .$ Thus P is a prime ideal of H.

Conversely, suppose that P is a prime ideal of H and $α \in [0, 1) .$ Clearly, $P_{α}^{1}$ is a proper fuzzy ideal of H. Now we show that $P_{α}^{1}$ is prime L-fuzzy prime ideal. Suppose not. Then there exist fuzzy ideals ξ and ν of H such that

\begin{array}{l} ξ \cap ν \subseteq P_{α}^{1} and ξ ⊈ P_{α}^{1} and ν ⊈ P_{α}^{1} . \end{array}

Then there exist $s, t \in H$ such that

\begin{array}{l} ξ (s) > P_{α}^{1} (s) and ν (t) > P_{α}^{1} (t) . \end{array}

This implies that $P_{α}^{1} (s) = P_{α}^{1} (t) = α$ and so $s \notin P$ and $t \notin P$ . Since P is a prime ideal of a bounded $\lor$ -semihoop with DNP, we have $s \land t \notin P$ and so $P_{α}^{1} (s \land t) = α .$ Now we have

\begin{array}{l} (ξ \cap ν) (s \land t) & = ξ (s \land t) \land ν (s \land t) \geq ξ (s) \land ν (t) > α \land α \\ = α = α_{P} (s \land t), \end{array}

which contradicts our assumption $ξ \cap ν \subseteq P_{α}^{1} .$ Hence $P_{α}^{1}$ is a prime fuzzy ideal of H.

Theorem 3.12.

Let ξ be a fuzzy ideal of H, where H is a bounded $\lor$ -semihoop with DNP. Then ξ is a prime fuzzy ideal of H if and only if there exist prime ideal of P of H such that $ξ = P_{α}^{1}$ , where $α \in [0, 1)$ .

Proof.

Let H be a bounded $\lor$ -semihoop with DNP. Suppose that ξ is a prime fuzzy ideal of H. Since ξ is proper it assumes at least two values. Since $ξ (0) = 1,$ 1 is necessarily in $Im (ξ)$ . Suppose that $α, β \in Im (ξ)$ other than 1. Then there exist $s, t \in H$ such that $ξ (s) = α$ and $ξ (t) = β$ . Now we claim that $α = β$ . Now put $P = ξ_{1} = {u \in H : ξ (u) = 1}$ . Consider the fuzzy ideals $χ_{(s]}$ and $P_{α}^{1} .$ Now we claim to show that $χ_{(s]} \cap P_{α}^{1} \subseteq ξ$ . For any $u \in H$ , if $u \notin (s]$ , then it is clear that $(χ_{(a]} \cap P_{α}^{1}) (u) \leq ξ (u) .$ Let $u \in (s]$ . Now, in this case, if $u \in P$ , we have

\begin{array}{l} (χ_{(s]} \cap P_{α}^{1}) (u) = χ_{(s]} (u) \land P_{α}^{1} (u) = 1 = ξ (u) . \end{array}

and if $u \notin P$ , then we have

\begin{array}{l} (χ_{(s]} \cap P_{α}^{1}) (u) = χ_{(s]} (u) \land P_{α}^{1} (u) = 1 \land α = α = ξ (s) \leq ξ (u) . \end{array}

Therefore in either cases, we have $(χ_{(s]} \cap P_{α}^{1}) (u) \leq ξ (u)$ , for all $u \in H$ and so

\begin{array}{l} χ_{(s]} \cap P_{α}^{1} \subseteq ξ . \end{array}

But as ξ is a prime fuzzy ideal of H, we have

\begin{array}{l} χ_{(a]} \subseteq ξ or P_{α}^{1} \subseteq ξ \end{array}

But as $χ_{(s]} (s) = 1 > α = ξ (s)$ , we have $χ_{(s]} ⊈ ξ$ . Therefore $P_{α}^{1} \subseteq ξ$ . In particular, since $t \notin P$ , we get that $α = P_{α}^{1} (t) \leq ξ (t) = β$ . In similar fashion, we can show that $β \leq α$ and hence $α = β$ . So ξ assumes exactly one value say α other than 1 and hence $μ = P_{α}^{1}$ .

Again to show P is a prime ideal, let $R, S$ be ideals of H such that $R \cap S \subseteq P$ . Then since $R \cap S \subseteq P \Rightarrow χ_{R} \cap χ_{S} = χ_{R \cap S} \subseteq χ_{P} \subseteq P_{α}^{1} = ξ$ and ξ is a prime fuzzy ideal, we have either

\begin{array}{l} χ_{R} \subseteq ξ = P_{α}^{1} or χ_{S} \subseteq ξ = P_{α}^{1} \end{array}

This implies that $R \subseteq P or S \subseteq P$ . and hence either P is a prime ideal of H. The converse part of this theorem follows from the Theorem 3.11.

Theorem 3.13.

Let H be H is a bounded $\lor$ -semihoop with DNP. Then if ξ is a prime fuzzyy ideal of H, then $ξ (s \land t) = ξ (s)$ or $ξ (t),$ for all $s, t \in H$ .

Proof.

Suppose that ξ is a prime ideal of $H .$ Then $ξ = P_{α}^{1}$ for some prime ideal P of H and $α \in [0, 1) .$ Let $s, t \in H .$ If $s \in P$ or $t \in P$ , then $s \land t \in P$ and so $ξ (s \land t) = 1 = ξ (s) or ξ (t)$ . If $s \notin P$ and $t \notin P$ , then $x \land y \notin I$ as a prime ideal and so $ξ (s \land t) = α = ξ (s) = ξ (t)$ . Hence in either cases, we have $ξ (s \land t) = ξ (s)$ or $ξ (t)$

Theorem 3.14.

Let H be a bounded semihoop and $s^{2} = s$ , for any $s \in H$ . If ξ fuzzy primary ideal of H, then $ξ (s \land t) = ξ (s) \lor ξ (t),$ for all $s, t \in H$ .

Proof.

Suppose that ξ a fuzzy primary ideal of H. Let $s, t \in H .$ Then, since $s ⊙ t \leq s \land t,$ we have $ξ (s \land t) \leq ξ (s ⊙ t) .$ Again since ξ is fuzzy primary ideal, there exists $n \in N$ such that $ξ (s ⊙ t) \leq ξ (s^{n}) \lor ξ (t^{n}) .$ Since $s^{2} = s,$ for all $s \in H,$ we get $s^{n} = s$ and $t^{n} = t .$ Therefore $ξ (s \land t) \leq ξ (s) \lor ξ (t)$ .

Definition 3.15.

A proper fuzzy ideal ξ of H. is called a fuzzy primary ideal, if for all $s, t \in H$ :

\begin{array}{l} ξ (s ⊙ t) \leq ξ (s^{n}) \lor ξ (t^{n}), for some n \in N . \end{array}

Example 3.16.

In Example 3.2, λ is a primary fuzzy ideal of H.

Theorem 3.17.

A fuzzy subset ξ of H is a fuzzy primary ideal of H if and only if $ξ^{- 1} ([α, 1])$ is a primary ideal of $H,$ for all $α \in [0, 1]$ .

Corollary 3.17.1.

I is a primary ideal of H if and only if χ_I is a fuzzy primary ideal.

4. Fuzzy filters of bounded semihoops

In this section, we define a fuzzy filter in a bounded Semihoop and investigate some of its properties.

Definition 4.1.

A fuzzy subset ζ of H is called a fuzzy filter of H, if for all $s, t \in H,$

$ζ (1) = 1;$
$ζ (s) \land ζ (t) \leq ζ (s ⊙ t);$
$ζ (s) \leq ζ (t)$ , whenever $s \leq t .$

$FF (H)$ will represent the class of all fuzzy filters of H. It is evident that the characteristic functions $χ_{{1}}$ and χ_H of ${1}$ and H respectively, belongs to $FF (H)$ and hence it is nonempty. The set of all filters of H is denoted by $F (H) .$

Example 4.2.

Let $H = {0, s, t, 1}$ be a chain. Define binary operations $^{'} ⊙^{'}$ and $^{'} \to^{'}$ on H by the following tables:

\begin{array}{l} \begin{array}{l} \begin{array}{l} ⊙ & 0 & s & t & 1 \\ 0 & 0 & 0 & 0 & 0 \\ s & 0 & 0 & s & s \\ t & 0 & s & t & t \\ 1 & 0 & s & t & 1 \end{array} & \begin{array}{l} \to & 0 & s & t & 1 \\ 0 & 1 & 1 & 1 & 1 \\ s & s & 1 & 1 & 1 \\ t & 0 & s & 1 & 1 \\ 1 & 0 & s & t & 1 \end{array} \end{array} \end{array}

Then $(H, ⊙, \to, \land, 1)$ is a bounded semihoop, where $s \land t = \min {s, t},$ for every $s, t \in H$ . Define a fuzzy subset of ζ on H by: $ζ (0) = 0.4 = ζ (s), ζ (1) = ζ (t) = 1 .$ Then ζ is a fuzzy filter of $H .$

Theorem 4.3.

$ζ \in FF (H)$ if and only if $ζ^{- 1} ([α, 1]) \in FF (H),$ for all $α \in [0, 1]$ .

Corollary 4.3.1.

$F \in F (H)$ if and only if $χ_{{F}} \in FF (H)$ .

Theorem 4.4.

A fuzzy subset ζ of H is a fuzzy filter if and only if it satisfies the following conditions:

$ζ (1) = 1;$
$ζ (s) \land ζ (s \to t) \leq ζ (t),$ for every $s, t \in H$ .

Proof.

Suppose that ζ is a fuzzy filter of $H .$ Then, by definition, $ζ (1) = 1 .$ Let $s, t \in H .$ Then, since $(s ⊙ (s \to t)) \leq t,$ we have $ζ (s) \land ζ (s \to t) \leq ζ (s ⊙ (s \to t)) \leq ζ (t) .$

Conversely suppose that ζ is a fuzzy subset of H satisfying the given condition. Then by (1), $ζ (1) = 1.$ Let $s, t \in H .$ Now

\begin{array}{l} ζ (s) \land ζ (t) & = ζ (s) \land ζ (t) \land 1 \\ = ζ (s) \land ζ (t) \land ζ (t \to (s \to (s ⊙ t))) \\ \dots (since t \to (s \to (s ⊙ t)) = 1) \\ \leq ζ (s) \land ζ (s \to (s ⊙ t)) \\ \leq ζ (s ⊙ t) . \end{array}

Again let $s, t \in H$ such that $s \leq t .$ Then $s \to t = 1$ . Now $ζ (s) = ζ (s) \land 1 = ζ (s) \land ζ (s \to t) \leq ζ (t) .$ Therefore ζ is a fuzzy filter of $H .$

Theorem 4.5.

Let $ζ \in FF (H), S \subseteq H$ and $s \in H .$ Then the following conditions hold:

$t \in [s) \Rightarrow ζ (s) \leq ζ (t)$ ;
If $t \in [S)$ , then there exist $s_{1}, \dots, s_{n} \in S$ such that $⋀_{i = 1}^{n} ζ (s_{i}) \leq ζ (t) .$

Proof.

(1) Let $t \in [s)$ . Then $s^{n} \leq t,$ for some $n \in N$ and so $ζ (s) = ζ (s) \land \dots \land ζ (s) \leq φ (s^{n}) \leq φ (t);$

(2) Let $S \subseteq H$ and $t \in [S)$ , then $\exists s_{1}, \dots, s_{n} \in S$ such that $s_{1} ⊙ \dots ⊙ s_{n} \leq t$ and so $ζ (s_{1}) \land \dots \land ζ (s_{n}) \leq ζ (s_{1} ⊙ \dots ⊙ s_{n}) \leq ζ (t)$

Theorem 4.6.

$ζ \in FF (H)$ if and only if for any finite subset S of $H,$ $\underset{s \in S}{⋀} ζ (s) \leq ζ (t), for\ all t \in [S) .$

Proof.

Suppose that $ζ \in FF (H)$ and S is a finite subset of $H .$ Put $α = \underset{s \in S}{⋀} ζ (s) .$ Then $α \leq ζ (s),$ for all $s \in S$ and hence $S \subseteq ζ^{- 1} ([α, 1]),$ the α-level subset of ζ. By Theorem 4.3, $ζ^{- 1} ([α, 1])$ is a filter of H containing S. Therefore $[S) \subseteq ζ^{- 1} ([α, 1])$ and hence $α \leq ζ (t),$ for all $t \in [S) .$ That is,

\begin{array}{l} \underset{s \in S}{⋀} ζ (s) \leq ζ (t), for\ all t \in [S) . \end{array}

Conversely suppose that φ satisfies the given conditions. Now, since $[\emptyset) = {1}$ , by hypothesis, we have $1 = \underset{s \in \emptyset}{⋀} ζ (s) \leq ζ (1)$ and hence $ζ (1) = 1.$ Let $s, t \in H .$ Put $S = {s, s \to t} .$ Then since $s ⊙ (s \to t) \leq t,$ we have $t \in [S)$ and hence $ζ (s) \land ζ (s \to t) = \underset{s \in S}{⋀} ζ (s) \leq ζ (t) .$ Therefore, by Theorem 4.4, ζ is a fuzzy filter of $H .$

Theorem 4.7.

If ${ζ_{i} : i \in Ω}$ is a class of fuzzy filters of H, then $\cap_{i \in Ω} ζ_{i}$ is a fuzzy filter of H.

From Theorem 4.7, it can be easily checked that the class $FF (H)$ of all fuzzy filters of H is closed under arbitrary intersection. It follows from this fact that for any fuzzy subset ζ in H, we can find the smallest fuzzy filter of H containing ζ which we call the fuzzy filter of H generated by ζ and will be denoted by $[ζ)$ . It can be easily checked that the class $FF (H)$ of all fuzzy filters of H is closed under arbitrary intersection. It follows from this fact that for any fuzzy subset ζ in H, we can find the smallest fuzzy filter of H containing ζ which we call the fuzzy filter of H generated by ζ and will be denoted by $[ζ)$ .

Now we give a characterization of any fuzzy filter generated by a fuzzy subset of H in terms of filters generated by its level subset.

Theorem 4.8.

For any fuzzy subset ζ of H, then the fuzzy subset given by:

\begin{array}{l} [ζ) (s) = ⋁ {α \in [0, 1] : s \in [ζ^{- 1} ([α, 1])}, for\ all s \in H . \end{array}

is a fuzzy filter of H generated by ζ, where $[ζ^{- 1} ([α, 1])$ is a filter generated by the set $ζ^{- 1} ([α, 1])$ .

Theorem 4.9.

Let B be any subset of H. Then $[χ_{B}) = χ_{[B)}$ .

In the following we give an algebraic characterization of fuzzy filter of H generated by a fuzzy subset of H.

Theorem 4.10.

Let ζ be a fuzzy subset of H. Then, the fuzzy subset $\overset{―}{ζ}$ defined by: $\overset{―}{ζ} (0) = 1$ and for t ≠ 0,

\begin{array}{l} \overset{―}{ζ} (t) = ⋁ {⋀_{i = 1}^{n} φ (s_{i}) : s_{1}, \dots, s_{n} \in S, s_{1} ⊙ \dots ⊙ s_{n} \leq t} \end{array}

is the fuzzy filter of H generated by ζ.

Definition 4.11.

Let H be a bounded semihoop with DNP. Then a proper fuzzy filter ζ of H is called a prime fuzzy ideal of H, if for any fuzzy filters $τ, ρ \in FF (H),$

\begin{array}{l} τ \cap ρ \subseteq ζ implies\ either τ \subseteq ζ or ρ \subseteq ζ . \end{array}

Lemma 4.12.

Let F be a filter of H and $α \in [0, 1)$ . Then the fuzzy subset $F_{α}^{1}$ of H defined by:

\begin{array}{l} F_{α}^{1} (s) = {\begin{cases} 1 & if s \in F \\ α & if s \notin F \end{cases}, for all s \in H is a fuzzy filter of H . \end{array}

In the following we characterize prime fuzzy filter in a bounded $\lor$ -semihoop with DNP H in terms of prime filters of H and elements in $[0, 1)$ .

Theorem 4.13.

Let ζ be a fuzzy filter of H, where H is a bounded $\lor$ -semihoop with DNP. Then ζ is a prime fuzzy filter of H if and only if there exist prime filter Q of H such that $ζ = Q_{α}^{1}$ , where $α \in [0, 1)$ .

Theorem 4.14.

Let H be a bounded $\lor$ -semihoop with DNP. Then if ζ is a prime fuzzyy filter of H, then $ζ (s \lor t) = ζ (s)$ or $ζ (t),$ for all $s, t \in H$ .

Definition 4.15.

Let ζ be a proper fuzzy filter of a semihoop H. ζ is called a fuzzy primary filter, if for every $s, t \in H$ , $ζ ((s ⊙ t)^{*}) \leq ζ ((s^{n})^{*}) \lor ζ ((t^{n})^{*}),$ for some positive integer n.

Example 4.16.

In Example 4.2, ζ is a fuzzy primary filter.

Theorem 4.17.

A fuzzy subset ζ of H is a fuzzy primary filters of H if and only if $ζ^{- 1} ([α, 1])$ is a primary filter of $H,$ for all $α \in [0, 1]$ .

Theorem 4.18.

Let H be a bounded $\lor$ -semihoop with $(DNP),$ and $u^{2} = u,$ for every $u \in H$ . If ζ is a fuzzy primary filter of H, then $ζ (s \lor t) = ζ (s) \lor ζ (t),$ for all $s, t \in H$ .

5. Fuzzy congruences relation on semihoops

In this section we define the notion of fuzzy congruence relation in bounded semihoops and study their relation with fuzzy ideals (filters). By a fuzzy relation on H, we mean a fuzzy subset of H × H. For any $α \in [0, 1]$ and a fuzzy relation Φ on H, the set $Φ^{- 1} ([α, 1] = {(s, t) \in H \times H : α \leq Φ (s, t) \leq 1}$ is called the α-level relation of Φ on H.

Definition 5.1.

A fuzzy relation Φ on H is called a fuzzy equivalence relation on H if

$Φ (s, s) = 1$ (reflexive),
$Φ (s, t) = Φ (t, s)$ for all $s, t \in H$ (symmetric),
$Φ (s, t) \land Φ (t, u) \leq Φ (s, u)$ for all $s, t, u \in H$ (transitive).

Definition 5.2.

Let $A = {⊙, \to, \land} .$ Then a fuzzy relation Φ on H is said to be compatible, if

\begin{array}{l} Φ (s, t) \land Φ (u, v) \leq Φ (s * u, t * v) for all s, t, u, v \in H . \\ and for all * \in A . \end{array}

Definition 5.3.

A compatible fuzzy equivalence relation on H is called a fuzzy congruence relation on H.

The class of all fuzzy congruence relations on H is denoted by $F Con (H)$ and it is clear that $F Con (H)$ is a complete lattice. For any $s \in H$ and $Φ \in F Con (H)$ , define a fuzzy subset $[s]_{Φ}$ of H by:

\begin{array}{l} [s_{Φ}] (t) = Φ (s, t), for all t \in H . \end{array}

We call $[s]_{Φ}$ a fuzzy congruence class of Φ determined by $s .$ Let us put

\begin{array}{l} H / Φ = {[s]_{Φ} : s \in H} \end{array}

Define binary operations $⊙, \to$ and $\land$ on $H / Φ$ by:

\begin{array}{l} [h]_{Φ} ⊙ [g]_{Φ} & = [h ⊙ g]_{Φ}, [h]_{Φ} \land [g]_{Φ} \\ = [h \land g]_{Φ} and [h]_{Φ} \to [g]_{Φ} = [h \to g]_{Φ} \end{array}

Then it is routine to verify that $(H / Φ, ⊙, \to, \land, [0]_{Φ}, [1]_{Φ})$ is a bounded semihoop algebra and it is called the quotient bounded semihoop algebra of H modulo Φ.

For any fuzzy subset ξ of H, let us define a fuzzy relation $Φ_{ξ}$ on H as follows:

\begin{array}{l} Φ_{ξ} (s, t) = ξ ((s \to t)^{*}) \land ξ ((t \to s)^{*}), for all s, t \in H . \end{array}

Theorem 5.4.

If ξ be a fuzzy ideal of H, then the fuzzy relation $Φ_{ξ}$ defined above is a fuzzy congruence relation of H.

Proof.

It is easy to check that $Φ_{ξ}$ is a fuzzy equivalence relation on H. Since $s \to t \leq (s \land u) \to (t \land u),$ for any $u \in H$ , we have $((s \land u) \to (t \land u))^{*} \leq (s \to t)^{*}$ . Thus, as ξ is a fuzzy ideal, we have $ξ ((s \to t)^{*}) \leq ξ (((s \land u) \to (t \land u))^{*}) .$ Similarly we have $ξ ((t \to s)^{*}) \leq ξ (((t \land u) \to (s \land u))^{*}) .$

\begin{array}{l} Φ_{ξ} (s, t) & = ξ ((s \to t)^{*}) \land ξ ((t \to s))^{*} \\ \leq ξ (((s \land u) \to (t \land u))^{*}) \land ξ (((t \land u) \to (s \land u))^{*}) \\ = Φ_{ξ} (s \land u, t \land u) \end{array}

Similarly, we can show that $Φ_{ξ} (s, t) \leq Φ_{ξ} (s ⊙ u, t ⊙ u)$ and $Φ_{ξ} (s, t) \leq Φ_{ξ} (s \to u, t \to u)$ . Therefore, $Φ_{ξ}$ is a fuzzy congruence of H.

Define a binary operation $^{'} \leq^{'}$ on $H / Φ_{ξ}$ by:

\begin{array}{l} [s]_{Φ_{ξ}} & \leq [t]_{Φ_{ξ}} if and only if ξ ((s \to t)^{*}) = 1, \\ for all [s]_{Φ_{ξ}}, [t]_{Φ_{ξ}} \in H / Φ_{ξ} . \end{array}

Then it can and it can be easily verified that $^{'} \leq^{'}$ is a partial order and $(H / Φ_{ξ}, ⊙, \to, \land, [0]_{Φ_{ξ}}, [1]_{Φ_{ξ}})$ is a bounded semihoop.

Theorem 5.5.

A proper fuzzy ideal ξ of H is a fuzzy primary ideal if and only if for any $[s]_{Φ_{ξ}}, [t]_{Φ_{ξ}} \in H / Φ_{ξ},$ $[s]_{Φ_{ξ}} ⊙ [t]_{Φ_{ξ}} = [0]_{Φ_{ξ}}$ implies $([s]_{Φ_{ξ}})^{n} = [0]_{Φ_{ξ}}$ or $([t]_{Φ_{ξ}})^{n} = [0]_{Φ_{ξ}},$ for some $n \in N .$

Proof.

Suppose that ξ is a fuzzy primary ideal of H and $[s]_{Φ_{ξ}} ⊙ [t]_{Φ_{ξ}} = [0]_{Φ_{ξ}}$ . Then $[s ⊙ t]_{Φ_{ξ}} = [0]_{Φ_{ξ}} .$ This implies that $ξ ((s ⊙ t) \Rightarrow 0)^{*}) = 1 .$ That is, $ξ ((s ⊙ t)^{* *}) = 1.$ Thus, by Corollary 3.4.1, we have $ξ (s ⊙ t) = 1 .$ Thus since $ξ (s ⊙ t) \leq ξ (s^{n}) \lor ξ (t^{n}), for some n \in N,$ we have $ξ (s^{n}) = 1$ or $ξ (t^{n}) = 1.$ Then, by Corollary 3.4.1, we have $ξ ((s^{n})^{* *}) = 1$ or $ξ ((t^{n})^{* *})) = 1.$ If $ξ ((s^{n})^{* *}) = 1$ , then $ξ ((s^{n} \to 0)^{*}) = ξ ((s^{n})^{* *}) = 1$ and hence $[s^{n}]_{Φ_{ξ}} \leq [0]_{Φ_{ξ}} .$ Again since $ξ ((0 \to s^{n})^{*}) = ξ (1^{*}) = ξ (0) = 1,$ we have $[0]_{Φ_{ξ}} \leq [s^{n}]_{Φ_{ξ}} .$ Thus if $ξ ((s^{n})^{* *}) = 1$ , we get $([s]_{Φ_{ξ}})^{n} = [0]_{Φ_{ξ}} .$ If $ξ ((t^{n})^{* *}) = 1$ , by similar way we can show that $([t]_{Φ_{ξ}})^{n} = [0]_{Φ_{ξ}} .$

Conversely suppose that ξ is a proper fuzzy ideal satisfying the given condition. Let $ξ ((s ⊙ t)) = 1,$ for some $s, t \in H .$ Then, by Corollary 3.4.1, we have $ξ ((s ⊙ t)^{* *}) = 1.$ Now, since $ξ (((s ⊙ t) \Rightarrow 0)^{*}) = ξ ((s ⊙ t)^{* *}) = 1$ , we have $[s]_{Φ_{ξ}} ⊙ [t]_{Φ_{ξ}} = [s ⊙ t]_{Φ_{ξ}} \leq [0]_{Φ_{ξ}} .$ Again, since $ξ ((0 \to (s ⊙ t))^{*}) = ξ (1^{*}) \allowbreak = ξ (0) = 1$ , we have $[0]_{Φ_{ξ}} \leq [s ⊙ t]_{Φ_{ξ}} = [s]_{Φ_{ξ}} ⊙ [t]_{Φ_{ξ}}$ . Thus $[s]_{Φ_{ξ}} ⊙ [t]_{Φ_{ξ}} = [0]_{Φ_{ξ}} .$ Thus, by hypothesis, there exists $n \in N$ such that $([s]_{Φ_{ξ}})^{n} = [0]_{Φ_{ξ}}$ or $([t]_{Φ_{ξ}})^{n} = [0]_{Φ_{ξ}} .$ If $([s]_{Φ_{ξ}})^{n} = [0]_{Φ_{ξ}},$ then $1 = ξ ((s^{n} \to 0)^{*}) = ξ ((s^{n})^{* *}) = ξ (s^{n}) .$ By similar way, we can prove that $ξ (t^{n}) = 1$ Therefore, in any case, we have $ξ (s ⊙ t) \leq ξ (s^{n}) \lor ξ (t^{n}), for some n \in N .$ . Therefore, ξ is a fuzzy primary ideal.

Now we discuss fuzzy congruence relation related to fuzzy (primary) filters. Let ζ is a fuzzy filter of a semihoop $H .$ We define a fuzzy binary relation $Ψ_{ζ}$ on H by:

\begin{array}{l} Ψ_{ζ} (s, t) = ζ (s \to t) \land ζ (t \to s), for all s, t \in H . \end{array}

Then $Ψ_{ζ}$ is a fuzzy congruence relation on $H .$

Let $H / Ψ_{ζ} = {[s_{Φ_{ζ}}] : s \in H},$ where $s_{Φ_{ζ}}$ is a fuzzy subset of H defined by : $s_{Φ_{ζ}} (t) = Φ_{ζ} (s, t), for\ all s \in H .$ Then the binary relation ”≤” on $H / Ψ_{ζ}$ which is defined by:

\begin{array}{l} s_{Φ_{ζ}} \leq t_{Φ_{ζ}} if\ and\ only\ if ζ (s \to t) = 1, \end{array}

is an order relation on $H / Ψ_{ζ} .$ Now, $(H / Ψ_{ζ}, ⊙, $ \break $ \to, \land, [1]_{Ψ_{ζ}})$ is a semihoop, where for any $s, t \in H :$

\begin{array}{l} [h]_{Ψ_{ζ}} ⊙ [g]_{Ψ_{ζ}} & = [h ⊙ g]_{Ψ_{ζ}}, [h]_{Ψ_{ζ}} \land [g]_{Ψ_{ζ}} \\ = [h \land g]_{Ψ_{ζ}} and [h]_{Ψ_{ζ}} \to [g]_{Ψ_{ζ}} \\ = [h \to g]_{Ψ_{ζ}} \end{array}

Theorem 5.6.

Let ζ be a proper fuzzy filter of H. Then ζ is a fuzzy primary filter if and only if $[s]_{Ψ_{ζ}} ⊙ [t]_{Ψ_{ζ}} = [0]_{Ψ_{ζ}}$ implies that there exists a natural number n such that $([s]_{Φ_{ζ}})^{n} = [0]_{Φ_{ζ}}$ or $([t]_{Φ_{ζ}})^{n} = 0_{Φ_{ζ}}$ , for every $s_{Φ_{ζ}}, t_{Φ_{ζ}} \in H / Ψ_{ζ}$ .

Proof.

Suppose that ζ is a fuzzy primary filter of a semihoop H and $[s]_{Ψ_{ζ}} ⊙ [t]_{Ψ_{ζ}} = [0]_{Ψ_{ζ}}$ . Then $[s ⊙ t]_{Ψ_{ζ}} = [0]_{Ψ_{ζ}} .$ This implies that $ζ ((s ⊙ t) \Rightarrow 0) = 1 .$ That is, $ζ ((s ⊙ t)^{*}) = 1.$ Thus, as ζ is a fuzzy primary filter, we have $1 = ζ ((s ⊙ t)^{*}) \leq ζ ((s^{n})^{*}) \lor ζ ((t^{n})^{*}),$ for some positive integer $n .$ If $ζ ((s^{n})^{*}) = 1$ , then $ζ ((s^{n} \to 0) = 1$ and hence $([s]_{Φ_{ζ}})^{n} \leq [0]_{Φ_{ζ}} .$ Again since $ζ ((0 \to s^{n})) = ζ (1) = 1,$ we have $[0]_{Φ_{ζ}} \leq ([s]_{Φ_{ζ}})^{n}$ . Therefore $([s]_{Φ_{ζ}})^{n} = [0]_{Φ_{ζ}} .$ By similar way, if $ζ ((t^{n})^{*}) = 1,$ we have, $([t]_{Φ_{ζ}})^{n} = [0]_{Φ_{ζ}} .$

Conversely suppose that ζ is a proper fuzzy filter satisfying the given condition. Let $ζ ((s ⊙ t)^{*}) = 1,$ for some $s, t \in H .$ Then $ζ ((s ⊙ t) \Rightarrow 0) = 1$ and hence $[s]_{Ψ_{ζ}} ⊙ [t]_{Ψ_{ζ}} = [s ⊙ t]_{Ψ_{ζ}} \leq [0]_{Ψ_{ζ}} .$ Again, since $ζ ((0 \to (s ⊙ t)) = ζ (1) = 1$ , we have $[0]_{Ψ_{ζ}} \leq [s ⊙ t]_{Ψ_{ζ}} = [s]_{Ψ_{ζ}} ⊙ [t]_{Ψ_{ζ}}$ . Thus $[s]_{Ψ_{ζ}} ⊙ [t]_{Ψ_{ζ}} = [0]_{Ψ_{ζ}} .$ Thus, by hypothesis, there exists $n \in N$ such that $([s]_{Ψ_{ζ}})^{n} = [0]_{Ψ_{ζ}}$ or $([t]_{Ψ_{ζ}})^{n} = [0]_{Ψ_{ζ}} .$ If $([s]_{Ψ_{ζ}})^{n} = [0]_{Ψ_{ζ}},$ then $ζ (s^{n} \to 0) = 1$ and so $ζ ((s^{n})^{*}) = ζ (s^{n} \to 0) = 1 .$ By similar way, we can prove that if $([t]_{Ψ_{ζ}})^{n} = [0]_{Ψ_{ζ}},$ then $ζ (t^{n}) = 1$ Therefore, in any case, we have $ζ ((s ⊙ t)^{*}) \leq ζ ((s^{n})^{*}) \lor ζ ((t^{n})^{*}) for some n \in N .$ . Therefore ζ is a fuzzy primary filter.

Conclusions and discussion

In this study, we have explored the concept of fuzzy ideals and fuzzy filters within a bounded semihoop, delving into their properties and characteristics. Additionally, we have examined various characterizations of fuzzy ideals and filters derived from a given fuzzy subset. Our investigation also encompassed prime fuzzy ideals, fuzzy primary ideals, prime fuzzy filters, fuzzy primary filters, along with their respective characterizations. Furthermore, we have delved into the fuzzy congruence relation in connection to fuzzy (primary) ideals and filters. These findings not only enrich the field of fuzzy set theory but also offer valuable insights into the broader exploration of bounded semihoops and their associated structures. Looking ahead, we plan to delve into the study of soft semihoops and their corresponding structures in future research endeavors.

The study of fuzzy ideals and filters in bounded semihoops offers a unique opportunity to bridge the gap between crisp mathematical structures and fuzzy logic. By delving into the properties, characterizations, and relationships of these fuzzy constructs, we can gain insights into how they interact within the framework of bounded semihoops.

Disclosure statement

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

References

Addis, G. M. (2022). L-fuzzy congruence classes in universal algebras. International Journal of Intelligent Systems, 37(1), 386–423. https://doi.org/10.1002/int.22631
Web of Science ®Google Scholar
Aglianó, P., Ferreirim, I. M. A., & Montagna, F. (2007). Basic hoops: An algebraic study of continuous t-norm. Studia Logica, 87(1), 73–98. https://doi.org/10.1007/s11225-007-9078-1
Google Scholar
Alaba, B. A., & Addis, G. M. (2019a). L-fuzzy ideals in universal algebras. Annals of Fuzzy Mathematics and Informatics, 17(1), 31–39. https://doi.org/10.30948/afmi.2019.17.1.31
Google Scholar
Alaba, B. A., & Addis, G. M. (2019b). L-fuzzy prime ideals in universal algebras. Advances in Fuzzy Systems, 2019, 1–7. https://doi.org/10.1155/2019/5925036
Web of Science ®Google Scholar
Alaba, B. A., & Alemayehu, T. G. (2019a). e-fuzzy filters of MS-algebras. Korean Journal of Mathematics, 27, 1159–1180. https://doi.org/10.11568/kjm.2019.27.4.1159
Google Scholar
Alaba, B. A., & Alemayehu, T. G. (2019b). β-fuzzy filters of MS-algebras. Korean Journal of Mathematics, 27, 595–612. https://doi.org/10.11568/kjm.2019.27.3.595
Google Scholar
Alemayehu, T. G., Engidaw, D. A., & Addis, G. M. (2021). L-Fuzzy congruences and L-fuzzy kernel ideals in Ockham algebras. Journal of Mathematics, 2021, Article ID 6644443, 1–12. https://doi.org/10.1155/2021/6644443
Web of Science ®Google Scholar
Alemayehu, T. G., Engidaw, D. A., & Addis, G. M. (2022). Kernel L-ideals and L-congruence on a subclass of Ockham Algebras. Journal of Mathematics, 2022, Article ID 7668044, 9. https://doi.org/10.1155/2022/7668044
Web of Science ®Google Scholar
Borzooei, R. A., & Aaly Kologani, M. (2015). Local and perfect semihoops. Journal of Intelligent and Fuzzy Systems, 29(1), 223–234. https://doi.org/10.3233/IFS-151589
Web of Science ®Google Scholar
Bosbach, B. (1969). Komplementäre Halbgruppen. Axiomatik und Arithmetik. Fundamenta Mathematicae, 64(3), 257–720. https://doi.org/10.4064/fm-64-3-257-287
Google Scholar
Bosbach, B. (1970). Komplementäre Halbgruppen Kongruenzen und Quotienten. Fundamenta Mathematicae, 69(1), 1–14. https://doi.org/10.4064/fm-69-1-1-14
Google Scholar
Esteva, F., Godo, L., Hájek, P., & Montagan, F. (2003). Hoops and fuzzy logic. Journal of Logic and Computation, 13(4), 532–555. https://doi.org/10.1093/logcom/13.4.532
Web of Science ®Google Scholar
Lu, J., Zhu, L., & Gao, W. (2023). Remarks on bipolar cubic fuzzy graphs and its chemical applications. International Journal of Mathematics and Computer in Engineering, 1(1), 1–10. https://doi.org/10.2478/ijmce-2023-0001
Google Scholar
NiuIdeal, H. L., Xing, X. L., & Wang, J. T. (2020). Ideal theory on bounded semihoops. Italian Journal of Pure and Applied Mathematics- N, 44, 911–925.
Google Scholar
Norahun, W. Z. (2020). µ-Fuzzy filters in distributive lattices. Advances in Fuzzy Systems, 2020, Article ID 8841670, 9. https://doi.org/10.1155/2020/8841670
Web of Science ®Google Scholar
Rosenfeld, A. (1971). Fuzzy groups. Journal of Mathematical Analysis and Applications, 35(3), 512–517. https://doi.org/10.1016/0022-247X(71)90199-5
Web of Science ®Google Scholar
Zadeh, L. A. (1965). Fuzzy sets. Information & Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
Google Scholar

Fuzzy ideals and fuzzy filters of bounded semihoops

ABSTRACT

1. Introduction

2. Preliminaries

(Esteva et al., Citation2003)

(Esteva et al., Citation2003)

(Borzooei & Aaly Kologani, Citation2015)

(NiuIdeal et al., Citation2020)

(Borzooei & Aaly Kologani, Citation2015)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

3. Fuzzy ideals of on a bounded semihoops

4. Fuzzy filters of bounded semihoops

5. Fuzzy congruences relation on semihoops

Conclusions and discussion

Disclosure statement

References

Information for

Open access

Opportunities

Help and information

Fuzzy ideals and fuzzy filters of bounded semihoops

ABSTRACT

1. Introduction

2. Preliminaries

(Esteva et al., Citation2003)

(Esteva et al., Citation2003)

(Borzooei & Aaly Kologani, Citation2015)

(NiuIdeal et al., Citation2020)

(Borzooei & Aaly Kologani, Citation2015)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

(NiuIdeal et al., Citation2020)

3. Fuzzy ideals of on a bounded semihoops

4. Fuzzy filters of bounded semihoops

5. Fuzzy congruences relation on semihoops

Conclusions and discussion

Disclosure statement

References

Related research

To cite this article:

Download citation

Information for

Open access

Opportunities

Help and information

Keep up to date

Your download is now in progress and you may close this window

Login or register to access this feature