Full article: Characterisation of p-ideals in Hemirings on the Basis of Hemiregularity, Prime and Normal Fuzzy Ideals

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Abstract

In p-hemiregular hemirings, some properties of ‘prime fuzzy p-ideals’ are examined. In the case of fuzzy p-ideal, p-hemiregular hemiring is also characterised. It is showed that in the case of hemiring $H$ , fuzzy subset φ is a ‘prime fuzzy left p-ideal’ in $H$ if and only if φ is having two values, set having all values of in $H$ so that φ(p) equals to one is a ‘prime left(right) p-ideal’ in $H$ , φ(0) = 1. In hemirings for ‘maximal fuzzy left(right) p-ideals’, some related properties are also studied.

AMS Mathematics Subject Classifications (2010):

Keywords:

1 Introduction

Classical mathematics cannot be strongly used to solve complex problems in environmental sciences, social sciences, engineering, economics etc. due to diﬀerent uncertainties in above-mentioned fields. So, among diﬀerent changes in mathematics and sciences, one of the biggest changes in this century is the introduction of the concept of uncertainty. This change in science is visible by the development of the traditional view to alternative view, where according to the traditional view, uncertainty is annoying in science and it should be escaped straight out. On the other hand, according to the alternative view, we cannot avoid uncertainty and it should be considered also science should work for certainty in all its materialisations. Thus, uncertainty is considered as unscientific which shows that uncertainty is not only necessary but it has a great usage and is crucial for science according to the modern or alternative view. The first step of evolution from the traditional view to alternative view of uncertainty has been started at the end of the nineteenth century, when physics got bothered with mechanisms at the molecular level. Basically, the demand to study physical mechanisms at the molecular level in diﬀerent way lead to the evolution of related statistical methods which brought out applications not only in the study of molecular mechanisms but also in the other fields like the actuarial profession, design of large telephone exchanges etc. Fuzzy sets were firstly introduced by Zadeh [Citation1], which are sets with bounds that are not explicit. In this type of sets, membership is not an entity of confirmation rather it is an entity of degree. The importance of his paper is that he not only claimed probability theory as the solitary operator for uncertainty but also to the basics of probability theory. The evolution from membership to membership and vice versa expressed by fuzzy sets has a great usage. It not only tells us eﬀective ways of measuring uncertainties but also by using it we can express the uncertain notions in natural language. Fuzzy set theory greatly fascinated mathematicians working in diﬀerent areas of mathematics like group theory, topology, vector spaces, functional analysis etc. So they worked a lot to prolong the results and concepts of algebra to fuzzy set theory. An extension to these results and concept of fuzzy sets applied in algebra is discussed in detail by the authors in [Citation2–6]. In diﬀerent fields such as generalised fuzzy algebra, the usage of rings and semirings has been suﬃciently studied in the literature. In algebraic research to realise the importance of rings, the work of many great and leading algebraists in this area is suﬃcient. Some of them who proved some useful results are [Citation6–19]. When we apply the theory of rings to random semirings, much of the theory of rings work. Much progress is done in generalizing the theory of semirings to ring theory by many other analysts. Since the fast progress in computer science was longing for abstract mathematical grounding. So in the 1980s, in computer science, semiring theory added a lot. Semirings contributed largely in providing reasonable generalisation of ring theory. Above all, the theory of algebras over commutative rings can be derived in every respect to the theory of algebras over commutative semirings. We also recognise the importance of ideals in algebra specifically in rings. In the structural theory of semirings, ideals play a vital role and therefore have great usage for a number of purposes. Yet, their usage is rather reserved in demanding to get correlations of semiring theorems for rings because they generally do not synchronise to ideals in rings. Of course, in the case of ideals in rings, there are many results which probably have no correlations with semirings. As ideals are of much importance in mathematics, therefore they are also studied at large scale by diﬀerent mathematicians such as [Citation20] who not only studied properties of ideals in detail but also suggested the basic concepts of ideals, translation ideals and maximal ideals. They are also studied thoroughly by [Citation21, Citation22]. To some degree, there are many results in semirings which are not interconnected using ideals; so, correspondingly, there are many modified notions of ideals like k-ideals, h-ideals, p-ideals etc. which semiring theory recommends. Furthermore, the theory which was profitable in algebra to conclude many vital concepts was fuzzy set theory taken up by Zadeh [Citation1]. Many scholars have worked on fuzzy set theory in semirings and made known the notions of ‘fuzzy semirings’, ‘fuzzy ideals’, ‘fuzzy k-ideals’, ‘fuzzy h-ideals’ and ‘L-fuzzy ideals’ in semirings and got so many useful results. Touqeer et al. gave the concept of α-ideals and different decision making techniques [Citation23–28]. Ghosh [Citation12, Citation29] made known the concepts of ‘fuzzy k-ideals’, ‘prime fuzzy k-ideals’ and ‘semiregularity of semirings’. ‘Fuzzy k-ideals’ which are idempotent in semirings are portrayed by Mordeson [Citation30]. Moreover, he also explained thoroughly k-regular semirings in the form of ‘fuzzy left k- ideals’. Further ‘k-ideals’ were more comprehensively studied by [Citation31–36].

2 Preliminaries

A hemiring ( $H$ , +,) refers to a non-empty set H with ‘+’ and ‘ ’ as the two binary operations defined on ϱ such that it is an additively commutative semiring with additive identity. About rings, semirings and basic ideals in these algebras, one can see [Citation20,Citation21,Citation27,Citation37].

Definition 2.1

If be any left ideal in H then it is called left k-ideal in H if q, r ∈ , p ∈ $H$ further p + q = r shows that p ∈ . Right k-ideals can be defined in a similar way.

Definition 2.2

If be any left ideal in $H$ then it is said to be left h-ideal in $H$ so that (∀ p, r ∈ $H$ )(∀ , ∈ )(p + + r = + r) implies p . Right h-ideals can be defined in a similar manner. From now onwards, $H$ will denote hemiring unless otherwise specified.

3 Fuzzy p-ideals

Definition 3.1

For any non-empty set of X which is left ideal is called left p-ideal when (p · r) · (q · r), q ∈ and p, r ∈ $H$ then $p + (p \cdot r) \cdot (q \cdot r) + r = q + r$ implies p ∈ ..

Definition 3.2

In hemiring $H$ , a fuzzy left p-ideal can be defined as ϱ which is fuzzy left ideal so that $(\forall (p \dots r) \cdot (q \dots r), q, p, r \in H) (p + (p \dots r) \cdot (q \dots r) + r = q + r)$ implies ϱ(p) min ϱ((p r) (q r)), ϱ(q). Fuzzy right p-ideals can be defined in a similar way.

Definition 3.3

p-product of ϱ and p can be defined as $\begin{aligned} (ϱ_{p}^{\circ} p) (p_{1} \cdot p_{2}) = sup_{p_{1} \cdot p_{2} + ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})) \cdot q_{1} + r_{1} \cdot r_{2} = ((p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2})) \cdot q_{2} + r_{1} \cdot r_{2}} \\ min {ϱ ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})), ϱ ((p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2})), p (q_{1}), p (q_{2}} \end{aligned}$

4 p-hemiregularity

Definition 4.1

$H$ is said to be p-hemiregular if for every ∈ $H$ ∃ , ∈ $H$ so that $\overset{ˇ}{s} + \overset{ˇ}{s} ((\overset{ˇ}{s} \dots \overset{ˇ}{r}) \cdot (\overset{ˇ}{t} \dots \overset{ˇ}{r})) \overset{ˇ}{s} + \overset{ˇ}{r} = \overset{ˇ}{s} \overset{ˇ}{t} \overset{ˇ}{s} + \overset{ˇ}{r}$

Definition 4.2

In $H$ when (p · r) · (q · r), q ∈ and p, r ∈ $H$ , p-closure of can be defined as $\bar{\overset{ˇ}{G}} = p \in H | p + (p \cdot r) \cdot (q \cdot r) + r = q + r$

Lemma 4.3

Let $\overset{ˇ}{G}, \overset{ˇ}{H} \subseteq H \subseteq$ $H$ then for $H$ $\bar{\overset{ˇ}{G} \overset{ˇ}{H}}$ must be equal to $\bar{\bar{\overset{ˇ}{G}} \bar{\overset{ˇ}{H}}}$

Proof As $\overset{ˇ}{G}$ contain in $\bar{\overset{ˇ}{G}}$ and $\overset{ˇ}{H}$ contain in $\bar{\overset{ˇ}{H}}$ then clearly $\overset{ˇ}{G} \overset{ˇ}{H}$ contain in $\bar{\overset{ˇ}{G}} \bar{\overset{ˇ}{H}}$ and through upon $\bar{\overset{ˇ}{G} \overset{ˇ}{H}}$ contain in $\bar{\bar{\overset{ˇ}{G}} \bar{\overset{ˇ}{H}}}$ . For the proof of converse inclusion p contains in $\bar{\overset{ˇ}{G}}$ and q contains in $\bar{\overset{ˇ}{H}}$ then in $H \exists (p_{i} \cdot r_{i}) \cdot (q_{i} \cdot r_{i}) \in \overset{ˇ}{G}, q_{i} \in \overset{ˇ}{H}, p, r_{1}, r_{2}$ , so that $p + (p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1}) + r_{1} = (p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2}) + r_{1}$ and $q + q_{1} + r_{2} = q_{2} + r_{2}$ by putting $r = p r_{2} + 2 ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})) r_{2} + r_{1} q + 2 r_{1} q_{1} + r_{1} r_{2}$ by $r \in H$ and $\begin{aligned} ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{2} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r = ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + p r_{2} \\ + 2 ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} q + 2 r_{1} q_{1} + r_{1} r_{2} \\ = ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{2} + p r_{2} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} q + r_{1} q_{1} + r_{1} r_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{2} + p r_{2} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} (q + q_{1} + r_{2}) \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{2} + p r_{2} + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) r_{2} \\ + r_{1} (r_{2} + q_{2}) + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = ((p_{2} \dots r_{2}) \dots (q_{2} \dots r_{2})) q_{2} + p r_{2} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} r_{2} + r_{1} q_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = ((p_{2} \dots r_{2}) \dots (q_{2} \dots r_{2})) q_{2} + (p + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) + r_{1}) r_{2} + r_{1} q_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = (((p_{2} \dots r_{2}) \dots (q_{2} \dots r_{2})) q_{2} + ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) + r_{1}) r_{2} + r_{1} q_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = ((p_{2} \dots r_{2}) \dots (q_{2} \dots r_{2})) q_{2} + ((p_{2} \dots r_{2}) \dots (q_{2} \dots r_{2})) r_{2} + r_{1} q_{2} + r_{1} r_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = ((p_{2} \dots r_{2}) \dots (q_{2} \dots r_{2})) (q_{2} + r_{2}) + r_{1} (q_{2} + r_{2}) + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} \\ + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = (((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) + r_{1}) (q_{2} + r_{2}) + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) q_{1} \\ + r_{1} q_{1} + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) r_{2} \\ = (p + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) + r_{1}) (q + q_{1} + r_{2}) + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} \\ + r_{1} q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} \\ = p q + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) q_{1} + p q_{1} + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) q + p r_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} q + r_{1} q_{1} + r_{1} r_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r_{1} q_{1} + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) r_{2} \\ = p q + (((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) q_{1} + p q_{1} + r_{1} q_{1}) + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) q \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + (((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2}) + (p r_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} q + r_{1} q_{1} + r_{1} r_{2}) \\ = p q + (((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) + p + r_{1}) q_{1} + ((p_{1} \dots r_{1}) \dots (q_{1} \dots r_{1})) (q + q_{1} + r_{2}) \\ + (p r_{2} + ((p_{1} \dots r_{1}) (q_{1} \dots r_{1})) r_{2} + r_{1} q + r_{1} q_{1} + r_{1} r_{2}) \\ = p q + (((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) + r_{1}) q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) (q_{2} + r_{2}) + p r_{2} \\ + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} q + r_{1} q_{1} + r_{1} r_{2}) \\ = p q + ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{2} + p r_{2} \\ + 2 ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) r_{2} + r_{1} q + 2 r_{1} q_{1} + r_{1} r_{2} \\ = p q + ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{2} \\ + r s o ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{2} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1} + r \\ = p q + ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{1} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{2} + r \end{aligned}$ Hence, we can derive pq $\in \bar{\overset{ˇ}{G}} \bar{\overset{ˇ}{H}}$ as $((p_{i} \cdot r_{i}) \cdot (q_{i} \cdot r_{i})) \in \overset{ˇ}{G} \overset{ˇ}{H}$ and $r \in H$ , which shows that pq $\in \bar{\overset{ˇ}{G} \overset{ˇ}{H}}$ for any p $\in \bar{\overset{ˇ}{G}}$ , q $\in \bar{\overset{ˇ}{H}}$ . Now suppose randomly $r^{'}$ $\in \bar{\overset{ˇ}{G}} \bar{\overset{ˇ}{H}}$ . Then $r^{'} = \sum_{i = 1}^{n} p_{i} q_{i}$ for any $p_{i} \in \bar{\overset{ˇ}{G}}$ and $q_{i} \in \bar{\overset{ˇ}{H}}$ , therefore $r^{'} \in \bar{\overset{ˇ}{G} \overset{ˇ}{H}}$ , i.e. $\bar{\overset{ˇ}{G}} \bar{\overset{ˇ}{H}} \subseteq \bar{\overset{ˇ}{G} \overset{ˇ}{H}}$ ; hence, $\bar{\bar{\overset{ˇ}{G}} \bar{\overset{ˇ}{H}}} \subseteq \bar{\bar{\overset{ˇ}{G} \overset{ˇ}{H}}} = \bar{\overset{ˇ}{G} \overset{ˇ}{H}}$ so clearly $\bar{\bar{\overset{ˇ}{G} \overset{ˇ}{H}}} = \bar{\overset{ˇ}{G} \overset{ˇ}{H}}$ .

Lemma 4.4

For right and left p-ideal $\overset{ˇ}{G}$ and $\overset{ˇ}{H}$ respectively of $H$ , $\bar{\overset{ˇ}{G} \overset{ˇ}{H}} \subseteq \overset{ˇ}{G} \cap \overset{ˇ}{H}$ .

Proof Suppose

p + \sum_{i = 1}^{m} ((p_{i} \cdot r_{i}) \cdot (q_{i} \cdot r_{i})) q_{i} + r

\sum_{j = 1}^{n} ((p {^{'}}_{j} \cdot r {^{'}}_{j} \cdot (q {^{'}}_{j} \cdot r {^{'}}_{j})) q {^{'}}_{j} + r

where

p \in \bar{\overset{ˇ}{G} \overset{ˇ}{H}}

for some

(p_{i} \cdot r_{i}) \cdot (q_{i} \cdot r_{i}), (p^{'}_j \cdot r^{'}_j) \cdot (q {^{'}}_{j} \cdot r {^{'}}_{j}) \in \overset{ˇ}{G}

and

q_{i}, q {^{'}}_{j} \in \overset{ˇ}{H}

and

r \in H

. Because

\overset{ˇ}{G}

is a right p-ideal in

H

and

(H, +)

is commutative semigroup. Therefore, elements

\sum_{i = 1}^{m} ((p_{i} \cdot r_{i}) \cdot (q_{i} \cdot r_{i})) q_{i}

\sum_{j = 1}^{n} ((p {^{'}}_{j} \cdot r {^{'}}_{j}) \cdot (q {^{'}}_{j} \cdot r {^{'}}_{j})) q {^{'}}_{j} \in \overset{ˇ}{G}

and as a result,

p \in \overset{ˇ}{G}

. Correspondingly, we can also prove that

p \in \overset{ˇ}{H}

, therefore

p \in

\overset{ˇ}{G} \cap \overset{ˇ}{H}

clearly

\bar{\overset{ˇ}{G} \overset{ˇ}{H}} \subseteq \overset{ˇ}{G} \cap \overset{ˇ}{H}

Lemma 4.5

For any right left p-ideals $\overset{ˇ}{G} a n d \overset{ˇ}{H}$ , respectively, $\bar{\overset{ˇ}{G} \overset{ˇ}{H}} = \overset{ˇ}{G} \cap \overset{ˇ}{H}$ iff $H$ is semi regular.

Proof Let ∈ $\overset{ˇ}{G} \cap \overset{ˇ}{H}$ and $H$ be p-hemiregular then for every ∈ $H$ ∃, c ∈ $H$ so that $\overset{ˇ}{s} + \overset{ˇ}{s} ((\overset{ˇ}{s} \cdot c) \cdot (\overset{ˇ}{t} \cdot c)) \overset{ˇ}{s} + c = \overset{ˇ}{s} \overset{ˇ}{t} \overset{ˇ}{s} + c$ because is right p-ideal in $H$ we will have (_i · c_i)· (i · c_i) ∈ and (_i · c_i) · (i · c_i) ∈ .

Therefore, $p \in \overset{ˇ}{G} \cap \overset{ˇ}{H}$ then by 4.4 clearly $\bar{\overset{ˇ}{G} \overset{ˇ}{H}} \subseteq \overset{ˇ}{G} \cap \overset{ˇ}{H}$ .

Conversely, assume that ∈ $H$ so as we know it is easy to prove $H$ + $N_{\circ} \overset{ˇ}{s}$ for $N_{\circ}$ = {0, 1, 2, … } be the principle right ideal in $H$ which is generated by . As a result, $\bar{(\overset{ˇ}{s} H + N_{\circ} \overset{ˇ}{s})}$ is a right p-ideal in $H$ thus $\bar{(\overset{ˇ}{s} H + N_{\circ} \overset{ˇ}{s})} = \bar{(\overset{ˇ}{s} H + N_{\circ} \overset{ˇ}{s})} \cap H = \bar{\bar{(\overset{ˇ}{s} H + N_{\circ} \overset{ˇ}{s})} H} = \bar{(\overset{ˇ}{s} H + N_{\circ} \overset{ˇ}{s}) H} = \bar{\overset{ˇ}{s} H}$ as $H$ itself is p-ideal trivially. Therefore $\overset{ˇ}{s} = \overset{ˇ}{s} \cdot 0 + 1 \cdot \overset{ˇ}{s} \in \overset{ˇ}{s} H + N_{\circ} \overset{ˇ}{s} \subseteq \bar{(\overset{ˇ}{s} H + N_{\circ} \overset{ˇ}{s})} = \bar{\overset{ˇ}{s} H}$ clearly $\overset{ˇ}{s} \in \bar{H \overset{ˇ}{s}}$ whence $\overset{ˇ}{s} \in \bar{\overset{ˇ}{s} H} \cap \bar{H \overset{ˇ}{s}} = \bar{\bar{\overset{ˇ}{s} H} \cdot \bar{H \overset{ˇ}{s}}} = \bar{\overset{ˇ}{s} H H \overset{ˇ}{s}} \subseteq \bar{\overset{ˇ}{s} H \overset{ˇ}{s}}$ As $\bar{\overset{ˇ}{s} H} and \bar{H \overset{ˇ}{s}}$ are right and left p-ideals in $H$ respectively. This follows that for every

$\in H \exists \overset{ˇ}{t}, c \in H$ so that

$\overset{ˇ}{s} + \overset{ˇ}{s} ((\overset{ˇ}{s} \cdot c) \cdot (\overset{ˇ}{t} \cdot c)) \overset{ˇ}{s} + c = \overset{ˇ}{s} \overset{ˇ}{t} \overset{ˇ}{s} + c$ therefore $H$ is a p-hemiregular hemiring.

Theorem 4.6

$(ϱ_{p}^{\circ} x_{i})$ = ϱ ∩ X_i when ϱ be ‘fuzzy right p-ideal’ and X_i be ‘fuzzy left p-ideal’ if and only if $H$ be p-hemiregular.

Proof

Assume that $H$ be any p-hemiregular hemiring then $(ϱ_{p}^{\circ} x_{i})$ = ϱ ⊆ X_i by Lemma 2.7 for some ∈ $H$ ∃, c ∈ so that $\overset{ˇ}{s} + \overset{ˇ}{s} ((\overset{ˇ}{s} \cdot c) \cdot (\overset{ˇ}{t} \cdot c)) \overset{ˇ}{s} + c = \overset{ˇ}{s} \overset{ˇ}{t} \overset{ˇ}{s}$ Thus $\begin{aligned} (ϱ_{p}^{\circ} x_{i}) (\overset{ˇ}{s}) & = sup_{\overset{ˇ}{s} + (\overset{ˇ}{s} ((\overset{ˇ}{s} \cdot c) \cdot (\overset{ˇ}{t} \cdot c)) \overset{ˇ}{s}) + c = \overset{ˇ}{s} \overset{ˇ}{t} \overset{ˇ}{s} + c} (min {ϱ ((\overset{ˇ}{s}) ((\overset{ˇ}{s} \cdot c) \cdot (\overset{ˇ}{t} \cdot c))), ϱ (\overset{ˇ}{s} \overset{ˇ}{t}), x_{i} (\overset{ˇ}{s})}) \\ \geq min {ϱ ((\overset{ˇ}{s}) ((\overset{ˇ}{s} \cdot c) \cdot (\overset{ˇ}{t} \cdot c))), ϱ (\overset{ˇ}{s} \overset{ˇ}{t}), x_{i} (\overset{ˇ}{s})} \\ \geq min {ϱ (\overset{ˇ}{s}), x_{i} (\overset{ˇ}{s})} \\ = (ϱ \cap x_{i}) (\overset{ˇ}{s}) \end{aligned}$ therefore $ϱ \cap x_{i} \subseteq ϱ_{p}^{\circ} x_{i}$ hence $ϱ_{p}^{\circ} x_{i} = ϱ \cap x_{i}$ . Conversely, suppose and are right and left p-ideals, respectively, in $H$ . So, as we know, it is easy to analyse characteristic functions of $\overset{ˇ}{G} and \overset{ˇ}{H}$ which are $χ_{\overset{ˇ}{G}} and χ_{\overset{ˇ}{H}}$ to be ‘fuzzy right p-ideal’ and ‘fuzzy left p-ideal’, respectively.

Further by 4.4

\bar{\overset{ˇ}{G} \overset{ˇ}{H}} \subseteq \overset{ˇ}{G} \cap \overset{ˇ}{H}

. Suppose

\overset{ˇ}{s} \in \overset{ˇ}{G} \cap \overset{ˇ}{H}

then

χ_{\overset{ˇ}{G}} (\overset{ˇ}{s}) = 1 = χ_{\overset{ˇ}{H}}

therefore

(χ_{\overset{ˇ}{G}}_{p}^{\circ} χ_{\overset{ˇ}{H}}) (\overset{ˇ}{s}) = (χ_{\overset{ˇ}{G}} \cap χ_{\overset{ˇ}{H}}) (\overset{ˇ}{s}) = min {χ_{\overset{ˇ}{G}} (\overset{ˇ}{s}), χ_{\overset{ˇ}{H}}) (\overset{ˇ}{s})} = 1

Thus

min {χ_{\overset{ˇ}{G}} ({\overset{ˇ}{s}}_{1}), χ_{\overset{ˇ}{H}}) ({\overset{ˇ}{t}}_{1}, χ_{\overset{ˇ}{G}} ({\overset{ˇ}{s}}_{2}), χ_{\overset{ˇ}{H}}) ({\overset{ˇ}{t}}_{2})} = 1

for some ₁, ₂, 1, 2

{\overset{ˇ}{s}}_{1} \dots {\overset{ˇ}{s}}_{2} + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) \cdot {\overset{ˇ}{t}}_{1} + r_{1} \cdot r_{2} = ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) \cdot {\overset{ˇ}{t}}_{2} + r_{1} \dots r_{2}

But clearly, then χ(_i) = 1 = χ(i) for any i = 1, 2, it shows _i ∈ and i ∈ . Hence,

\overset{ˇ}{s} \in \bar{\overset{ˇ}{G} \overset{ˇ}{H}}

therefore

\overset{ˇ}{G} \cap \overset{ˇ}{H}

4.5 helps to complete proof.

5 Normal Fuzzy Left p-ideals

Definition 5.1

In $H$ , ‘normal fuzzy left p-ideal’ can be defined as ϱ which is fuzzy left p-ideal if there is any p ∈ $H$ so that ϱ(p) equals to one.

Proposition 5.2

In a hemiring , $H$ there is given any fuzzy left p-ideal ϱ, suppose ϱ⁺ be a fuzzy set of $H$ defined as ϱ⁺(p) = ϱ(p) + 1 − ϱ(0)∀p ∈ $H$ . Then ϱ⁺ be a normal fuzzy left p-ideal in $H$ which consists of ϱ.

Proof We have ϱ⁺(0) = ϱ(0) + 1 − ϱ(0) = 1 and $\begin{aligned} ϱ^{+} (p + q) & = ϱ (p + q) + 1 - ϱ (0) \end{aligned}$ $\begin{aligned} \geq min {ϱ (p), ϱ (q)} + 1 - ϱ (0) \end{aligned}$ $\begin{aligned} = min {ϱ (p) + 1 - ϱ (0), ϱ (q) + 1 - ϱ (0)} \end{aligned}$ $\begin{aligned} = min {ϱ^{+} (p), ϱ^{+} (q)} \end{aligned}$

∀ p, q ∈ $H$ . Correspondingly $ϱ^{+} (p q) = ϱ (p q) + 1 - ϱ (0) \geq ϱ (q) + 1 - ϱ (0) = ϱ^{+} (q)$ whence ϱ⁺ is a fuzzy left ideal in H. Suppose (p · r) · (q · r), q, p, r ∈ $H$ so that $p + (p \dots r) \cdot (q \dots r) + r = q + r$ then $\begin{aligned} ϱ^{+} (p) & = ϱ (p) + 1 - ϱ (0) \\ \geq min {ϱ ((p \dots r) \dots (q \dots r)), ϱ (q)} + 1 - ϱ (0) \\ = min {ϱ ((p \dots r) \cdot (q \dots r)) + 1 - ϱ (0), ϱ (q) + 1 - ϱ (0)} \\ min {ϱ^{+} ((p \dots r) \cdot (q \dots r)), ϱ^{+} (q)} \end{aligned}$

Hence, clearly ϱ ⊆ ϱ⁺ and ϱ⁺ be a normal fuzzy left p-ideal in $H$ .

Corollary 5.3

Suppose ϱ and ϱ⁺ are as in 5.2. Let if there exists p∈ $H$ so that ϱ⁺(p) = 0 then ϱ(p) = 0.

Proof For any left p- ideal in $H$ , clearly χ is a normal fuzzy left p-ideal in $H$ . It is so clear ϱ⁺= ϱ iff ϱ be a ‘normal fuzzy left p-ideal’ in $H$ .

Theorem 5.4

In a hemiring, suppose ϱ be a ‘fuzzy left p-ideal’ and suppose f: [0, ϱ(0) →[0, 1] is any increasing function. Fuzzy set ϱ_f: → [0, 1] defined as ϱ_f (p) = f (ϱ(p)) be a fuzzy left p-ideal in . Particularly ϱ_f is normal if f (ϱ(0)) = 1 only when so that ϱ ⊆ ϱ_f.

Proof Clearly, we have $ϱ_{f} (p + q) \geq f (min {ϱ (p), ϱ (q)}$ $ϱ_{f} (p + q) \geq f (min {ϱ (p), ϱ (q)}$ $= min {f (ϱ (p)), f (ϱ (q))} = min {ϱ_{f} (p), ϱ_{f} (q)}$ , ∀p, q ∈ $H$ . Correspondingly $ϱ_{f} (p q) = f (ϱ (p q)) \geq f (ϱ (q)) = ϱ_{f} (q)$ Therefore, ϱ_f be a fuzzy left ideal in $H$ . If for any (p · r) · (q · r), q, p, r ∈ $H$ are so then $p + (p \dots r) \cdot (q \dots r) + r = q + r$ then $ϱ_{f} (p) = f (ϱ (p)) \geq f (min {ϱ ((p \cdot r) \cdot (q \cdot r)), ϱ (q)})$ $min {f (ϱ ((p \dots r) \cdot (q \dots r))), f (ϱ (q))} = min {ϱ_{f} ((p \dots r) \cdot (q \dots r)), ϱ_{f} (q)}$

Hence, ϱ_f be a fuzzy left p-ideal in $H$ . Then clearly ϱ is normal if f (ϱ(0)) = 1. Suppose f (l) = f (ϱ(p)) ≥ ϱ(p) for p ∈ $H$ which clearly proves ϱ ⊆ ϱ_f.

Theorem 5.5

Suppose ϱ∈ $N (\hat{m})$ be variable if it is a maximal element in ( $N (\hat{m})$ ,⊆). So ϱ will take just two values that are 0 and 1.

Proof. We have ϱ(0) = 1, as ϱ is normal. Suppose ϱ(p) /= 1 for any p ∈ $H$ . We claim ϱ(p) = 0. If not then ∃ p_° ∈ so that 0 < ϱ(p_°) < 1. A fuzzy set X_i defined on $H$ putting X_i(p) = $\frac{(ϱ (p) + ϱ (p_{\circ})}{2} \forall p \in H$ then it is obvious that X_i is well defined and we will have $\begin{aligned} x_{i} (p + q) & = \frac{ϱ (p + q) + ϱ (p_{\circ})}{2} \\ \geq \frac{(min {ϱ (p), ϱ (q)} + ϱ (p_{\circ})}{2} \\ = min {\frac{(ϱ (p) + ϱ (p_{\circ}))}{2}, \frac{(ϱ (q) + ϱ (p_{\circ}))}{2}} \\ = min {x_{i} ((p \cdot r) \cdot (q \cdot r)), x_{i} (q)}, \end{aligned}$ , ∀ p, q ∈ $H$ . Hence, X_i be a ‘fuzzy left p-ideal’ in $H$ . X_i⁺ be a ‘maximal fuzzy left p-ideal’ in $H$ by 5.2. Note that $\begin{aligned} {x_{i}}^{+} (p_{\circ}) & = x_{i} (p_{\circ}) + 1 - x_{i} (0) \\ = \frac{ϱ (p_{\circ}) + ϱ (p_{\circ})}{2} + 1 - \frac{ϱ (0) + ϱ (p_{\circ})}{2} \end{aligned}$ and $\begin{aligned} {x_{i}}^{+} (p_{\circ}) < 1 = {x_{i}}^{+} (0) \end{aligned}$ Therefore, ${x_{i}}^{+}$ is variable and ϱ is no more a maximal element in $N (\hat{m})$ , which is clearly a contradiction.

6 Prime Fuzzy Left p-ideals

Definition 6.1

In $H$ , φ is called ‘prime fuzzy left(right) p-ideal’ if φ is variable function and for any ϱ and X_i to be two fuzzy left(right) p-ideals in $H$ , $ϱ_{p}^{\circ} x_{i}$ ⊆ φ which shows that ϱ ⊆ φ or X_i ⊆ φ.

Theorem 6.2

A fuzzy subset φ is a ‘prime fuzzy left(right) p-ideal’ in $H$ if and only if (i) $H$ = {p ∈ $H$ |φ(p) = φ(0)}be a prime left(right) p-ideal in $H$ (ii) ℑ(φ) = {φ(p)|p ∈ $H$ } consists of two elements exactly. (iii) φ(0) must be equal to one.

Proof. This theorem will just be proved for left p-ideals. Because right p-ideals can be proved similarly. It is simple to evaluate $φ^{\circ}$ as a ‘prime left p-ideal’. Let us consider ℑ(φ) has more than two values. Then there must exist two elements (p · r) · (q · r), q ∈ $H$ \ $φ^{\circ}$ so that φ((p · r) · (q · r)) /= φ(q) without having loss in generality we may suppose that φ((p · r) · (q · r)) < φ(q). As φ is a fuzzy left p-ideal and q does not contains in $φ (\circ)$ . It shows that φ((p · r) · (q · r)) < φ(q) < φ(0) therefore ∃r, l ∈ [0, 1] then (1) $φ ((p \dots r) \cdot (q \dots r)) < r < φ (q) < l < φ (0)$ (1) Suppose X_i and ω are fuzzy left p-ideals so that we can define them as X_i = $r χ_{〈 (p \cdot r) \cdot (q \cdot r) 〉}$ and $ω = \hat{l} χ_{〈 q 〉}$ where $χ_{〈 (p \cdot r) \cdot (q \cdot r) 〉}, χ_{〈 q 〉}$ be characteristic function of ideals which are generated by (p · r) · (q · r) and q, respectively. Then, for each p ∈ $H$ which cannot be expressed by the form (p₁ · p₂) + ((p₁ · r₁) · (q₁ · r₁)) · q₁ + r₁ · r₂ = ((p₂ · r₂) · (q₂ · r₂)) · q₂ + r₁ · r₂, where r ∈ $H$ , (p₁ · r₁) · (q₁ · r₁), (p₂ · r₂) · (q₂ · r₂) ∈ 〈(p · r) · (q · r)〉 and q₁, q₂ ∈ 〈q〉 we will have on $φ (x_{i}_{p}^{\circ} ω (p_{1} \cdot p_{2}) = 0$ the other hand

$(x_{i}_{p}^{\circ} ω (p_{1} \cdot p_{2}) = sup_{(p_{1} \cdot p_{2}) + ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})) \cdot q_{1} + r_{1} \cdot r_{2} = ((p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2})) \cdot q_{2} + r_{1} \cdot r_{2}} (min {x_{i} ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})), x_{i} ((p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2})), ω (q_{1}), ω (q_{2})})$

As φ is a ‘fuzzy left p-ideal’ from $(p_{1} \dots p_{2}) + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) \cdot q_{1} + r_{1} \dots r_{2} = ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) \cdot q_{2} + r_{1} \dots r_{2}$ it shows that $φ (p) \geq min {φ (((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) q_{1}), φ (((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) q_{2})}$ $\geq min {φ (q_{1}, φ (q_{2}} \geq r$

So $(x_{i}_{p}^{\circ} ω) (p_{1} \cdot p_{2}) \leq φ (p)$ hence $(x_{i}_{p}^{\circ} ω)$ ⊆ φ which shows X_i ⊆ φ or ω ⊆ φ as φ is a fuzzy prime left p-ideal. Hence, $x_{i} ((p \cdot r) \cdot (q \cdot r)) = r \leq φ ((p \cdot r) \cdot (q \cdot r))$ or $ω (q) = \hat{l} \leq φ (q)$ clearly which contradicts to 1. As a resul,t ℑ (ϱ) exactly consists of two elements. To verify (iii), let φ be a ‘prime fuzzy left p-ideal’ and φ(0) must be not equal to one. So by (ii), $Im (φ) = {κ_{1}, κ_{2}}$ for $0 \leq κ_{1} < κ_{2} < 1$ . As $φ (0) = φ (0, p) \geq φ (p) \forall p \in H$ , we have φ(0)=κ₂

Therefore $φ (p) = \{\begin{array}{ll} κ_{2} & if p \in φ^{\circ}, \\ κ_{1} & otherwise \end{array}$ Let for fixed (p · r) · (q · r) ∈ φ° and q ∈ $H$ \ φ° be two fuzzy subsets $ϱ (p) = \{\begin{array}{ll} \hat{l} & if p \in 〈 (p \cdot r) \cdot (q \cdot r) 〉, \\ 0 & otherwise \end{array}$ $x_{i} (p) = \{\begin{array}{ll} r & if p \in 〈 q 〉, \\ 0 & otherwise \end{array}$

for $0 \leq κ_{1} < r < κ_{2} < t \leq 1$ . Clearly, ϱ and X_i be fuzzy left p-ideals in $H$ . If p is not able to satisfy the equality $(p_{1} \dots p_{2}) + ((p_{1} \dots r_{1}) \cdot (q_{1} \dots r_{1})) \cdot q_{1} + r_{1} \dots r_{2} = ((p_{2} \dots r_{2}) \cdot (q_{2} \dots r_{2})) \cdot q_{2} + r_{1} \dots r_{2}$ for (p₁ · r₁) · (q₁ · r₁), (p₂ · r₂) · (q₂ · r₂) ∈ 〈(p · r) · (q · r)〉, q₁, q₂ ∈ 〈q〉 and r ∈ $H$ then on the other hand $(x_{i}_{p}^{\circ} ω (p_{1} \cdot p_{2}) = 0$ $\begin{aligned} (ϱ_{p}^{\circ} x_{i} (p_{1} \cdot p_{2}) = sup_{p_{1} \cdot p_{2}) + ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})) \cdot q_{1} + r_{1} \cdot r_{2} = ((p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2})) \cdot q_{2} + r_{1} \cdot r_{2}} \\ (min {ϱ ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})), ϱ ((p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2})), x_{i} (q_{1}, x_{i} (q_{2}}) min {t, r} = r \end{aligned}$ By (i), $φ^{\circ}$ is a ‘prime left p-ideal’. Let ((p₁ ·r₁)·(q₁ ·r₁)), ((p₂ ·r₂)·(q₂ ·r₂)) ∈ $φ^{\circ}$ 〈(p·r)·(q ·r)〉 then ((p₁ · r₁) · (q₁ · r₁)), ((p₂ · r₂) · (q₂ · r₂)) ∈ $φ^{\circ}$ because (p · r)· (q · r) ∈ $φ^{\circ}$ and 〈(p · r)· (q · r)〉 ⊆ $φ^{\circ}$ .

This shows that p ∈ $φ^{\circ}$ therefore $φ (p) = κ_{2 > r = (ϱ_{p}^{\circ} x_{i}) (p_{1} \cdot p_{2})}$ thus $ϱ_{p}^{\circ} x_{i} \subseteq φ$ . But $ϱ ((p \dots r) \cdot (q \dots r)) = t > κ_{2} = φ ((p \dots r) \dots (q \dots r))$ and $X_{i} (q) = r > κ_{1} = φ (q)$ which implies ϱ $⊈$ φ and X_i $⊈$ φ. Which contradicts the given assumption which is φ is a ‘prime fuzzy left p-ideal’ in $H$ . Thus, φ(0) is equal to one. Conversely, the above-mentioned conditions are supposed to satisfy. Then φ(0) = 1 and ℑ (ϱ) = {κ, 1} for any $0 \leq κ < 1$

Further $φ (p + q) \geq min {φ (p), φ (q)}$ for some p, q ∈ $H$ as φ(p + q) < min{φ(p), φ(q)} shows φ(p) = φ(q) = 1 that is p, q ∈ $φ^{\circ}$ hence p + q ∈ $φ^{\circ}$ and as a result φ(p + q) = 1 which cannot be possible. Correspondingly, φ(pq) ≥ φ(q) as φ(q) = 1 shows that pq ∈ $φ^{\circ}$ hence φ(pq) = 1. Which shows that φ is a ‘fuzzy left ideal’ in $H$ . In fact, φ be a ‘fuzzy left p-ideal’. It is clearly prime. Of course, if there are two fuzzy left p-ideals $x_{i} ⊈ φ$ so that $ϱ_{p}^{\circ} x_{i} \subseteq φ$ then $ϱ (p_{\circ} > φ (p_{\circ}))$ and for $x_{i} (q_{\circ} > φ (q_{\circ}))$ any $p_{\circ}, q_{\circ} \in H$ . It is possible only if $φ (p_{\circ})$ = $φ (q_{\circ})$ = κ, i.e. $p_{\circ}, q_{\circ}$ ∈/ $φ^{\circ}$ as $φ^{\circ}$ is prime then there must exists r ∈ $H$ so that $p_{\circ} r q_{\circ} \notin φ^{\circ}$ otherwise $p_{\circ} H q_{\circ} \subseteq φ_{\circ}$ whence $(H p_{\circ}) (H q_{\circ}) \subseteq φ^{\circ}$ . So, $\bar{(H p_{\circ}) (H q_{\circ})} \subseteq \bar{φ^{\circ}} = φ^{\circ}$ because φ^ˆ is a left p-ideal in $H$ . Furthermore, $\bar{(H p_{\circ})} \bar{(H q_{\circ})} \subseteq \bar{\bar{(H p_{\circ})} \bar{(H q_{\circ})}} = \bar{(H p_{\circ}) (H q_{\circ})}$ by 4.3. Therefore, $\bar{(H p_{\circ})} \bar{(H q_{\circ})} \subseteq φ^{\circ}$ and as a result $\bar{(H p_{\circ})} \subseteq φ^{\circ}$ or $\bar{(H q_{\circ})} \subseteq φ^{\circ}$ . In the first case, $\bar{〈 p_{\circ} 〉〈 p_{\circ} 〉} \subseteq \bar{H p_{\circ}}$ hence $\bar{〈 p_{\circ} 〉} \subseteq φ^{\circ}$ , thus $p_{\circ} \in 〈 p_{\circ} 〉$ for $〈 p_{\circ} 〉 \subseteq \bar{〈 p_{\circ} 〉} \subseteq φ^{\circ}$ which is a clearly contradiction. The second case also gives contradiction.

Suppose $(p \cdot r) \cdot (q \cdot r) = p_{\circ} r q_{\circ}$ then $φ ((p \cdot r) \cdot (q \cdot r)) = κ$ as a result of assumption (2) $(ϱ_{p}^{\circ} x_{i}) (a_{1} \cdot a_{2}) \leq φ (a) = κ$ (2) Clearly $(p \cdot r) \cdot (q \cdot r) + p_{\circ} r q_{\circ} = 2 p_{\circ} q_{\circ}$ Therefore $(p \cdot r) \cdot (q \cdot r) + p_{\circ} (r q_{\circ}) + r = (2 p_{\circ}) (r q_{\circ}) + r$ for some $r \in H$ thus for $(p \cdot r) \cdot (q \cdot r) = p_{\circ} r q_{\circ}$ we will have $\begin{aligned} (ϱ_{p}^{\circ} x_{i}) ({\overset{ˇ}{s}}_{1} \cdot {\overset{ˇ}{s}}_{2}) = sup_{({\overset{ˇ}{s}}_{1} \cdot {\overset{ˇ}{s}}_{2}) + ((p_{1} \cdot r_{1}) \cdot (q_{1} \cdot r_{1})) \cdot q_{1} + r_{1} \cdot r_{2} = ((p_{2} \cdot r_{2}) \cdot (q_{2} \cdot r_{2})) \cdot q_{2} + r_{1} \cdot r_{2}} \\ (min {ϱ (p_{1}), ϱ (p_{2}), x_{i} (q_{1}), x_{i} (q_{2})}) \\ \geq min {ϱ (p_{\circ}), ϱ (2 p_{\circ}), x_{i} (r q_{\circ})} \\ \geq min {ϱ (p_{\circ}), x_{i} (r q_{\circ})} \\ \geq min {ϱ (p_{\circ}), x_{i} (q_{\circ})} > κ \end{aligned}$ As $ϱ (p_{\circ}) > κ$ and $x_{i} (q_{\circ}) > κ$ this contradicts 2. Therefore, for any fuzzy left p-ideals ϱ and X_i in $H ϱ_{p}^{\circ} x_{i} \subseteq φ$ shows ϱ ⊆ φ or X_i ⊆ ζ, which clearly completes our proof.

7 Conclusion

The work done on fuzzy sets in semirings and hemirings must be connected in future with building a fuzzy spectrum in hemirings and in examining semiprime fuzzy p-ideals. In this paper, we examined the fact that key results in semirings related to fuzzy sets are similar but not the same as that of the corresponding results in hemirings. So, fuzzy sets of diﬀerent types must be studied in rings, semirings and hemirings. The results which we obtained in this paper can be consumed to examine the graph that whether it is balanced or clusterable and also to examine social network problems.

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

Additional information

Notes on contributors

Muhammad Touqeer

Muhammad Touqeer did his PhD from University of Punjab, Lahore in 2015 and is currently working as Assistant Professor of Mathematics in the Department of Basic Sciences, University of Engineering and Technology Taxila, Pakistan. His field of interest includes Fuzzy sets, Soft sets, Fuzzy decision making and BCK/BCI algebras.

Atiqa Maryam

Atiqa Maryam did his MS-Mathematics from University of Engineering and Technology Taxila, Pakistan in 2019 and is currently working as Lecturer (Visiting) in Mathematics Department, University of Education Jauhrabad, Pakistan. Her field of interest is Fuzzy Algebra.

Muhammad Nauman Saeed

Muhammad Nauman Saeed did his MS- Mathematics from University of Engineering and Technology Taxila, Pakistan in 2019 and is currently working as SESE-Mathematics in Govt. High School Verowal, Pakistan. His field of interest includes Fuzzy sets, Soft sets, Fuzzy soft set decision making problems.

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Characterisation of p-ideals in Hemirings on the Basis of Hemiregularity, Prime and Normal Fuzzy Ideals

Abstract

1 Introduction

2 Preliminaries

3 Fuzzy p-ideals

4 p-hemiregularity

5 Normal Fuzzy Left p-ideals

6 Prime Fuzzy Left p-ideals

7 Conclusion

Disclosure statement

Notes on contributors

Muhammad Touqeer

Atiqa Maryam

Muhammad Nauman Saeed

References

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Help and information

Characterisation of p-ideals in Hemirings on the Basis of Hemiregularity, Prime and Normal Fuzzy Ideals

Abstract

1 Introduction

2 Preliminaries

3 Fuzzy p-ideals

4 p-hemiregularity

5 Normal Fuzzy Left p-ideals

6 Prime Fuzzy Left p-ideals

7 Conclusion

Disclosure statement

Additional information

Notes on contributors

Muhammad Touqeer

Atiqa Maryam

Muhammad Nauman Saeed

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