A Novel Study of Fuzzy Bi-Ideals in Ordered Semirings

Abstract

In this study, by generalizing the notion of fuzzy bi-ideals of ordered semirings, the notion of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals is established. We prove that $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals are fuzzy bi-ideals but that the converse is not true, and an example is provided to support this proof. A condition is given under which fuzzy bi-ideals of ordered semirings coincide with $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals. An equivalent condition and certain correspondences between bi-ideals and $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals are presented. Moreover, the $({\kappa }^{*},\kappa )$-lower part of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals is described and depicted in terms of several classes of ordered semirings. Furthermore, it is shown that the ordered semiring is bi-simple if and only if it is $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-simple.

1. Introduction

The concept of an “ordered semiring” was first used by Gan and Jiang [1] in connection to a semiring with a compatible partial order relation. They also proposed the idea of ideals in ordered semirings. Good et al. [2] developed the concept of bi-ideals in semigroups. Following that, Lajos et al. [3] established bi-ideals in associative rings. Bi-ideals of ordered semirings were described and characterized in terms of regularity, and the relationship between bi-ideals and quasi-ideals was characterized by Palakawong et al. [4]. Senarat et al. [5] developed the terms-ordered k-bi-ideal, strong-prime-ordered k-bi-ideal, and prime-ordered k-bi-ideal in ordered semirings. By expanding on the idea of bi-ideals of ordered semirings, Davvaz et al. [6] introduced the concept of bi-hyperideals in ordered semi-hyperrings. The notions of $(m,n)$-bi-hyperideals and Prime $(m,n)$-bi-hyperideals were established and inter-related properties were considered by Omidi and Davvaz [7]. The characterization of ordered h-regular semirings was considered by Anjum et al. [8]. In [9], Patchakhieo and Pibaljommee characterized ordered k-regular semirings using ordered k-ideals. The ordered intra-k-regular semirings have been introduced and defined in different ways by Ayutthaya and Pibaljommee [10]. Omidi and Davvaz [7] considered the concepts of $(m,n)$-bi-hyperideals and Prime $(m,n)$-bi-hyperideals and established inter-related features. Anjum et al. [8] proposed characterizing ordered h-regular semirings. By using ordered k-ideals, Patchakhieo and Pibaljommee described ordered k-regular semirings in [9]. The ordered intra-k-regular semirings have been presented and characterized in various ways by Ayutthaya and Pibaljommee [10].

Fuzzy sets to semirings were initially discussed by Ahsan et al. in [11] and Kuroki [12] applied the idea to semigroups. Mandal [13] pioneered the study of ideals and interior ideals in ordered semirings, as well as their characterizations in the sense of regularity. He developed the concepts of fuzzy bi-ideals and fuzzy quasi-ideals in ordered semirings in [14]. Gao et al. [15] presented semisimple fuzzy ordered semirings and weakly regular fuzzy ordered semirings in terms of different kinds of fuzzy ideals. Saba et al. [16] initiated the study of ordered semirings based on single-valued neutrosophic sets. Several characterizations of regular and intra-regular ordered semigroups in terms of $(\in ,\in \vee q)$-fuzzy generalized bi-ideals were presented by Jun et al. [17], who also proposed the idea of $(\alpha ,\beta )$-fuzzy generalized bi-ideal in ordered semigroups. Similar semiring concepts, such as $(\in ,\in \vee q)$-fuzzy bi-ideals on semirings, were investigated by Hedayati [18]. Additionally, other ideas connected to our research in several domains have been examined in [19,20,21,22,23,24,25].

In this study, we describe a novel form of fuzzy ideal in ordered semirings. The concept of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal is presented. We show that any fuzzy bi-ideal is the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal, but the converse assertion is invalid, and an example is shown. A criterion for an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal to be a fuzzy bi-ideal is given. Furthermore, some correspondences between bi-ideal and $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal are included. Furthermore, regularly ordered semirings are described in terms of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals and their $(\kappa *,\kappa )$-lower parts. The structure of the paper is as follows: Section 2 highlights some of the ideas and properties of ordered semirings, ideals, fuzzy subsets, and fuzzy subsemirings that are necessary to generate our key results. Section 3 focuses on the concept of the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal of ordered semirings. Section 4 examines the $(\kappa *,\kappa )$-lower part of the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal. Section 5 contains instructions for some potential future research work.

2. Preliminaries

An ordered semiring $({\rm Y},+,·,\le )$ is a semiring with compatible order relation $“\le ”$, i.e., $\wp \le \varrho \Rightarrow \wp \tau \le \varrho \tau ,\tau \wp \le \tau \varrho$ and $\wp +\tau \le \varrho +\tau ,\tau +\wp \le \tau +\varrho$, $\forall \wp ,\varrho ,\tau \in {\rm Y}$.

If $\wp +\varrho =\varrho +\wp$, $\forall \wp ,\varrho \in {\rm Y}$, then ${\rm Y}$ is said to be additively commutative. An element $0\in {\rm Y}$ is an absorbing zero if $0\wp =0=\wp 0$ and $\wp +0=\wp =0+\wp$, $\forall \wp \in {\rm Y}$.

For $P\subseteq {\rm Y}$, we define $\left(P\right]=\{\wp \in {\rm Y}|\wp \le \varrho \mathrm{for\; some}\wp \in P\}.$ For $(\varnothing \ne )P,Q\subseteq {\rm Y}$, $PQ$ is defined as $\{\wp \varrho |\wp \in P\mathrm{and}\varrho \in Q\}$.

A subset ($\varnothing \ne )\Sigma$ of ${\rm Y}$ is said to be a sub-semiring if $\Sigma \Sigma \subseteq \Sigma$ and $\Sigma +\Sigma \subseteq \Sigma$. Additionally, $\Sigma$ refers to the left (resp. right) ideal of ${\rm Y}$ if $\Sigma +\Sigma \subseteq \Sigma$ and ${\rm Y}\Sigma \subseteq \Sigma (\mathrm{resp}.\Sigma {\rm Y}\subseteq \Sigma )$, and $(\Sigma ]\subseteq \Sigma$. If it is both the left and right ideals of ${\rm Y}$, it is referred to as an ideal. A sub-semiring P of ${\rm Y}$ is called a bi-ideal (in brief, BI) of ${\rm Y}$ if $P{\rm Y}P\subseteq P$ and $\left(P\right]\subseteq P$.

A mapping ${\tilde{\lambda }}^f:{\rm Y}\to [0,1]$ is said to be fuzzy set (in brief, FS) of ${\rm Y}$. For the FSs ${\tilde{\lambda }}^f$ and ${\tilde{\pounds }}^f$ of ${\rm Y}$, ${\tilde{\lambda }}^f\cap {\tilde{\pounds }}^f, {\tilde{\lambda }}^f\cup {\tilde{\pounds }}^f, {\tilde{\lambda }}^f+{\tilde{\pounds }}^f$ and ${\tilde{\lambda }}^f\circ {\tilde{\pounds }}^f$ are described as:

$$\begin{array}{cc}\hfill ({\tilde{\lambda }}^f\cap {\tilde{\pounds }}^f)(\wp )& ={\tilde{\lambda }}^f(\wp )\wedge {\tilde{\pounds }}^f(\wp )=\min \{{\tilde{\lambda }}^f(\wp ),{\tilde{\pounds }}^f(\wp )\},\hfill \\ \hfill ({\tilde{\lambda }}^f\cup {\tilde{\pounds }}^f)(\wp )& ={\tilde{\lambda }}^f(\wp )\vee {\tilde{\pounds }}^f(\wp )=\max \{{\tilde{\lambda }}^f(\wp ),{\tilde{\pounds }}^f(\wp )\},\hfill \end{array}$$

$$({\tilde{\lambda }}^f+{\tilde{\pounds }}^f)(\wp )=\left\{\begin{array}{cc}{\displaystyle \underset{\wp \le \varrho +\varkappa }{\bigvee }}{\tilde{\lambda }}^f\left(\varrho \right)\wedge {\tilde{\pounds }}^f\left(\varkappa \right),\hfill & \hfill \\ 0,\mathrm{if}\wp \mathrm{can}\mathrm{not}\mathrm{be}\mathrm{written}\mathrm{as}\wp \le \varrho +\varkappa ,\hfill & \hfill \end{array}\right.$$

and

$$({\tilde{\lambda }}^f\circ {\tilde{\pounds }}^f)(\wp )=\left\{\begin{array}{cc}{\displaystyle \underset{\wp \le \varrho \varkappa }{\bigvee }}{\tilde{\lambda }}^f\left(\varrho \right)\wedge {\tilde{\pounds }}^f\left(\varkappa \right),\hfill & ,\hfill \\ 0,\mathrm{if}\wp \mathrm{cannot}\mathrm{be}\mathrm{written}\mathrm{as}\wp \le \varrho \varkappa .\hfill & ,\hfill \end{array}\right.$$

For $\Omega \subseteq {\rm Y}$, the characteristic function ${\chi }_{\Omega }^f$ is defined as:

$${\chi }_{\Omega }^f(\wp )=\left\{\begin{array}{cc}1,\hfill & if\wp \in \Omega ;\hfill \\ 0,\hfill & if\wp \notin \Omega .\hfill \end{array}\right.$$

Define ⪯ on the set $\mathcal{F}({\rm Y})$ of all FSs of ${\rm Y}$ by

$${\tilde{\lambda }}^f⪯{\tilde{\pounds }}^f\iff {\tilde{\lambda }}^f(\wp )\le {\tilde{\pounds }}^f(\wp ),\forall \wp \in {\rm Y}.$$

If ${\tilde{\lambda }}^f,{\tilde{\pounds }}^f\in \mathcal{F}({\rm Y})$ such that ${\tilde{\lambda }}^f⪯{\tilde{\pounds }}^f$, then $\forall {\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$, ${\tilde{\lambda }}^f\circ {\tilde{\lambda }}^f⪯{\tilde{\pounds }}^f\circ {\tilde{\lambda }}^f$ and ${\tilde{\lambda }}^f\circ {\tilde{\lambda }}^f⪯{\tilde{\lambda }}^f\circ {\tilde{\pounds }}^f$. We represent by $1^f$ the FS of ${\rm Y}$ given by $1^f:{\rm Y}\to [0,1]|r\mapsto 1^f\left(r\right)=1$.

Let $P,Q\subseteq {\rm Y}$. Then $P\subseteq Q\iff {\chi }_P^f⪯{\chi }_Q^f$; ${\chi }_P^f\cap {\chi }_Q^f={\chi }_{P\cap Q}^f$; ${\chi }_P^f\circ {\chi }_Q^f={\chi }_{\left(PQ\right]}$.

A FS ${\tilde{\lambda }}^f$ is called a:

• Fuzzy subsemiring of ${\rm Y}$ if ${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)$ and ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)$, $\forall \wp ,\varrho \in {\rm Y}$.
• Fuzzy left (resp. right) ideal (in brief, FL(R)I) of ${\rm Y}$ if $\wp \le \varrho \Rightarrow {\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right),$ ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)$ and ${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f\left(\varrho \right)$ (resp. ${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )$), $\forall \wp ,\varrho \in {\rm Y}$.
• Fuzzy ideal of ${\rm Y}$ if ${\tilde{\lambda }}^f$ is both fuzzy right and left ideals of ${\rm Y}$.
• Fuzzy bi-ideal (in brief, FBI) if it is fuzzy subsemiring and $\wp \le \varrho \Rightarrow {\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)$ and ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)$, $\forall \wp , t, \varrho , \in {\rm Y}$.

3. $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-Fuzzy Bi-Ideals of Ordered Semirings

In this section, the concept of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals of ${\rm Y}$ is introduced.

Let $\wp \in {\rm Y}$ and $\iota \in (0,1]$. The ordered fuzzy point (OFP) ${\wp }_{\iota }$ of ${\rm Y}$ is defined by

$${\wp }_{\iota }\left(\varkappa \right)=\left\{\begin{array}{cc}\iota ,\hfill & \mathrm{if}\varkappa \in (\wp ];\hfill \\ 0,\hfill & \mathrm{if}\varkappa \notin (\wp ].\hfill \end{array}\right.$$

For ${\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$, ${\wp }_{\iota }\in {\tilde{\lambda }}^f$ represents for ${\wp }_{\iota }\subseteq {\tilde{\lambda }}^f$. Thus ${\wp }_{\iota }\in {\tilde{\lambda }}^f\iff {\tilde{\lambda }}^f(\wp )\ge \iota$.

Definition 1.

An OFP ${\wp }_{\iota }$ of Υ is said to be $({\kappa }^{*},q)$-quasi-coincident with a FS ${\tilde{\lambda }}^f$ of Υ for ${\kappa }^{*}\in (0,1]$, denoted as ${\wp }_{\iota }({\kappa }^{*},q){\tilde{\lambda }}^f$, and defined as:

$${\tilde{\lambda }}^f(\wp )+\iota >{\kappa }^{*}.$$

For the OFP ${\wp }_{\iota }$, we define

(1)

${\wp }_{\iota }({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$, if ${\tilde{\lambda }}^f(\wp )+\iota +\kappa >{\kappa }^{*}$;

(2)

${\wp }_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$, if ${\wp }_{\iota }\in {\tilde{\lambda }}^f$ or ${\wp }_{\iota }({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$;

(3)

${\wp }_{\iota }\overline{\alpha }{\tilde{\lambda }}^f$, if ${\wp }_{\iota }\alpha {\tilde{\lambda }}^f$ does not hold for $\alpha \in \{({\kappa }^{*},q_{\kappa }),\in \vee ({\kappa }^{*},q_{\kappa })\}$;

for $1\ge {\kappa }^{*}>\kappa \ge 0$.

Definition 2.

A FS ${\tilde{\lambda }}^f$ of Υ is said to be an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal (in brief, $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI) of Υ if:

(1)

$\wp \le \varrho$, ${\varrho }_{\iota }\in {\tilde{\lambda }}^f\Rightarrow {\wp }_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$,

(2)

${\wp }_{\iota }\in {\tilde{\lambda }}^f$ and ${\varrho }_{\theta }\in {\tilde{\lambda }}^f\Rightarrow {(\wp +\varrho )}_{\iota \wedge \theta }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$,

(3)

${\wp }_{\iota }\in {\tilde{\lambda }}^f$ and ${\varrho }_{\theta }\in {\tilde{\lambda }}^f\Rightarrow {(\wp \varrho )}_{\iota \wedge \theta }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$, and

(4)

$t\in {\rm Y}$, ${\wp }_{\iota }\in {\tilde{\lambda }}^f$, ${\varrho }_{\iota }\in \lambda \Rightarrow {(\wp t\varrho )}_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$.

$\forall \iota ,\theta \in (0,1]$ and $\wp ,t,\varrho \in {\rm Y}$.

Example 1.

On ${\rm Y}=\{{\wp }_1,{\wp }_2,{\wp }_3\}$, define the opertaions and order relation as

$$\begin{gathered} \begin{array}{ccc}\begin{array}{cccc}+& {\wp }_1& {\wp }_2& {\wp }_3\\ {\wp }_1& {\wp }_1& {\wp }_2& {\wp }_3\\ {\wp }_2& {\wp }_2& {\wp }_2& {\wp }_2\\ {\wp }_3& {\wp }_3& {\wp }_2& {\wp }_2\end{array}& & \begin{array}{cccc}·& {\wp }_2& {\wp }_3& {\wp }_3\\ {\wp }_1& {\wp }_1& {\wp }_1& {\wp }_1\\ {\wp }_2& {\wp }_1& {\wp }_2& {\wp }_2\\ {\wp }_3& {\wp }_1& {\wp }_3& {\wp }_3\end{array}\end{array} \\ \le :=\{({\wp }_1,{\wp }_1),({\wp }_2,{\wp }_2),({\wp }_3,{\wp }_3),({\wp }_1,{\wp }_2),({\wp }_1,{\wp }_3)\}. \end{gathered}$$

Then $({\rm Y},+,·,\le )$ is an ordered semiring. Define an FS ${\tilde{\lambda }}^f$ of Υ as

$${\tilde{\lambda }}^f\left(\varkappa \right)=\left\{\begin{array}{cc}0.5,\hfill & if\varkappa ={\wp }_1;\hfill \\ 0.4,\hfill & if\varkappa ={\wp }_2;\hfill \\ 0.3,\hfill & if\varkappa ={\wp }_3.\hfill \end{array}\right.$$

${\tilde{\lambda }}^f$ is the $(\in ,\in \vee (0.2,q_{0.6}))$-FBI of Υ and can be easily verified.

Lemma 1.

Each FBI of Υ is the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ.

Proof. 

Straightforward. □

Remark 1.

In general, the converse of Lemma 1 does not hold. It is illustrated by the following example:

Example 2.

Define operations and ordered relations on ${\rm Y}=\{{\wp }_1,{\wp }_2,{\wp }_3\}$ as follows:

$$\begin{gathered} \begin{array}{ccc}\begin{array}{cccc}+& {\wp }_1& {\wp }_2& {\wp }_3\\ {\wp }_1& {\wp }_1& {\wp }_2& {\wp }_3\\ {\wp }_2& {\wp }_2& {\wp }_2& {\wp }_3\\ {\wp }_3& {\wp }_3& {\wp }_3& {\wp }_3\end{array}& & \begin{array}{cccc}·& {\wp }_1& {\wp }_2& {\wp }_3\\ {\wp }_1& {\wp }_1& {\wp }_1& {\wp }_1\\ {\wp }_2& {\wp }_1& {\wp }_2& {\wp }_2\\ {\wp }_3& {\wp }_1& {\wp }_2& {\wp }_2\end{array}\end{array} \\ \le :=\{({\wp }_1,{\wp }_1),({\wp }_2,{\wp }_2),(b,{\wp }_3),({\wp }_1,{\wp }_2),({\wp }_2,{\wp }_3)\}. \end{gathered}$$

Then, $({\rm Y},+,·,\le )$ is an ordered semiring. Define the FS ${\tilde{\lambda }}^f$ of Υ as

$${\tilde{\lambda }}^f\left(\varkappa \right)=\left\{\begin{array}{cc}0.6,\hfill & if\varkappa ={\wp }_1;\hfill \\ 0.5,\hfill & if\varkappa ={\wp }_2;\hfill \\ 0.7,\hfill & if\varkappa ={\wp }_3.\hfill \end{array}\right.$$

It can be easily verified that ${\tilde{\lambda }}^f$ is the $(\in ,\in \vee (0.9,q_{0.1}))$-fuzzy bi-deal of Υ but not an FBI of Υ as follows: ${\wp }_1\le {\wp }_3\nRightarrow {\tilde{\lambda }}^f\left({\wp }_1\right)\ge {\tilde{\lambda }}^f\left({\wp }_3\right)$.

Theorem 1.

An FS ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ⇔

(1)

$\wp \le \varrho$$\Rightarrow {\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$

(2)

${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$,

(3)

${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, and

(4)

${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$,

$\forall \wp ,t,\varrho \in {\rm Y}$.

Proof. 

(⇒) Let $\wp ,\varrho \in {\rm Y}$ such that $\wp \le \varrho$. If ${\tilde{\lambda }}^f(\wp )<{\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, then ∃$\iota \in (0,1]$ such that ${\tilde{\lambda }}^f(\wp )<\iota \le {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. So $s_{\iota }\in {\tilde{\lambda }}^f$, but ${(\wp )}_{\iota }\overline{\in \vee ({\kappa }^{*},q_{\kappa })}{\tilde{\lambda }}^f$, which is a contradiction. Therefore ${\tilde{\lambda }}^f(\wp )\ge min\{{\tilde{\lambda }}^f\left(\varrho \right),\frac{{\kappa }^{*}-\kappa }{2}\}$. Next, if ${\tilde{\lambda }}^f(\wp +\varrho )<{\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, for some $\wp ,\varrho \in {\rm Y}$, then ${\tilde{\lambda }}^f(\wp +\varrho )<\iota \le {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, for some $\iota \in (0,1]$. Thus, ${\wp }_{\iota },{\varrho }_{\iota }\in {\tilde{\lambda }}^f$, but ${(\wp +\varrho )}_{\iota }\overline{\in \vee ({\kappa }^{*},q_{\kappa })}{\tilde{\lambda }}^f$, which is a contradiction. Therefore, ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Similarly, ${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp ,\varrho \in {\rm Y}$. Again, if ${\tilde{\lambda }}^f(\wp t\varrho )<{\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, for some $\wp ,t,\varrho \in {\rm Y}$, then ${\tilde{\lambda }}^f(\wp t\varrho )<\iota \le {\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}$ for some $\iota \in (0,1]$. Thus, ${\wp }_{\iota },{\varrho }_{\iota }\in {\tilde{\lambda }}^f$, but ${(\wp t\varrho )}_{\iota }\overline{\in \vee ({\kappa }^{*},q_{\kappa })}{\tilde{\lambda }}^f$, again a contradiction. Consequently, ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}$.

(⇐) Take any $\wp ,\varrho \in {\rm Y}$ and $\iota ,\theta \in (0,1]$ such that $\wp \le \varrho$ and ${\varrho }_{\theta }\in {\tilde{\lambda }}^f$. Then, ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$, and it follows that ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge \iota \wedge \frac{{\kappa }^{*}-\kappa }{2}$. If $\iota \le \frac{{\kappa }^{*}-\kappa }{2}$, then ${\tilde{\lambda }}^f(\wp )\ge \iota$ implies ${\wp }_{\iota }\in {\tilde{\lambda }}^f$. Again, if $\iota >\frac{{\kappa }^{*}-\kappa }{2}$, then ${\tilde{\lambda }}^f(\wp )\ge \frac{{\kappa }^{*}-\kappa }{2}$. Thus, ${\tilde{\lambda }}^f(\wp )+\iota >\frac{{\kappa }^{*}-\kappa }{2}+\frac{{\kappa }^{*}-\kappa }{2}={\kappa }^{*}-\kappa$, so ${\wp }_{\iota }({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. Therefore, ${\wp }_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. Again, take any ${\wp }_{\theta }\in {\tilde{\lambda }}^f$ and ${\varrho }_{\theta }\in {\tilde{\lambda }}^f$. Then, ${\tilde{\lambda }}^f(\wp )\ge \iota$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$. Therefore, ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge \iota \wedge \theta \wedge \frac{{\kappa }^{*}-\kappa }{2}$. Now, if $\iota \wedge \theta \le \frac{{\kappa }^{*}-\kappa }{2}$, then ${\tilde{\lambda }}^f(\wp +\varrho )\ge \iota \wedge \theta$ implies ${(\wp +\varrho )}_{\iota \wedge \theta }\in {\tilde{\lambda }}^f$. Again, if $\iota \wedge \theta >\frac{{\kappa }^{*}-\kappa }{2}$, then ${\tilde{\lambda }}^f(\wp +\varrho )\ge \frac{{\kappa }^{*}-\kappa }{2}$. Therefore, ${\tilde{\lambda }}^f(\wp +\varrho )+\iota \wedge \theta >\frac{{\kappa }^{*}-\kappa }{2}+\frac{{\kappa }^{*}-\kappa }{2}={\kappa }^{*}-\kappa$ implies that ${(\wp +\varrho )}_{\iota \wedge \theta }({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. Therefore, ${(\wp +\varrho )}_{\iota \wedge \theta }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. Similarly, ${(\wp \varrho )}_{\iota \wedge \theta }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$ for any ${\wp }_{\theta }\in {\tilde{\lambda }}^f$ and ${\varrho }_{\theta }\in {\tilde{\lambda }}^f$. Further, take any $t\in {\rm Y}$ and ${\wp }_{\iota },{\varrho }_{\iota }\in {\tilde{\lambda }}^f$, $\forall \iota \in (0,1]$. Then ${\tilde{\lambda }}^f(\wp )\ge \iota$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$. Therefore, ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge \iota \wedge \frac{{\kappa }^{*}-\kappa }{2}$. Now if $\iota \le \frac{{\kappa }^{*}-\kappa }{2}$, then ${\tilde{\lambda }}^f(\wp t\varrho )\ge \iota$ implies ${(\wp t\varrho )}_{\iota }\in {\tilde{\lambda }}^f$. If $\iota >\frac{{\kappa }^{*}-\kappa }{2}$, then ${\tilde{\lambda }}^f(\wp t\varrho )\ge \frac{{\kappa }^{*}-\kappa }{2}$. Thus ${\tilde{\lambda }}^f(\wp t\varrho )+\iota >\frac{{\kappa }^{*}-\kappa }{2}+\frac{{\kappa }^{*}-\kappa }{2}={\kappa }^{*}-\kappa$ i.e., ${(\wp t\varrho )}_{\iota }({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. Therefore, ${(\wp t\varrho )}_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$, as required. □

Theorem 2.

If ${\tilde{\lambda }}^f(\in \mathcal{F}({\rm Y})$) is $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ with ${\tilde{\lambda }}^f(\wp )<\frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp \in {\rm Y}$. Then ${\tilde{\lambda }}^f$ is an FBI of Υ.

Proof. 

Suppose that $\wp ,\varrho \in {\rm Y}$ such that $\wp \le \varrho$. Since ${\tilde{\lambda }}^f$ is $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI, ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. By hypothesis, ${\tilde{\lambda }}^f\left(\varrho \right)<\frac{{\kappa }^{*}-\kappa }{2}$; thus, it implies ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)$. Again, for any $\wp ,\varrho \in {\rm Y}$, we have

$${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$$

and

$${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}.$$

Since ${\tilde{\lambda }}^f\left(\varrho \right)<\frac{{\kappa }^{*}-\kappa }{2}$ and ${\tilde{\lambda }}^f(\wp )<\frac{{\kappa }^{*}-\kappa }{2}$, so

$${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)$$

and

$${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right).$$

Finally, take any $\wp ,t,\varrho \in {\rm Y}$. Since ${\tilde{\lambda }}^f$ is $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI, by Theorem 1 and the hypothesis

$${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp ),{\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}={\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right),$$

as required. □

Theorem 3.

Let $(\varnothing \ne )\Omega \subseteq {\rm Y}$. Then Ω is a BI of Υ⇔${\chi }_{\Omega }^f$, an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI.

Proof. 

Straightforward. □

Theorem 4.

An FS ${\tilde{\lambda }}^f$ is the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ⇔$U({\tilde{\lambda }}^f;\iota )(\ne \varnothing )(\iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}])$, a BI of Υ.

Proof. 

(⇒) Let $\wp \in {\rm Y}$ and $\varrho \in U({\tilde{\lambda }}^f;\iota )$ be such that $\wp \le \varrho$. Then, ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$. By Theorem 1, ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge \iota \wedge \frac{{\kappa }^{*}-\kappa }{2}=\iota$. Therefore, $\wp \in U({\tilde{\lambda }}^f;\iota )$. Let $\wp ,\varrho \in U({\tilde{\lambda }}^f;\iota )$, where $\iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}]$. Then ${\tilde{\lambda }}^f(\wp )\ge \iota$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$. By Theorem 1, ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge \iota \wedge \frac{{\kappa }^{*}-\kappa }{2}=\iota$. Therefore, $\wp +\varrho \in U({\tilde{\lambda }}^f;\iota )$. Similarly, $\wp \varrho \in U({\tilde{\lambda }}^f;\iota )$ for $\wp ,\varrho \in U({\tilde{\lambda }}^f;\iota )$. Let $\wp ,\varrho \in U({\tilde{\lambda }}^f;\iota )$ and $t\in {\rm Y}$. Then, ${\tilde{\lambda }}^f(\wp )\ge \iota$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$. So, by Theorem 1, ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge \iota \wedge \iota \wedge \frac{{\kappa }^{*}-\kappa }{2}=\iota$. Thus ${\tilde{\lambda }}^f(\wp t\varrho )\ge \iota$. Therefore $\wp t\varrho \in U({\tilde{\lambda }}^f;\iota )$. Hence $U({\tilde{\lambda }}^f;\iota )$ is a BI.

(⇐) Take any $\wp ,\varrho \in {\rm Y}$ with $\wp \le \varrho$. If ${\tilde{\lambda }}^f(\wp )<{\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, then for some $\iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}]$, ${\tilde{\lambda }}^f(\wp )<\iota \le {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. So $\varrho \in U({\tilde{\lambda }}^f;\iota )$, but $\wp \notin U({\tilde{\lambda }}^f;\iota )$, which is a contradiction. Thus ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp ,\varrho \in {\rm Y}$ with $\wp \le \varrho$. Again, if ${\tilde{\lambda }}^f(\wp +\varrho )<{\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, for some $\wp ,\varrho \in {\rm Y}$, then ${\tilde{\lambda }}^f(\wp +\varrho )<\iota \le {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, for some $\iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}]$. Thus, $\wp ,\varrho \in U({\tilde{\lambda }}^f;\iota )$, but $\wp +\varrho \notin U({\tilde{\lambda }}^f;\iota )$, a contradiction. Therefore, ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp ,\varrho \in {\rm Y}$. Similarly, ${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp ,\varrho \in {\rm Y}$. Further, if ${\tilde{\lambda }}^f(\wp t\varrho )<{\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, for some $\wp ,t,\varrho \in {\rm Y}$. Then, $\exists \iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}]$ such that ${\tilde{\lambda }}^f(\wp t\varrho )<\iota \le {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$ implies ${\wp }_{\iota },{\varrho }_{\iota }\in U({\tilde{\lambda }}^f;\iota )$, but ${(\wp t\varrho )}_{\iota }\notin U({\tilde{\lambda }}^f;\iota )$, again a contradiction. Therefore ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp ,\varrho \in {\rm Y}$, as required. □

Example 3.

Define the operations $(+,·)$ and order relation ≤ on ${\rm Y}=\{{\wp }_1,{\wp }_2,{\wp }_3,\tau \}$ in the following ways:

$$\begin{gathered} \begin{array}{ccc}\begin{array}{ccccc}+& {\wp }_1& {\wp }_2& {\wp }_3& {\wp }_4\\ {\wp }_1& {\wp }_1& {\wp }_2& {\wp }_3& {\wp }_4\\ {\wp }_2& {\wp }_2& {\wp }_2& {\wp }_3& {\wp }_4\\ {\wp }_3& {\wp }_3& {\wp }_3& {\wp }_3& {\wp }_4\\ {\wp }_4& {\wp }_4& {\wp }_4& {\wp }_4& {\wp }_4\end{array}& & \begin{array}{ccccc}·& {\wp }_1& {\wp }_2& {\wp }_3& {\wp }_4\\ {\wp }_1& {\wp }_1& {\wp }_1& {\wp }_1& {\wp }_1\\ {\wp }_2& {\wp }_1& {\wp }_2& {\wp }_2& {\wp }_2\\ {\wp }_3& {\wp }_1& {\wp }_2& {\wp }_2& {\wp }_2\\ {\wp }_4& {\wp }_1& {\wp }_2& {\wp }_2& {\wp }_2\end{array}\end{array} \\ \le :=\{({\wp }_1,{\wp }_1),({\wp }_2,{\wp }_2),({\wp }_3,{\wp }_3),({\wp }_1,{\wp }_2),({\wp }_2,{\wp }_3),({\wp }_3,c)\} \end{gathered}$$

Then, $({\rm Y},+,·,\le )$ is an ordered semiring. Now define an FS ${\tilde{\lambda }}^f$ of Υ as ${\tilde{\lambda }}^f\left({\wp }_1\right)=0.5, {\tilde{\lambda }}^f\left({\wp }_2\right)=0.4, {\tilde{\lambda }}^f\left({\wp }_3\right)=0.1$ and ${\tilde{\lambda }}^f\left(\tau \right)=0.3$. Therefore,

$$U({\tilde{\lambda }}^f;\iota )=\left\{\begin{array}{cc}{\rm Y},\hfill & \mathit{if}{\wp }_1<\iota \le 0.1;\hfill \\ \{{\wp }_1,{\wp }_2,\tau \},\hfill & \mathit{if}0.1<\iota \le 0.3;\hfill \\ \{{\wp }_1,{\wp }_2\},\hfill & \mathit{if}0.3<\iota \le 0.4;\hfill \\ \left\{{\wp }_1\right\},\hfill & \mathit{if}0.4<\iota \le 0.5;\hfill \\ \varnothing ,\hfill & \mathit{if}0.5<\iota \le 1.\hfill \end{array}\right.$$

By Theorem 4, ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ as $U({\tilde{\lambda }}^f;\iota )$ is a BI of Υ, $\forall \iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}]$, with ${\kappa }^{*}=1$ and $\kappa =0$.

Definition 3.

Let ${\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$. The set

$${\left[{\tilde{\lambda }}^f\right]}_{\iota }=\{\wp \in {\rm Y}|{\wp }_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f\},$$

where $\iota \in (0,1],$ is said to be an $(\in \vee ({\kappa }^{*},q_{\kappa }))$-level subset of ${\tilde{\lambda }}^f$.

Theorem 5.

Let ${\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$ such that $\wp \le \varrho$ implies ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)$. Then. ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ⇔$\forall \iota \in (0,1]$, the $(\in \vee ({\kappa }^{*},q_{\kappa }))$-level subset ${\left[{\tilde{\lambda }}^f\right]}_{\iota }$ of ${\tilde{\lambda }}^f$ is a bi-deal of Υ.

Proof. 

(⇒) Take any $\wp \in {\rm Y}$ and $\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$ such that $\wp \le \varrho$. As $\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$, we have ${\varrho }_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$ implies ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$ or ${\tilde{\lambda }}^f\left(\varrho \right)+\iota +\kappa >{\kappa }^{*}$. By hypothesis, we have ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$ or ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\ge {\kappa }^{*}-\iota -\kappa$. Thus, ${\wp }_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. Therefore, $\wp \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$. Next, take any $\wp ,\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$. Then, ${\wp }_{\iota },{\varrho }_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$; that is, ${\tilde{\lambda }}^f(\wp )\ge \iota$ or ${\tilde{\lambda }}^f(\wp )+\iota +\kappa >{\kappa }^{*}$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$ or ${\tilde{\lambda }}^f\left(\varrho \right)+\iota +\kappa >{\kappa }^{*}$.

Case (i).

Let ${\tilde{\lambda }}^f(\wp )\ge \iota$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$. If $\iota >\frac{{\kappa }^{*}-\kappa }{2}$; then,

$$\begin{array}{cc}\hfill {\tilde{\lambda }}^f(\wp +\varrho )& \ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge \iota \wedge \iota \wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\frac{{\kappa }^{*}-\kappa }{2},\hfill \end{array}$$

and, so, ${(\wp +\varrho )}_{\iota }({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. If $\iota \le \frac{{\kappa }^{*}-\kappa }{2}$, then

$$\begin{array}{cc}\hfill {\tilde{\lambda }}^f(\wp +\varrho )& \ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge \iota \wedge \iota \wedge \frac{{\kappa }^{*}-\kappa }{2}=\iota ,\hfill \end{array}$$

and so ${(\wp +\varrho )}_{\iota }\in {\tilde{\lambda }}^f$. Hence, ${(\wp +\varrho )}_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$.

Case (ii).

Let ${\tilde{\lambda }}^f(\wp )\ge \iota$ and ${\tilde{\lambda }}^f\left(\varrho \right)+\iota +\kappa >{\kappa }^{*}$. If $\iota >\frac{{\kappa }^{*}-\kappa }{2}$, then

$$\begin{array}{cc}\hfill {\tilde{\lambda }}^f(\wp +\varrho )& \ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & ={\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & >({\kappa }^{*}-\iota -\kappa )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & ={\kappa }^{*}-\iota -\kappa ,\hfill \end{array}$$

that is, ${\tilde{\lambda }}^f(\wp +\varrho )+\iota +\kappa >{\kappa }^{*}$, and thus ${(\wp +\varrho )}_{\iota }({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$. If $\iota \le \frac{{\kappa }^{*}-\kappa }{2}$, then

$$\begin{array}{cc}\hfill {\tilde{\lambda }}^f(\wp +\varrho )& \ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge \iota \wedge ({\kappa }^{*}-\iota -\kappa )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\iota ,\hfill \end{array}$$

and so ${(\wp +\varrho )}_{\iota }\in {\tilde{\lambda }}^f$. Hence, ${(\wp +\varrho )}_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$.

Case (iii).

Let ${\tilde{\lambda }}^f(\wp )+\iota +\kappa >{\kappa }^{*}$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge \iota$. Proof is analogous to case proof (ii).

Case (iv).

Let ${\tilde{\lambda }}^f(\wp )+\iota +\kappa >{\kappa }^{*}$ and ${\tilde{\lambda }}^f\left(\varrho \right)+\iota +\kappa >{\kappa }^{*}$. Proof is analogous to previous two cases.

Thus for all cases, we have ${(\wp +\varrho )}_{\iota }\in \vee ({\kappa }^{*},q_{\kappa }){\tilde{\lambda }}^f$, and thus $\wp +\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$. Similarly, for any $t\in {\rm Y}$ and $\wp ,\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$, we have $\wp \varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$ and $\wp t\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$. Hence, ${\left[{\tilde{\lambda }}^f\right]}_{\iota }$ is a BI of ${\rm Y}$.

(⇐) Let ${\tilde{\lambda }}^f(\wp )<{\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, for some $\wp ,\varrho \in {\rm Y}$. Then, $\iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}]$ such that ${\tilde{\lambda }}^f(\wp )<\iota \le {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Thus, it follows that $\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$ but $\wp \notin {\left[{\tilde{\lambda }}^f\right]}_{\iota }$, which is a contradiction, and hence ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Let ${\tilde{\lambda }}^f(\wp +\varrho )<{\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$ for some $\wp ,\varrho \in {\rm Y}$. Then ∃$\iota \in (0,\frac{{\kappa }^{*}-\kappa }{2}]$ such that ${\tilde{\lambda }}^f(\wp +\varrho )<\iota \le {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Thus, it follows that $\wp ,\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$ but $\wp +\varrho \notin {\left[{\tilde{\lambda }}^f\right]}_{\iota }$, which is a contradiction. Therefore, ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp ,\varrho \in {\rm Y}$. Similarly, ${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, $\forall \wp ,\varrho \in {\rm Y}$. Next, suppose that ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$ for some $\wp ,t,\varrho \in {\rm Y}$. It follows that $\wp ,\varrho \in {\left[{\tilde{\lambda }}^f\right]}_{\iota }$ but $\wp t\varrho \notin {\left[{\tilde{\lambda }}^f\right]}_{\iota }$ which is again a contradiction. Thus ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, as required. □

4. Lower Part of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI

The concept of the lower part of the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$ is defined and characterized.

Definition 4.

The $({\kappa }^{*},\kappa )$-lower part $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$ of ${\tilde{\lambda }}^f$ is defined as

$$\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp )={\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2},$$

$\forall \wp \in {\rm Y}$ and $1\ge {\kappa }^{*}>\kappa \ge 0$.

The $({\kappa }^{*},\kappa )$-lower part ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$ of the characteristic function ${\chi }_{\Omega }^f$ is defined for $\Omega \subseteq R$ as

$${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )=\left\{\begin{array}{cc}\frac{{\kappa }^{*}-\kappa }{2},\hfill & if\wp \in \Omega ;\hfill \\ 0,\hfill & if\wp \notin \Omega .\hfill \end{array}\right.$$

Definition 5.

Let ${\tilde{\pounds }}^f,{\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$. Define ${\tilde{\pounds }}^f{(\cap )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$, ${\tilde{\pounds }}^f{(\cup )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$, and ${\tilde{\pounds }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$ as follows:

$$\begin{array}{cc}\hfill \left({\tilde{\pounds }}^f{(\cap )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(\wp )& =({\tilde{\pounds }}^f\cap {\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill \left({\tilde{\pounds }}^f{(\cup )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(\wp )& =({\tilde{\pounds }}^f\cup {\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill \left({\tilde{\pounds }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(\wp )& =({\tilde{\pounds }}^f\circ {\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill \left({\tilde{\pounds }}^f{(+)}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(\wp )& =({\tilde{\pounds }}^f+{\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \end{array}$$

$\forall \wp \in {\rm Y}$ and $1\ge {\kappa }^{*}>\kappa \ge 0$.

Lemma 2.

${\tilde{\pounds }}^f,{\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$. Then,

(1) 

${\left(\underline{{{\pounds }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\kappa }^{{\kappa }^{*}}=\underline{{{\pounds }^f}_{\kappa }^{{\kappa }^{*}}}$ and $\underline{{{\pounds }^f}_{\kappa }^{{\kappa }^{*}}}\subseteq {\tilde{\pounds }}^f$;

(2) 

If ${\tilde{\pounds }}^f\subseteq {\tilde{\lambda }}^f$, and ${\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$, then ${\tilde{\pounds }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\subseteq {\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$ and ${\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\pounds }}^f\subseteq {\tilde{\lambda }}^f{\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$;

(3) 

If ${\tilde{\pounds }}^f\subseteq {\tilde{\lambda }}^f$, and ${\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$, then ${\tilde{\pounds }}^f{(+)}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\subseteq {\tilde{\lambda }}^f{(+)}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$ and $\lambda {(+)}_{\kappa }^{{\kappa }^{*}}{\tilde{\pounds }}^f\subseteq \lambda {(+)}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$;

(4) 

${\tilde{\pounds }}^f{(\cap )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f=\underline{{{\pounds }^f}_{\kappa }^{{\kappa }^{*}}}\cap \underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$;

(5) 

${\tilde{\pounds }}^f{(\cup )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f=\underline{{{\pounds }^f}_{\kappa }^{{\kappa }^{*}}}\cup \underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$;

(6) 

${\tilde{\pounds }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f=\underline{{{\pounds }^f}_{\kappa }^{{\kappa }^{*}}}\circ \underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$;

(7) 

${\tilde{\pounds }}^f{(+)}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f=\underline{{{\pounds }^f}_{\kappa }^{{\kappa }^{*}}}+\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$.

Proof. 

Straightforward. □

Lemma 3.

Let $\Sigma ,\Omega \subseteq {\rm Y}$. Then,

(1) 

${\chi }_{\Sigma }{(+)}_{\kappa }^{{\kappa }^{*}}{\chi }_{\Omega }={\left(\underline{{\chi }_{\kappa }^{{\kappa }^{*}}}\right)}_{\Sigma +\Omega }$;

(2) 

${\chi }_{\Sigma }{(\cap )}_{\kappa }^{{\kappa }^{*}}{\chi }_{\Omega }={\left(\underline{{\chi }_{\kappa }^{{\kappa }^{*}}}\right)}_{\Sigma \cap \Omega }$;

(3) 

${\chi }_{\Sigma }{(\cup )}_{\kappa }^{{\kappa }^{*}}{\chi }_{\Omega }={\left(\underline{{\chi }_{\kappa }^{{\kappa }^{*}}}\right)}_{\Sigma \cup \Omega }$;

(4) 

${\chi }_{\Sigma }{(\circ )}_{\kappa }^{{\kappa }^{*}}{\chi }_{\Omega }={\left(\underline{{\chi }_{\kappa }^{{\kappa }^{*}}}\right)}_{(\Sigma \Omega ]}$.

Proof. 

Straightforward. □

Lemma 4.

If ${\tilde{\lambda }}^f$ is the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ, then $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$ is an FBI of Υ.

Proof. 

Let $\wp ,\varrho \in {\rm Y}$ be such that $\wp \le \varrho$. Then, ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Thus, it implies ${\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, and, so, $\left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)(\wp )\ge \left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)\left(\varrho \right)$. Next suppose that $\wp ,\varrho \in {\rm Y}$. Since ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. It follows that ${\tilde{\lambda }}^f(\wp +\varrho )\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\wedge \frac{{\kappa }^{*}-\kappa }{2}$, and hence, $\left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)(\wp +\varrho )\ge \left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)(\wp )\wedge \left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)\left(\varrho \right)$. Similarly, $\left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)(\wp \varrho )\ge \left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)(\wp )\wedge \left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)\left(\varrho \right)$, $\forall \wp ,\varrho \in {\rm Y}$. Let $\wp ,t,\varrho \in {\rm Y}$; we have ${\tilde{\lambda }}^f(\wp t\varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Then ${\tilde{\lambda }}^f(\wp \varrho )\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$, and so $\left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)(\wp t\varrho )\ge \left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)(\wp )\wedge \left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)\left(\varrho \right)$. Therefore, $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$ is an FBI of ${\rm Y}$. □

Lemma 5.

Let $(\varnothing \ne )\Omega \subseteq S$. Then, Ω is a BI of Υ⇔${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$, the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ.

Proof. 

Let $\wp ,\varrho \in {\rm Y}$ and $\iota ,\theta \in (0,1]$ be such that ${\wp }_{\iota }\in {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$ and ${\varrho }_{\theta }\in {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. Then, ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )\ge \iota >0$ and ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)\ge \theta >0$. Therefore, $\wp ,\varrho \in \Omega$. As $\Omega$ is a BI of ${\rm Y}$, $\wp +\varrho \in \Omega$. Thus ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp +\varrho )=\frac{{\kappa }^{*}-\kappa }{2}$. If $\omega \wedge \theta \le \frac{{\kappa }^{*}-\kappa }{2}$, then ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp +\varrho )\ge \omega$, so we have ${(\wp +\varrho )}_{\iota \wedge \theta }\in {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. If $\iota \wedge \theta >\frac{{\kappa }^{*}-\kappa }{2}$, then ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp +\varrho )+\iota \wedge \theta >\frac{{\kappa }^{*}-\kappa }{2}+\frac{{\kappa }^{*}-\kappa }{2}={\kappa }^{*}-\kappa$. So ${(\wp +\varrho )}_{\iota \wedge \theta }({\kappa }^{*},q_{\kappa }){\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. Similarly, ${\wp }_{\iota }\in {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$ and ${\varrho }_{\theta }\in {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$ imply ${(\wp \varrho )}_{\iota \wedge \theta }({\kappa }^{*},q_{\kappa }){\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. Therefore, ${(\wp +\varrho )}_{\iota \wedge \theta }\in \vee ({\kappa }^{*},q_{\kappa }){\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. Let $\wp ,\varrho ,t\in {\rm Y}$ and $\iota \in (0,1]$ be such that ${\wp }_{\iota },{\varrho }_{\theta }\in {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. Then, $\wp ,\varrho \in \Omega$, ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )\ge \iota ,{\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )\ge \theta$. Since $\Omega$ is a BI of ${\rm Y}$, we have $\wp t\varrho \in \Omega$. Thus, ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp t\varrho )\ge \frac{{\kappa }^{*}-\kappa }{2}$. If $\iota \wedge \theta \le \frac{{\kappa }^{*}-\kappa }{2}$, then ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp t\varrho )\ge \iota \wedge \theta$. Therefore ${(\wp t\varrho )}_{\iota \wedge \theta }\in {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. Again, if $\iota \wedge \theta >\frac{{\kappa }^{*}-\kappa }{2}$, then ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp t\varrho )+\iota \wedge \theta >\frac{{\kappa }^{*}-\kappa }{2}+\frac{{\kappa }^{*}-\kappa }{2}={\kappa }^{*}-\kappa$. So ${(\wp t\varrho )}_{\iota \wedge \theta }({\kappa }^{*},q_{\kappa }){\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$. Thus, ${(\wp t\varrho )}_{\iota \wedge \theta }\in \vee ({\kappa }^{*},q_{\kappa }){\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$, as required.

Let $\wp \in {\rm Y}$ and $\varrho \in \Omega$ such that $\wp \le \varrho$. Then ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)=\frac{{\kappa }^{*}-\kappa }{2}$. Since ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$, and $\wp \le \varrho$, we have ${\left(\underline{f_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )\ge {\left(\underline{f_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}=\frac{{\kappa }^{*}-\kappa }{2}$. Thus, ${\left(\underline{f_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )=\frac{{\kappa }^{*}-\kappa }{2}$ and so $\wp \in \Omega$. Let $\wp ,\varrho \in \Omega$. Then, ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )=\frac{{\kappa }^{*}-\kappa }{2}$ and ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)=\frac{{\kappa }^{*}-\kappa }{2}$. Since ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$, we have ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp +\varrho )\ge {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )\wedge {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}=\frac{{\kappa }^{*}-\kappa }{2}$ and also ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp \varrho )\ge {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )\wedge {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}=\frac{{\kappa }^{*}-\kappa }{2}$. Thus, it implies ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp +\varrho )=\frac{{\kappa }^{*}-\kappa }{2}$ and ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp \varrho )=\frac{{\kappa }^{*}-\kappa }{2}$. Therefore, $\wp +\varrho ,\wp \varrho \in \Omega$. Let $\wp ,\varrho \in \Omega$ and $t\in {\rm Y}$. Then ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )=\frac{{\kappa }^{*}-\kappa }{2}$ and ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)=\frac{{\kappa }^{*}-\kappa }{2}$. Now, ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp t\varrho )\ge {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp )\wedge {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}=\frac{{\kappa }^{*}-\kappa }{2}$. Hence ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\Omega }(\wp t\varrho )=\frac{{\kappa }^{*}-\kappa }{2}$. Therefore $\wp t\varrho \in \Omega$. Hence, $\Omega$ is a BI of ${\rm Y}$. □

Theorem 6.

Let ${\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$. Then ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ⇔

(1)

${\tilde{\lambda }}^f{(+)}_k^{k^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$,

(2)

${\tilde{\lambda }}^f{(\circ )}_k^{k^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$,

(3)

${\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{1}^f{(\circ )}_k^{k^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$, and

(4)

$(\forall \wp ,\varrho \in {\rm Y})\wp \le \varrho$$\Rightarrow {\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$.

Proof. 

(⇒) Suppose that ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$. If ${\tilde{\lambda }}^f{(+)}_k^{k^{*}}{\tilde{\lambda }}^f=0$, then ${\tilde{\lambda }}^f{(+)}_k^{k^{*}}{\tilde{\lambda }}^f⪯{\tilde{\lambda }}^f$. Suppose that ${\tilde{\lambda }}^f{(+)}_k^{k^{*}}{\tilde{\lambda }}^f\ne 0$. Then, we have

$$\begin{array}{cc}\hfill \left({\tilde{\lambda }}^f{(+)}_k^{k^{*}}{\tilde{\lambda }}^f\right)(\wp )& =({\tilde{\lambda }}^f+{\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\underset{\wp \le \nu +\tau }{\bigvee }\{{\tilde{\lambda }}^f\left(\nu \right)\wedge {\tilde{\lambda }}^f\left(\tau \right)\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \le \underset{\wp \le \nu +\tau }{\bigvee }{\tilde{\lambda }}^f(\nu +\tau )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & ={\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp ).\hfill \end{array}$$

Thus, ${\tilde{\lambda }}^f{(+)}_k^{k^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$. Similarly, ${\tilde{\lambda }}^f{(\circ )}_k^{k^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$. Again, $\left({\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{1}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(\wp )=0$, then ${\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{1}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$. Suppose that $\left({\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{1}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(\wp )\ne 0$. Then, we have

$$\begin{array}{cc}\hfill & \left({\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{1}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(\wp )\hfill \\ \hfill & =({\tilde{\lambda }}^f\circ 1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\underset{\wp \le yz}{\bigvee }\left\{{\tilde{\lambda }}^f\left(y\right)\wedge \left\{\left\{\underset{z\le \nu \tau }{\bigvee }\{1^f\left(\nu \right)\wedge {\tilde{\lambda }}^f\left(\tau \right)\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\right\}\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\underset{\wp \le yz}{\bigvee }\underset{z\le \nu \tau }{\bigvee }\left\{{\tilde{\lambda }}^f\left(y\right)\wedge {\tilde{\lambda }}^f\left(\tau \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \le \underset{\wp \le yz}{\bigvee }\underset{z\le \nu \tau }{\bigvee }{\tilde{\lambda }}^f\left(y\nu \tau \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \le {\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp ).\hfill \end{array}$$

Therefore, ${\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{1}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$.

(⇐) Let $\wp ,\varrho \in {\rm Y}$. Then, by hypothesis, we have

$$\begin{array}{cc}\hfill {\tilde{\lambda }}^f(\wp +\varrho )& \ge \underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp +\varrho )\hfill \\ \hfill & \ge \left({\tilde{\lambda }}^f{(+)}_k^{k^{*}}{\tilde{\lambda }}^f\right)(\wp +\varrho )\hfill \\ \hfill & =\left\{\underset{\wp +\varrho \le \nu +\tau }{\bigvee }\{{\tilde{\lambda }}^f\left(\nu \right)\wedge {\tilde{\lambda }}^f\left(\tau \right)\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}.\hfill \end{array}$$

Similarly, by hypothesis, ${\tilde{\lambda }}^f(\wp \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$.

We also have

$$\begin{array}{cc}\hfill {\tilde{\lambda }}^f(\wp ts)& \ge \underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp ts)\hfill \\ \hfill & =({\tilde{\lambda }}^f\circ 1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f)(\wp ts)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\left\{\underset{\wp t\varrho \le pq}{\bigvee }{\tilde{\lambda }}^f\left(p\right)\wedge \left(1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)\left(q\right)\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge {\tilde{\lambda }}^f(\wp )\wedge \left(1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)\left(t\varrho \right)\hfill \\ \hfill & ={\tilde{\lambda }}^f(\wp )\wedge \left\{\underset{(u,v)\in A_{rs}}{\bigvee }{1}^f\left(u\right)\wedge {\tilde{\lambda }}^f\left(v\right)\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge {\tilde{\lambda }}^f(\wp )\wedge 1^f\left(t\right)\wedge {\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & ={\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2},\hfill \end{array}$$

as required. □

Theorem 7.

The following statements are equivalent in Υ:

(1)

Υ is regular.

(1)

${{\lambda }^f}_{\kappa }^{{\kappa }^{*}}⪯{\tilde{\lambda }}^f\circ 1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$ for any $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ.

Proof. 

Assume that ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$. If $\wp \in {\rm Y}$, then, as ${\rm Y}$ is regular, ∃$t\in {\rm Y}$ such that $\wp \le \wp t\wp$. Now, we have

$$\begin{array}{cc}\hfill ({\tilde{\lambda }}^f\circ 1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f)(\wp )& =({\tilde{\lambda }}^f\circ 1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\left\{\underset{\wp \le pq}{\bigvee }{\tilde{\lambda }}^f\left(p\right)\wedge \left(1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)\left(q\right)\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge {\tilde{\lambda }}^f(\wp )\wedge \left(1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f\right)(t\wp )\hfill \\ \hfill & ={\tilde{\lambda }}^f(\wp )\wedge \left\{\underset{t\wp \le uv}{\bigvee }{1}^f\left(u\right)\wedge {\tilde{\lambda }}^f\left(v\right)\right\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge {\tilde{\lambda }}^f(\wp )\wedge 1^f\left(t\right)\wedge {\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & ={\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}.\hfill \end{array}$$

Thus, ${{\lambda }^f}_{\kappa }^{{\kappa }^{*}}⪯{\lambda }^f\circ 1^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\lambda }^f$.

$\left(2\right)\Rightarrow \left(1\right)$. Let B be a BI of ${\rm Y}$. Then, by Lemma 5, ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_B$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$. Thus, by hypothesis, we have

$${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_B\subseteq {\chi }_B{(\circ )}_k^{k^{*}}{\chi }_I{(\circ )}_k^{k^{*}}{\chi }_B={\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{\left(BIB\right]}\subseteq {\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_{(\sum BIB]}.$$

So $B\subseteq (\sum BRB]$. Since B is BI, so $(\sum BRB]\subseteq B$. Thus $B=(\sum BRB]$. Hence, by ([9] Lemma 2.2), ${\rm Y}$ is regular. □

Theorem 8.

The following statements are equivalent in Υ:

(1)

Υ is regular and intra-regular.

(2)

${{\lambda }^f}_{\kappa }^{{\kappa }^{*}}={\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$ for any $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ.

Proof. 

(⇒) Suppose that ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$. As ${\rm Y}$ is regular and intra-regular, $a\le axa$ and $a\le y{a}^{2}z$. Therefore, $a\le \left(axya\right)\left(ayxa\right)$. We have

$$\begin{array}{cc}\hfill \left({\tilde{\lambda }}^f{(\circ )}_k^{k^{*}}{\tilde{\lambda }}^f\right)(\wp )& =({\tilde{\lambda }}^f+{\tilde{\lambda }}^f)(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\underset{\wp \le pq}{\bigvee }\{{\tilde{\lambda }}^f\left(p\right)\wedge {\tilde{\lambda }}^f\left(q\right)\}\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge {\tilde{\lambda }}^f\left(axya\right)\wedge {\tilde{\lambda }}^f\left(ayxa\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & \ge {\lambda }^f\left(a\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\hfill \\ \hfill & =\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(a\right).\hfill \end{array}$$

Thus $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}⪯{\tilde{\lambda }}^f{(\circ )}_k^{k^{*}}{\tilde{\lambda }}^f$. Since ${\tilde{\lambda }}^f$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI, so ${\tilde{\lambda }}^f{(\circ )}_k^{k^{*}}{\tilde{\lambda }}^f⪯\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}$. Hence ${{\lambda }^f}_{\kappa }^{{\kappa }^{*}}={\tilde{\lambda }}^f{(\circ )}_{\kappa }^{{\kappa }^{*}}{\tilde{\lambda }}^f$.

$\left(2\right)\Rightarrow \left(1\right)$. Let B be a BI of ${\rm Y}$. Then, by Lemma 5, ${\left(\underline{{{\chi }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_B$ is an $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$. Thus by hypothesis, we have

$${\left(\underline{{{\chi }^f}_k^{k^{*}}}\right)}_B\subseteq {{\chi }^f}_B{(\circ )}_k^{k^{*}}{{\chi }^f}_B={\left(\underline{{{\chi }^f}_k^{k^{*}}}\right)}_{\left(BB\right]}\subseteq {\left(\underline{{{\chi }^f}_k^{k^{*}}}\right)}_{(\sum BB]}.$$

Therefore, $B\subseteq (\sum BB]$. Since B is BI, so $(\sum BB]\subseteq B$. Thus $B=(\sum BB]$. Hence, by ([9], Theorem 3.12) ${\rm Y}$ is regular. □

Definition 6.

Let $t\in {\rm Y}$ and ${\tilde{\lambda }}^f\in \mathcal{F}({\rm Y})$. Define the following ${\mathcal{I}}_t$ of Υ as

$${\mathcal{I}}_t=\{\wp \in {\rm Y}|{\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\}.$$

Lemma 6.

Let ${\tilde{\lambda }}^f$ be the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of Υ. Then ${\mathcal{I}}_t$ ($\forall t\in {\rm Y}$) is the BI of Υ.

Proof. 

Let $t\in {\rm Y}$. As $t\in {\mathcal{I}}_t$, we have ${\mathcal{I}}_t\ne \varnothing$. Take any $\wp ,\varrho \in {\mathcal{I}}_t$. Then, ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(t\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge {\tilde{\lambda }}^f\left(t\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Since ${\tilde{\lambda }}^f$ is the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$, ${\tilde{\lambda }}^f(\wp +\varrho )\ge {\tilde{\lambda }}^f\left(r\right)\wedge {\tilde{\lambda }}^f\left(y\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge \frac{{\kappa }^{*}-\kappa }{2}$, so $\wp +\varrho \in {\mathcal{I}}_t$. By a similar argument, $\wp \varrho \in {\mathcal{I}}_t$.

Next, take any $\tau \in {\rm Y}$ and $\wp ,\varrho \in {\mathcal{I}}_t$. Then ${\tilde{\lambda }}^f(\wp )\ge {\tilde{\lambda }}^f\left(t\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$ and ${\tilde{\lambda }}^f\left(\varrho \right)\ge {\tilde{\lambda }}^f\left(t\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. By hypothesis, ${\tilde{\lambda }}^f(\wp \tau \varrho )\ge {\tilde{\lambda }}^f(\wp )\wedge {\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Therefore, ${\tilde{\lambda }}^f(\wp \tau \varrho )\ge {\tilde{\lambda }}^f\left(a\right)\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Thus $\wp \tau \varrho \in {\mathcal{I}}_t$. Additionally, for any $\wp \in {\rm Y}$ and $\varrho \in {\mathcal{I}}_t$ such that $\wp \le \varrho$, we have $\wp \in {\mathcal{I}}_t$. Hence, ${\mathcal{I}}_t$ is a BI of ${\rm Y}$. □

Definition 7.

An ordered semiring Υ is called $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-simple if every $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI is constant. That is, $\forall \wp ,\varrho \in {\rm Y}$; we have $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp )=\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(\varrho \right)$, for each $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI ${\tilde{\lambda }}^f$ of Υ.

Theorem 9.

The ordered semiring Υ is bi-simple ⇔ it is $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-simple.

Proof. 

(⇒) Let ${\tilde{\lambda }}^f$ be the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$ and $\wp ,\varrho \in {\rm Y}$. By Lemma 6, ${\mathcal{I}}_{\wp }$ is an left ideal of ${\rm Y}$. As ${\rm Y}$ is bi-simple, ${\mathcal{I}}_{\wp }=R$. So $\varrho \in {\rm Y}$. Thus, ${\tilde{\lambda }}^f\left(\varrho \right)\ge {\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}$. Therefore, $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(\varrho \right)={\tilde{\lambda }}^f\left(\varrho \right)\wedge \frac{{\kappa }^{*}-\kappa }{2}\ge {\tilde{\lambda }}^f(\wp )\wedge \frac{{\kappa }^{*}-\kappa }{2}=\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp )$. Similarly, $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(\varrho \right)\le \underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp )$. Thus, $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp )=\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(\varrho \right)$, as required.

(⇐) Assume that I is the proper BI of ${\rm Y}$. By Lemma 5, ${\left(\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\right)}_I$ is the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-FBI of ${\rm Y}$. As ${\rm Y}$ is $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-simple, $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}(\wp )=\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(\varrho \right)$, $\forall \wp ,\varrho \in {\rm Y}$. Let $p\in I$ and $q\in {\rm Y}$. Then, $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(p\right)=\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(q\right)$. As $p\in I$, we have $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(p\right)=\frac{{\kappa }^{*}-\kappa }{2}$. Therefore, $\underline{{{\lambda }^f}_{\kappa }^{{\kappa }^{*}}}\left(q\right)=\frac{{\kappa }^{*}-\kappa }{2}$, which implies that $q\in I$. Thus, $I={\rm Y}$, and hence ${\rm Y}$ is bi-simple. □

5. Conclusions

The notion of the $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideal, which is broader than the existing terminology, was introduced in this work. A condition is provided under which fuzzy bi-ideals and $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals coincide. Bi-ideals and $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals connections were taken into consideration. Regular and intra-regular ordered semirings were described in terms of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals and their $(\kappa *,\kappa )$-lower parts. Moreover, $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-simple ordered semirings were defined and characterized.