Approximations of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 67.1136 16.7875" width="67.1136pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.017,0,0,-0.017,14.998,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,21.787,0)"><use xlink:href="#g117-173"/></g><g transform="matrix(.017,0,0,-0.017,35.641,0)"><path id="g117-64" d="M574 548H512L295 69H293L76 548H14L263 0H325L574 548Z"/></g><g transform="matrix(.017,0,0,-0.017,45.714,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g><g transform="matrix(.012,0,0,-0.012,54.054,4.134)"><path id="g50-108" d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z"/></g><g transform="matrix(.017,0,0,-0.017,60.848,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Fuzzy Hyperideals in Ordered LA-Semihypergroups

Abughazalah, Nabilah; Yaqoob, Naveed; Ashraf, Samavia

doi:https://doi.org/10.1155/2021/9917947

Mathematical Problems in Engineering

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Volume 2021 | Article ID 9917947 | https://doi.org/10.1155/2021/9917947

Approximations of -Fuzzy Hyperideals in Ordered LA-Semihypergroups

Nabilah Abughazalah,¹Naveed Yaqoob,²and Samavia Ashraf²

Academic Editor: G. Muhiuddin

Received04 Apr 2021

Accepted24 Apr 2021

Published07 May 2021

Abstract

This paper concerns the relationship between rough sets, -fuzzy sets, and the LA-semihypergroups. We proved that the lower approximation (-approx) and the upper approximation (-approx) of different hyperideals of LA-semihypergroups become again hyperideal and also provided some examples. We also provided a result which shows that, if is not a hyperideal of an LA-semihypergroup , then its lower approximation is also not a hyperideal.

1. Introduction

Hyperstructures represent a natural extension of classical algebraic structures. Algebraic hyperstructure theory depends on hyperoperation and their properties. One of these theories is the semihypergroup theory. Semihypergroup theory was first introduced by French Mathematician Marty in 1934 [1]. The concept of semihypergroup is the generalization of the concept of semigroup. In semigroup, the composition of two elements is an element, while in a semihypergroup the composition of two elements is a nonempty set. In semihypergroup, closure and associative law hold. Many authors studied different aspects of semihypergroup (see [2–4]). Recently, Hila and Dine [5] introduced the notion of LA-sm-hyp-group (LA-semihypergroup), and then, Yaqoob, Corsini, and Yousafzai [6] extended the work of Hila and Dine and characterized intraregular left almost semihypergroups by their HP-ideals (hyperideals) using pure left identity. The concept of ordered semihypergroup was studied by Heidari and Davvaz [7] where they used a binary relation on semihypergroup such that the binary relation is a partial order and the structure is known as ordered semihypergroup [8].

The concept of fuzzy sets was first introduced by Zadeh [9] in 1964. Fuzzy set is a set having degrees of membership between 1 and 0. Fuzzy set theory is a classical application of set theory. Applications of fuzzy sets in some algebraic structures and other structures can be seen in [10–13]. Zdzisław Pawlak [14] introduced the notion of rough sets, in the year 1982. Rough set theory constitutes a sound basis for knowledge discovery databases. It offers mathematical tools to discover pattern hidden in data. It can be used for feature selection, feature extraction, decision rule generation, and paper extraction. Applications of rough sets in some algebraic structures and algebraic hyperstructures can be seen in [15–36].

2. Preliminaries and Notations

Let be a nonempty set. Then, the map is called HP-operation (hyperoperation) or join operation on the set , where denotes the set of all nonempty subsets of . A hypergroupoid is a set together with a (binary) HP-operation. For any nonempty subsets of of , we denote

Instead of and , we write and , respectively.

Definition 1. (see [5, 6]). A hypergroupoid is called an LA-sm-hyp-group if for every , we have .
The law is called a left invertive law.

Definition 2. (see [8]). Let be a nonempty set and be a partially ordered relation on . The triplet is called a partially ordered LA-sm-hyp-group if the following conditions are satisfied:(i)is an LA-sm-hyp-group(ii) is a partially ordered set(iii)For every , implies and , where means that for , there exist such that

Definition 3. (see [8]). If is a partially ordered LA-sm-hyp-group and , then is the subset of defined as follows:If and is a nonempty subset of , then .

Definition 4. (see [8]). A nonempty subset of a partially ordered LA-sm-hyp-group is called a LAS-sm-hyp-group (LA-subsemihypergroup) of if .

Definition 5. (see [8]). A nonempty subset of a partially ordered LA-sm-hyp-group is called a right (resp., left) HP-ideal (hyperideal) of if(i)(ii)For every and , implies If is both right HP-ideal and left HP-ideal of LA-sm-hyp-group , then is called a HP-ideal (or two-sided HP-ideal) of LA-sm-hyp-group .

Definition 6. (see [8]). A nonempty subset of a partially ordered LA-sm-hyp-group is called an interior HP-ideal of , if(i)(ii)For every and , implies

Definition 7. (see [8]). A nonempty subset of a partially ordered LA-sm-hyp-group is called a generalized bi-HP-ideal of , if(i)(ii)For every and , implies Definitions on -fuzzy HP-ideals in LA-sm-hyp-group can be found in [10].

3. Rough -Fuzzy LAS-sm-hyp-Groups

Definition 8. Let be a nonempty set and be a binary relation on . By , we mean the power set of . For all , we define and bywhere . and are called the -approx and the -approx operations, respectively.

Definition 9. Let be a PH-order relation (pseudohyperorder relation [36]) on a partially ordered LA-sm-hyp-group . Let be a fuzzy subset of LA-sm-hyp-group . For every in , we define --approx and the --approx of by

Theorem 1. Let be a PH-order relation on a partially ordered LA-sm-hyp-group . Let be an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group . Then, is an LAS-sm-hyp-group of LA-sm-hyp-group .

Proof. Let be an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group and be a PH-order relation on . We have to show that is an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group .(i)For each in and , we have . Now Thus, we get .(ii)Now, let in . Let Thus, we get . Hence is an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group .

Theorem 2. Let be a complete and symmetric PH-order relation on a partially ordered LA-sm-hyp-group . Let be an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group . Then, is an LAS-sm-hyp-group of LA-sm-hyp-group .

Proof. Let be an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group and be a complete and symmetric PH-order relation on . We have to show that is an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group .(i)For each in and , we have . Now Thus, we get (ii)Now let in . Let Thus, we get . Hence is an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group .

4. Rough -Fuzzy HP-Ideals

Theorem 3. Let be a PH-order on a partially ordered LA-sm-hyp-group . Let be an -fuzzy left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group . Then, is a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group .

Proof. Let be an -fuzzy bi-HP-ideal of LA-sm-hyp-group and be a PH-order relation on . We have to show that is an -fuzzy bi-HP-ideal of LA-sm-hyp-group .(i)For each in and , we have . Now Thus, we get .(ii)Now, let in . Let Thus, we get .(iii)Now, let in . Let Thus, we get . Hence, is an -fuzzy bi-HP-ideal of LA-sm-hyp-group . The other cases can be proved in a similar way.The following example shows that the converse of Theorem 3 is not true in general.

Example 1. Consider a set with the following hyperoperation“” defined in Table 1.
The order “” is . Then, becomes a partially ordered LA-sm-hyp-group. Let us define a fuzzy subset of LA-sm-hyp-group by and take and . Then, it is clear that is not an -fuzzy HP-ideal of LA-sm-hyp-group because for , in but and . Now define a PH-order relation on byBy the routine calculations, we getThus, we get . One can see that is an -fuzzy HP-ideal of LA-sm-hyp-group . Similarly, we can find the example for other cases.

Theorem 4. Let be a complete and symmetric PH-order relation on a partially ordered LA-sm-hyp-group . Let be an -fuzzy left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group . Then, is a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group .

Proof. Let be an -fuzzy bi-HP-ideal of LA-sm-hyp-group and be a complete and symmetric PH-order relation on . We have to show that is an -fuzzy bi-HP-ideal of LA-sm-hyp-group .(i)For each in and , we have . Now Thus, we get (ii)Now, let in . Let Thus, we get .(iii)Now, let in . Let Thus, we get . Hence is an -fuzzy bi-HP-ideal of LA-sm-hyp-group . The other cases can be proved in a similar way.The following theorem shows that, if is not a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group , then the -approx is also not a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group .

Theorem 5. Let be a complete and symmetric PH-order relation on a partially ordered LA-sm-hyp-group and is not an -fuzzy left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group . Then, is also not a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group .

Proof. Let be a complete and symmetric PH-order relation on . Suppose that is not an -fuzzy interior HP-ideal of LA-sm-hyp-group , that is,(i)If implies that (ii).(iii)Let , then . Now, Thus, we get .(iv)Now, let in . Let Thus, we get . Hence is not an interior HP-ideal of LA-sm-hyp-group . The other cases can be proved in a similar way.

5. Rough -Fuzzy Quasi-HP-Ideals

In this section, we first provide some results on -fuzzy quasi-HP-ideals and rough -fuzzy quasi-HP-ideals. For a partially ordered LA-sm-hyp-group , the fuzzy subset of LA-sm-hyp-group is defined as follows:

Definition 10. Let be the fuzzy subsets of a set . The product of the fuzzy subset and is defined as .

Definition 11. Let be a nonempty subset of an ordered LA-sm-hyp-group Then, the characteristic function of is a fuzzy subset of LA-sm-hyp-group , defined as follows:

Definition 12. Let be a fuzzy subset of LA-sm-hyp-group . Then, is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group if and only if(i), for all in (ii), for all in

Example 2. Consider a set with the following hyperoperation“” defined in Table 2.
The order “” is . Then, becomes a partially ordered LA-sm-hyp-group. Now define a fuzzy subset of LA-sm-hyp-group as and . If we take and , then is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .

Theorem 6. Let be an -fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group . Then the set is a quasi-HP-ideal of LA-sm-hyp-group , where .

Proof. Let be an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .(i)Let in , and in . Then, . As is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group and , so This implies that in .(ii)Now, we have to show that . Let in . Thus, in and in . So in and in for some in and in . Thus, and . SinceSimilarly, . Thus,Thus, in . Hence, is a quasi-HP-ideal of LA-sm-hyp-group .

Corollary 1. Let be an -fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group . Then the set is a quasi-HP-ideal of LA-sm-hyp-group .

Corollary 2. Let be an -fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group . Then, the set is a quasi-HP-ideal of LA-sm-hyp-group .

Lemma 1. A nonempty subset of a partially ordered LA-sm-hyp-group is a quasi-HP-ideal of LA-sm-hyp-group if and only if the characteristic function of is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .

Theorem 7. Every -fuzzy left (right) HP-ideal of LA-sm-hyp-group is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .

Proof. Let be an -fuzzy left HP-ideal of LA-sm-hyp-group and in . ThenThis implies thatHence, . Thus, is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group . The other case can be proved in a similar way.

Theorem 8. Every -fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group is an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group .

Proof. Suppose is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group . Let in . Now for every in , we haveSo , for all in . Thus, is an -fuzzy LAS-sm-hyp-group of LA-sm-hyp-group .

Theorem 9. Let be a PH-order on a partially ordered LA-sm-hyp-group . Let be an -fuzzy quasi-HP-ideal of LA-sm-hyp-group . Then, is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .

Proof. Let be an -fuzzy quasi-HP-ideal of LA-sm-hyp-group and be a PH-order relation on . We have to show that is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .(i)For each in and , we have . Now Thus, we get .(ii)Now let in . Let Thus , is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .

Theorem 10. Let be a complete and symmetric PH-order on a partially ordered LA-sm-hyp-group . Let be an -fuzzy quasi-HP-ideal of LA-sm-hyp-group . Then, is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .

Proof. Let be an -fuzzy quasi-HP-ideal of LA-sm-hyp-group and be a complete and symmetric PH-order relation on . We have to show that is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .(i)For each in and , we have . Now Thus, we get .(ii)Now let in . Let Thus, is an -fuzzy quasi-HP-ideal of LA-sm-hyp-group .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

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Copyright © 2021 Nabilah Abughazalah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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