[Retracted] Interval-Valued <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 13.7215 8.14084" width="13.7215pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g></svg>-</span>Polar Fuzzy Positive Implicative Ideals in <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M2" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 34.8213 11.6718" width="34.8213pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.017,0,0,-0.017,10.658,0)"><path id="g113-68" d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z"/></g><g transform="matrix(.017,0,0,-0.017,21.604,0)"><path id="g113-76" d="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"/></g></svg>-</span>Algebras

Muhiuddin, G.; Al-Kadi, D.; Mahboob, A.; Albjedi, Amjad

doi:https://doi.org/10.1155/2021/1042091

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article Retraction

!

This article has been Retracted. To view the article details, please click the ‘Retraction’ tab above.

Special Issue

Fuzzy Applications in Engineering and Risk Management

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 1042091 | https://doi.org/10.1155/2021/1042091

[Retracted] Interval-Valued -Polar Fuzzy Positive Implicative Ideals in -Algebras

G. Muhiuddin,¹D. Al-Kadi,²A. Mahboob,³and Amjad Albjedi¹

Academic Editor: Naeem Jan

Received14 Apr 2021

Revised05 May 2021

Accepted27 May 2021

Published30 Jun 2021

Abstract

In this paper, the notion of interval-valued -polar fuzzy positive implicative ideals in BCK-algebras is presented. Then, the relationships between interval-valued -polar fuzzy positive implicative ideals and interval-valued -polar fuzzy ideals are investigated. After that, the concepts of interval-valued -polar -fuzzy positive implicative ideals and interval-valued -polar -fuzzy ideals are defined and some equivalent conditions are provided. Furthermore, we show that interval-valued -polar -fuzzy positive implicative ideals are interval-valued -polar -fuzzy ideals, but the converse need not be true in general and an example is given in this aim.

1. Introduction

As an extension of fuzzy sets, Zadeh [1] defined fuzzy sets with an interval-valued membership function proposing the concept interval-valued fuzzy sets. This concept has been studied from various points of view in different algebraic structures as BCK-algebras and some of its generalization (see, for example, [2–7]), groups (see, for example, [8–10]), and rings (see, for example, [11–13]). Jun [14] studied interval-valued fuzzy ideals in BCI-algebras. Zhan et al. [15, 16] studied -fuzzy ideals of BCI-algebras. The concept of “quasi-coincidence” of an interval-valued fuzzy point together with “belongingness” within an interval-valued fuzzy set were used in the studies made by Ma et al. in [17, 18], where they discussed properties of some types of -interval-valued fuzzy ideals of BCI-algebras. Also, in [19–24], some more general ideas on bipolar fuzzy sets’ related ideals were considered.

The -polar fuzzy set, an extension of the bipolar fuzzy set, was introduced by Chen et al. [25] in 2014. When more than one agreement has to work with the -polar fuzzy model, it offers the system more accuracy, flexibility, and compatibility. The investigation of -polar fuzzy algebraic structures started with the idea of textitm-pF lie subalgebras proposed by Akram et al. [26]. Following that, Akram and Farooq [27] in lie subalgebras introduced the theory of m-pF lie ideal. A concept proposed by [28] for the m-pF subgroups. The notions of m-pF ideals and m-pF commutative ideals on BCK/BCI-algebras were introduced by Al-Masarwah and Ahmad [29]. The concepts of -fuzzy ideals and -fuzzy commutative ideals have been considered by Al-Masarwah and Ahmad in [30]. In [31], Muhiuddin et al. introduced and characterized the notion of -polar -fuzzy q-ideal in -algebras. Takallo et al. [32] proposed the notion of -fuzzy p-ideal in -algebras and studied related properties of -polar -fuzzy ideals and -polar -fuzzy p-ideals in -algebras. Recently, by generalizing the concept of -polar fuzzy positive implicative ideals of -algebras, Al-Masarwah et al. [33] introduced the notions of -fuzzy positive implicative ideals and -fuzzy positive implicative ideals in -algebras. Also, different kinds of concepts, related to this study, were investigated in various ways (see, for example, [34–40]).

In this paper, the notion of interval-valued -polar fuzzy positive implicative ideals in BCK-algebras is presented. We prove that every interval-valued -polar fuzzy positive implicative ideal of BCK-algebras is an interval-valued -polar fuzzy ideal but the converse statement is not true in general and an example is given in this aim. Moreover, the concepts of interval-valued -polar -fuzzy positive implicative ideals and interval-valued -polar -fuzzy ideals are defined and some equivalent conditions are provided. Furthermore, we show that interval-valued -polar -fuzzy positive implicative ideals are interval-valued -polar -fuzzy ideals, but converse need not be true in general and an example is given in this aim.

2. Preliminaries

An algebra of type (2, 0) is called a -algebra if, for all ,(i).(ii).(iii).(iv).(v) and imply , where can be presented by . Every -algebra satisfies the following axioms, for all :(1).(2).

A subset of is called a subalgebra if, for all , and is called an of if and, for all implies .

Definition 1. (see [33]). A subset of is called a positive implicative ideal of if :(i)(ii) and imply The interval number is the interval , where , and is the set of all interval numbers. For the interval numbers , , we describe(a)(b)(c) and (d) and A mapping is called an interval-valued fuzzy set of , where , for all , where and are fuzzy sets of with , for all .

Definition 2. A mapping is called an interval-valued m-polar fuzzy set (briefly, IVPF set) of and is defined aswhere is the projection mapping for . That is,for all , where and are fuzzy sets of with , for all and .
The projection map is order preserving and vice versa, i.e.,

Definition 3. (see [40]). An set of is called an ideal of if, for any ,(1)(2)That is,(1)(2),

Definition 4. (see [40]). The set , where is an set of is called the level cut subset of , .

Lemma 1. (see [40]). Every ideal of satisfies the following assertion, :

3. Interval-Valued -Polar Fuzzy Positive Implicative Ideals

Definition 5. An set of is called an ideal of if, for any ,(1)(2)That is,(1)(2),

Example 1. Consider a BCK-algebra with the Cayley table (Table 1).
Let be an set defined asIt is straightforward to check that is an ideal of .

Theorem 1. Every ideal of is an ideal of .

Proof. Let be an ideal of . Then, condition (1) of Definition 5 holds. By assumption, we havePut , soHence, is an ideal of .
As shown by the following example, the converse of the preceding Theorem 1 is not valid in general.

Example 2. Consider a BCK-algebra with the Cayley table (Table 2).
Now, define an set as follows:It is straightforward to check that is an ideal of , but it is not an ideal of since

Theorem 2. An set of is an ideal of ; it is an ideal of and .

Proof. () Suppose is an ideal of . By Theorem 1, is an ideal of . By assumption, we haveNow, replace by ; then,.
() Suppose that is an ideal of . Then, condition (1) of Definition 5 holds. As , so by Lemma 1, we haveNow, by assumption,Hence, is an ideal of .

Theorem 3. An set of is an ideal of is a positive implicative ideal of , .

Proof. () Suppose that is an ideal of . Let be such that . Then, , and we have . Let be such that and . Then, and . It follows from Definition 5 (2) thatThus, . Hence, is a positive implicative ideal of .
() Assume that is a positive implicative ideal of , . If for some , then such that . It implies that , a contradiction. Thus, , . Again, if , for some ; then,for some . It follows that and , but . This is a contradiction. Thus, . Hence, is an ideal of .

4. -Polar -Fuzzy Positive Implicative Ideals

An set of of the formis called an point, denoted as , with support and value . An point (1)Belongs to , written as , if , i.e., , (2)Is quasi-coincidence with , written as , if , i.e., , , where and in which and

Assume . We write(1) if does not hold(2) (resp. ) if or (resp. and )

Definition 6. An set of a -algebra is called an ideal of if(1)(2) and , , and

Example 3. Consider a BCK-algebra with the Cayley table (Table 3).
Define an set asChoose . Then, with direct computation, we find that is an ideal of .

Theorem 4. An set of is an ideal of .(1)(2),

Proof. () Suppose, on the contrary, that ; then, for some and . This implies that , but , a contradiction. Thus, .
Again, suppose the contrary that . Then, for some . This implies that and , but , a contradiction. Hence, .
() Suppose that such that . Then, . So,Now, if , then . Therefore, . On the contrary, if , then . So, . This implies that .
Hence, . Let and , , . Then, and . Thus,Now, if , then implies ; otherwise, when , then . So, we haveThis implies that . Hence, , as required.

Lemma 2. Let be an ideal of and such that . Then,

Proof. Let such that . Then, we haveHence,

Definition 7. An set of a -algebra is called an ideal of if(1)(2) and , and

Example 4. Consider a -algebra which is given in Example 2. Let be an set defined asChoose . Then, is an ideal of .

Theorem 5. An set of is an ideal of , :(1)(2)

Proof. () Suppose that is an ideal of . If , then such that . This implies that , but , a contradiction. Hence, .
If we assume that , then such that . This implies that and , but , a contradiction. Hence, .
() Let such that . Then, . So,Now, if , then . Therefore, . On the contrary, if , then . So, . This implies that . Hence, .
Let and , , and . Then, and . Thus,Now, if , then and ; otherwise, when , then . So, we haveThis implies that . Hence, , as required.

Theorem 6. Every ideal of is an ideal of .

Proof. Let be an ideal of . Then, condition (1) of Definition 6 holds. By assumption, we havePut , so,Thus,Hence, is an ideal of .
As shown by the following example, the converse of the preceding Theorem 6 is not valid in general.

Example 5. Reconsider the -algebras given in Example 2. Define an set asChoose . Clearly, is an of , but is not an ideal of because = .

Theorem 7. Let be an ideal of . Then, is an ideal of , .

Proof. () Assume is an ideal of . Now, replace by in Theorem 5 (2); then,. () Let be an ideal of . Then, condition (1) holds. As , . By Lemma 2, we haveSince is an ideal, soHence, is an ideal of .

Theorem 8. An set of is an ideal of a -algebra is a positive implicative ideal of , .

Proof. () Let for . Then, . It follows from Theorem 5 (i) thatThus, .
Next, suppose that and . Then, and . Again, it follows from Theorem 5 (ii) thatTherefore, . Hence, is a positive implicative ideal of .
() Suppose, on the contrary, that for some . Choose such thatIt follows that , but , a contradiction. Therefore, ,. Suppose thatfor some . Then, such that implies that and , but , which is not possible. Thus,Hence, by Theorem 5, is an ideal of .

5. Conclusion

We applied the theory of interval-valued fuzzy sets on positive implication ideals of BCK-algebras. In this aim, the concept of interval-valued -polar fuzzy positive implicative ideals in BCK-algebras is introduced. The related propertied of interval-valued -polar fuzzy positive implicative ideals and interval-valued -polar fuzzy ideals are investigated. In addition, the concepts of interval-valued -polar -fuzzy positive implicative ideals and interval-valued -polar -fuzzy ideals are defined and characterized. Furthermore, we have shown that interval-valued -polar -fuzzy positive implicative ideals are interval-valued -polar -fuzzy ideals, but converse is not valid and an illustration is provided in this support.

In future work, one may extend these concepts to various algebraic structures such as rings, hemirings, -semigroups, semihypergroups, semihyperrings, BL-algebras, MTL-algebras, R0-algebras, MV-algebras, EQ-algebras, and lattice implication algebras.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Taif University Researchers’ Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
S. R. Barbhuiya and K. D. Choudhury, “(, )-interval-valued fuzzy dot d-ideals of d-algebras,” Advanced Trends in Mathematics, vol. 3, pp. 1–15, 2015.
View at: Publisher Site | Google Scholar
Y. B. Yun, “On (, )-fuzzy ideals of BCK/-algebras,” Scientiae Mathematicae Japonicae, vol. 60, no. 3, pp. 101–105, 2004.
View at: Google Scholar
M. Akram, N. Yaqoob, and J. Kavikumar, “Interval-valued (, )-fuzzy KU-ideals of KU-algebras,” International Journal of Pure and Applied Mathematics, vol. 92, no. 3, pp. 335–349, 2014.
View at: Publisher Site | Google Scholar
S. M. Mostafa, M. A. Abd-Elnaby, and O. R. Elgendy, “Interval-valued fuzzy KU-ideals in KU-algebras,” International Mathematical Forum, vol. 6, no. 64, pp. 3151–3159, 2011.
View at: Google Scholar
Y. B. Jun, “Interval-valued fuzzy subalgebras/ideals in -algebras,” Scientiae Mathematicae, vol. 3, no. 3, pp. 435–444, 2000.
View at: Google Scholar
R. A. Borzooei, G. R. Rezaei, G. Muhiuddin, and Y. B. Jun, “Multipolar fuzzy a-ideals in -algebras,” International Journal of Machine Learning and Cybernetics, 2021, In press.
View at: Publisher Site | Google Scholar
M. Aslam, S. Abdullah, and S. Aslam, “Characterization of LA-semigroup by interval-valued (, )-fuzzy ideals,” Afrika Matematika, vol. 25, pp. 501–518, 2014.
View at: Publisher Site | Google Scholar
R. Biswas, “Rosenfeld’s fuzzy subgroups with interval-valued membership functions,” Fuzzy Sets and Systems, vol. 63, no. 1, pp. 87–90, 1994.
View at: Publisher Site | Google Scholar
N. Yaqoob, “Interval-valued intuitionistic fuzzy ideals of regular LA-semigroups,” Thai Journal of Mathematics, vol. 11, no. 3, pp. 683–695, 2013.
View at: Google Scholar
D. S. Lee and C. H. Park, “Interval-valued (, ) fuzzy ideals in rings,” International Mathematical Forum, vol. 4, no. 13, pp. 623–630, 2009.
View at: Google Scholar
A. Ahmad, M. Aslam, and S. Abdullah, “Interval-valued (, )-fuzzy hyperideals of semihyperring,” UPB Scientific Bulletin, Series A, vol. 75, no. 2, pp. 69–86, 2013.
View at: Google Scholar
Y. B. Jun and K. H. Kim, “Interval-valued fuzzy r-subgroups of near-rings,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 1, pp. 71–80, 2002.
View at: Google Scholar
Y. B. Jun, “Interval-valued fuzzy ideals in BCI-algebras,” The Journal of fuzzy Mathematics, vol. 9, pp. 807–814, 2001.
View at: Google Scholar
J. Zhan and Y. B. Jun, “On -fuzzy ideals of -algebras,” Neural Computing and Applications, vol. 20, no. 3, pp. 319–328, 2010.
View at: Publisher Site | Google Scholar
J. Zhan, Y. B. Jun, and B. Davvaz, “On -fuzzy ideals of -algebras,” Iranian Journal of Fuzzy Systems, vol. 6, no. 1, pp. 81–94, 2009.
View at: Google Scholar
X. Ma, J. Zhan, B. Davvaz, and Y. B. Jun, “Some kinds of (, )-interval-valued fuzzy ideals of BCI-algebras,” Information Sciences, vol. 178, no. 19, pp. 3738–3754, 2008.
View at: Publisher Site | Google Scholar
X. Ma, J. Zhan, and Y. B. Jun, “Some types of (, )-interval-valued fuzzy ideals of -algebras,” Iranian Journal of Fuzzy Systems, vol. 6, pp. 53–63, 2009.
View at: Google Scholar
M. S. Ali, K. Meenakshi, N. Gunasekaran, and M. Usha, “Finite-time passivity of discrete-time TS fuzzy neural networks with time-varying delays,” Iranian Journal of Fuzzy Systems, vol. 15, no. 4, pp. 93–107, 2018.
View at: Publisher Site | Google Scholar
M. Syed Ali, N. Gunasekaran, and Q. Zhu, “State estimation of T-S fuzzy delayed neural networks with markovian jumping parameters using sampled-data control,” Fuzzy Sets and Systems, vol. 306, pp. 87–104, 2017.
View at: Publisher Site | Google Scholar
G. Muhiuddin, “Bipolar fuzzy KU-subalgebras/ideals of KU-algebras,” Annals of Fuzzy Mathematics and Informatics, vol. 8, no. 3, pp. 409–418, 2014.
View at: Google Scholar
G. Muhiuddin, D. Al-Kadi, A. Mahboob, and K. P. Shum, “New types of bipolar fuzzy ideals of -algebras,” International Journal of Analysis and Applications, vol. 18, no. 5, pp. 859–875, 2020.
View at: Google Scholar
G. Muhiuddin, H. Harizavi, and Y. B. Jun, “Bipolar-valued fuzzy soft hyper BCK ideals in hyper BCK algebras,” Discrete Mathematics Algorithms and Applications, vol. 12, no. 2, p. 16, 2020.
View at: Publisher Site | Google Scholar
E. Yucel, M. Syed Ali, N. Gunasekaran, and S. Arik, “Sampled-data filtering of Takagi-Sugeno fuzzy neural networks with interval time-varying delays,” Fuzzy Sets and Systems, vol. 316, pp. 69–81, 2017.
View at: Publisher Site | Google Scholar
J. Chen, S. Li, S. Ma, and X. Wang, “-polar fuzzy sets: an extension of bipolar fuzzy sets,” The Scientific World Journal, vol. 2014, Article ID 416530, 8 pages, 2014.
View at: Publisher Site | Google Scholar
M. Akram, A. Farooq, and K. P. Shum, “On -polar fuzzy lie subalgebras,” Italian Journal of Pure and Applied Mathematics, vol. 36, pp. 445–454, 2016.
View at: Google Scholar
M. Akram and A. Farooq, “-polar fuzzy lie ideals of lie algebras,” Quasigroups and Related Systems, vol. 24, pp. 141–150, 2016.
View at: Google Scholar
A. Farooq, G. Ali, and M. Akram, “On m-polar fuzzy groups,” International Journal of Algebra and Statistics, vol. 5, no. 2, pp. 115–127, 2016.
View at: Publisher Site | Google Scholar
A. Al-Masarwah and A. G. Ahmad, “m-polar fuzzy ideals of BCK/BCI-algebras,” Journal of King Saud University-Science, vol. 31, no. 4, pp. 1220–1226, 2019.
View at: Publisher Site | Google Scholar
A. Al-Masarwah and A. G. Ahmad, “m-polar (α, β)-fuzzy ideals in BCK/BCI-algebras,” Symmetry, vol. 11, no. 1, p. 44, 2019.
View at: Publisher Site | Google Scholar
G. Muhiuddin, M. Mohseni Takallo, R. A. Borzooei, and Y. B. Jun, “-polar fuzzy q-ideals in BCI-algebras,” Journal of King Saud University-Science, vol. 32, no. 6, pp. 2803–2809, 2020.
View at: Publisher Site | Google Scholar
M. M. Takallo, S. S. Ahn, R. A. Borzooei, and Y. B. Jun, “Multipolar fuzzy p-ideals of-algebras,” Mathematics, vol. 7, no. 11, p. 1094, 2019.
View at: Publisher Site | Google Scholar
A. Al-Masarwah, A. G. Ahmad, G. Muhiuddin, and D. Al-Kadi, “Generalized -polar fuzzy positive implicative ideals of -algebras,” Journal of Mathematics, vol. 2021, Article ID 6610009, 10 pages, 2021.
View at: Google Scholar
D. Al-Kadi and G. Muhiuddin, “Bipolar fuzzy BCI-implicative ideals of BCI-algebras,” Annals of Communications in Mathematics, vol. 3, no. 1, pp. 88–96, 2020.
View at: Google Scholar
A. Al-masarwah, A. G. Ahmad, and G. Muhiuddin, “Doubt N-ideals theory in BCK-algebras based on N-structures,” Annals of Communication in Mathematics, vol. 3, no. 1, pp. 54–62, 2020.
View at: Google Scholar
G. Muhiuddin and A. M. Al-roqi, “Cubic soft sets with applications in BCK/BCI-algebras,” Annals of Fuzzy Mathematics and Informatics, vol. 8, no. 2, pp. 291–304, 2014.
View at: Google Scholar
G. Muhiuddin, A. M. Al-roqi, and S. Aldhafeeri, “Filter theory in MTL-algebras based on uni-soft property,” Bulletin of the Iranian Mathematical Society, vol. 43, no. 7, pp. 2293–2306, 2017.
View at: Google Scholar
G. Muhiuddin and A. M. Al-roqi, “Unisoft filters in R0-algebras,” Journal of Computational Analysis and Applications, vol. 19, no. 1, pp. 133–143, 2015.
View at: Google Scholar
G. Muhiuddin, “p-ideals of BCI-algebras based on neutrosophic N-structures,” Journal of Intelligent & Fuzzy Systems, vol. 40, no. 1, pp. 1097–1105, 2021.
View at: Publisher Site | Google Scholar
G. Muhiuddin, D. Al-Kadi, A. Mahboob, and A. Aljohani, “Generalized fuzzy ideals of BCI-algebras based on interval valued m-polar fuzzy structures,” International Journal of Computational Intelligence Systems, 2021.
View at: Google Scholar
G. Muhiuddin and D. Al-Kadi, “Interval valued -polar fuzzy ideals in BCK/BCI-algebras,” International Journal of Computational Intelligence Systems, vol. 14, no. 1, pp. 1014–1021, 2021.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 G. Muhiuddin 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.

PDF Download Citation

Download other formats

Order printed copies

Views

233

Downloads

479

Citations