Research Article
BibTex RIS Cite
Year 2019, Volume: 7 Issue: 2, 410 - 432, 15.10.2019

Abstract

References

  • [1] S. S. Ahn and Keumseong Bang, On fuzzy subalgebras in B-algebras, Commun. Korean Math. Soc. 18 (2003), no. 3, 429–437.
  • [2] M. A. Ansari, A. Haidar, and A. N. A. Koam, On a graph associated to UP-algebras, Math. Comput. Appl. 23 (2018), no. 4, 61.
  • [3] N. Dokkhamdang, A. Kesorn, and A. Iampan, Generalized fuzzy sets in UP-algebras, Ann. Fuzzy Math. Inform. 16 (2018), no. 2, 171–190.
  • [4] T. Guntasow, S. Sajak, A. Jomkham, and A. Iampan, Fuzzy translations of a fuzzy set in UP-algebras, J. Indones. Math. Soc. 23 (2017), no. 2, 1–19.
  • [5] A. Iampan, A new branch of the logical algebra: UP-algebras, J. Algebra Relat. Top. 5 (2017), no. 1, 35–54.
  • [6] A. Iampan, Introducing fully UP-semigroups, Discuss. Math., Gen. Algebra Appl. 38 (2018), no. 2, 297–306.
  • [7] Y. B. Jun, E. H. Roh, and H. S. Kim, On fuzzy B-algebras, Czech. Math. J. 52 (2002), no. 2, 375–384.
  • [8] W. Kaijae, P. Poungsumpao, S. Arayarangsi, and A. Iampan, UP-algebras characterized by their anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras, Ital. J. Pure Appl. Math. 36 (2016), 667–692.
  • [9] B. Kesorn, K. Maimun, W. Ratbandan, and A. Iampan, Intuitionistic fuzzy sets in UP-algebras, Ital. J. Pure Appl. Math. 34 (2015), 339–364.
  • [10] E. P. Klement, R. Mesiar, and E. Pap, On the relationship of associative compensatory operators to triangular norms and conorms, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4 (1996), no. 2, 129–144.
  • [11] E. P. Klement, R. Mesiar, and E. Pap, Triangular norms, Kluwer Academic Publishers, Netherlands, 2000.
  • [12] E. P. Klement, R. Mesiar, and E. Pap, Triangular norms. position paper I: basic analytical and algebraic properties, Fuzzy Sets Syst. 143 (2004), no. 1, 5–26.
  • [13] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA. 28 (1942), no. 12, 535–537.
  • [14] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World Scientific, Singapore, 1998.
  • [15] P. Mosrijai, A. Satirad, and A. Iampan, The new UP-isomorphism theorems for UP-algebras in the meaning of the congruence determined by a UP-homomorphism, Fundam. J. Math. Appl. 1(2018), no. 1, 12–17.
  • [16] P. Poungsumpao, W. Kaijae, S. Arayarangsi, and A. Iampan, Fuzzy UP-ideals and fuzzy UP-subalgebras of UP-algebras in term of level subsets, Int. J. Math. Comput. Sci. 14 (2019), no. 3, 647–674.
  • [17] C. Prabpayak and U. Leerawat, On ideals and congruences in KU-algebras, Sci. Magna 5 (2009), no. 1, 54–57.
  • [18] D. A. Romano, Notes on UP-ideals in UP-algebra, Commun. Adv. Math. Sci. 1(2018), no. 1, 35–38.
  • [19] D. A. Romano, Proper UP-filters of UP-algebra, Univ. J. Math. Appl. 1 (2018), no. 2, 98–100.
  • [20] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512–517.
  • [21] A. Satirad, P. Mosrijai, and A. Iampan, Formulas for finding UP-algebras, Int. J. Math. Comput. Sci. 14 (2019), no. 2, 403–409.
  • [22] A. Satirad, P. Mosrijai, and A. Iampan, Generalized power UP-algebras, Int. J. Math. Comput. Sci. 14 (2019), no. 1, 17–25.
  • [23] B. Schweizer and A. Sklar, Probabilistic metric spaces, Elsevier North-Holland, New York, 1983.
  • [24] B. Schweizer, Associative functions and statistical triangle inequalities, Publ. Math. Debrecen 8 (1961), 169–185.
  • [25] B. Schweizer, Associative functions and abstract semigroups, Publ. Math. Debrecen 10 (1963), 69–81.
  • [26] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), no. 1, 313–334.
  • [27] T. Senapati, Y. B. Jun, and K. P. Shum, Cubic set structure applied in UP-algebras, Discrete Math. Algorithms Appl. 10 (2018), no. 4, 1850049.
  • [28] T. Senapati, G. Muhiuddin, and K. P. Shum, Representation of UP-algebras in interval-valued intuitionistic fuzzy environment, Ital. J. Pure Appl. Math. 38 (2017), 497–517.
  • [29] T. Senapati, M. Bhowmik, and M. Pal, Fuzzy B-subalgebras of B-algebra with respect to t-norm, J. Fuzzy Set Valued Anal. 2012 (2012), 11.
  • [30] B.-S. Shieh, Infinite fuzzy relation equations with continuous t-norms, Inf. Sci. 178 (2008), no. 8, 1961–1967.
  • [31] J. Somjanta, N. Thuekaew, P. Kumpeangkeaw, and A. Iampan, Fuzzy sets in UP-algebras, Ann. Fuzzy Math. Inform. 12 (2016), no. 6, 739–756.
  • [32] M. Songsaeng and A. Iampan, N -fuzzy UP-algebras and its level subsets, J. Algebra Relat. Top. 6 (2018), no. 1, 1–24.
  • [33] S. Sripaeng, K. Tanamoon, and A. Iampan, On anti Q-fuzzy UP-ideals and anti Q-fuzzy UP-subalgebras of UP-algebras, J. Inf. Optim. Sci. 39 (2018), no. 5, 1095–1127.
  • [34] K. Tanamoon, S. Sripaeng, and A. Iampan, Q-fuzzy sets in UP-algebras, Songklanakarin J. Sci. Technol. 40 (2018), no. 1, 9–29.
  • [35] L. A. Zadeh, Fuzzy sets, Inf. Cont. 8 (1965), 338–353.

Fuzzy Sets in UP-algebras with Respect to A Triangular Norm

Year 2019, Volume: 7 Issue: 2, 410 - 432, 15.10.2019

Abstract

In this paper, we apply the notion of fuzzy sets with respect to a triangular norm to UP-algebras. We introduce the notions of $T$-fuzzy UP-subalgebras, $T$-fuzzy near UP-filters, $T$-fuzzy UP-filters, $T$-fuzzy UP-ideals, and $T$-fuzzy strongly UP-ideals, their properties are investigated and some useful examples are discussed. We discuss the relations between $T$-fuzzy UP-subalgebras (resp., $T$-fuzzy near UP-filters, $T$-fuzzy UP-filters, $T$-fuzzy UP-ideals, and $T$-fuzzy strongly UP-ideals) and a notion of UP-subalgebras (resp., near UP-filters, UP-filters, UP-ideals, strongly UP-ideals), and their level subsets and UP-homomorphisms are studied. Moreover, we have introduced the notion of fuzzy sets with respect to a triangular norm of anti-type in UP-algebras, and studied the properties as well as previous notions.

References

  • [1] S. S. Ahn and Keumseong Bang, On fuzzy subalgebras in B-algebras, Commun. Korean Math. Soc. 18 (2003), no. 3, 429–437.
  • [2] M. A. Ansari, A. Haidar, and A. N. A. Koam, On a graph associated to UP-algebras, Math. Comput. Appl. 23 (2018), no. 4, 61.
  • [3] N. Dokkhamdang, A. Kesorn, and A. Iampan, Generalized fuzzy sets in UP-algebras, Ann. Fuzzy Math. Inform. 16 (2018), no. 2, 171–190.
  • [4] T. Guntasow, S. Sajak, A. Jomkham, and A. Iampan, Fuzzy translations of a fuzzy set in UP-algebras, J. Indones. Math. Soc. 23 (2017), no. 2, 1–19.
  • [5] A. Iampan, A new branch of the logical algebra: UP-algebras, J. Algebra Relat. Top. 5 (2017), no. 1, 35–54.
  • [6] A. Iampan, Introducing fully UP-semigroups, Discuss. Math., Gen. Algebra Appl. 38 (2018), no. 2, 297–306.
  • [7] Y. B. Jun, E. H. Roh, and H. S. Kim, On fuzzy B-algebras, Czech. Math. J. 52 (2002), no. 2, 375–384.
  • [8] W. Kaijae, P. Poungsumpao, S. Arayarangsi, and A. Iampan, UP-algebras characterized by their anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras, Ital. J. Pure Appl. Math. 36 (2016), 667–692.
  • [9] B. Kesorn, K. Maimun, W. Ratbandan, and A. Iampan, Intuitionistic fuzzy sets in UP-algebras, Ital. J. Pure Appl. Math. 34 (2015), 339–364.
  • [10] E. P. Klement, R. Mesiar, and E. Pap, On the relationship of associative compensatory operators to triangular norms and conorms, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4 (1996), no. 2, 129–144.
  • [11] E. P. Klement, R. Mesiar, and E. Pap, Triangular norms, Kluwer Academic Publishers, Netherlands, 2000.
  • [12] E. P. Klement, R. Mesiar, and E. Pap, Triangular norms. position paper I: basic analytical and algebraic properties, Fuzzy Sets Syst. 143 (2004), no. 1, 5–26.
  • [13] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA. 28 (1942), no. 12, 535–537.
  • [14] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World Scientific, Singapore, 1998.
  • [15] P. Mosrijai, A. Satirad, and A. Iampan, The new UP-isomorphism theorems for UP-algebras in the meaning of the congruence determined by a UP-homomorphism, Fundam. J. Math. Appl. 1(2018), no. 1, 12–17.
  • [16] P. Poungsumpao, W. Kaijae, S. Arayarangsi, and A. Iampan, Fuzzy UP-ideals and fuzzy UP-subalgebras of UP-algebras in term of level subsets, Int. J. Math. Comput. Sci. 14 (2019), no. 3, 647–674.
  • [17] C. Prabpayak and U. Leerawat, On ideals and congruences in KU-algebras, Sci. Magna 5 (2009), no. 1, 54–57.
  • [18] D. A. Romano, Notes on UP-ideals in UP-algebra, Commun. Adv. Math. Sci. 1(2018), no. 1, 35–38.
  • [19] D. A. Romano, Proper UP-filters of UP-algebra, Univ. J. Math. Appl. 1 (2018), no. 2, 98–100.
  • [20] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512–517.
  • [21] A. Satirad, P. Mosrijai, and A. Iampan, Formulas for finding UP-algebras, Int. J. Math. Comput. Sci. 14 (2019), no. 2, 403–409.
  • [22] A. Satirad, P. Mosrijai, and A. Iampan, Generalized power UP-algebras, Int. J. Math. Comput. Sci. 14 (2019), no. 1, 17–25.
  • [23] B. Schweizer and A. Sklar, Probabilistic metric spaces, Elsevier North-Holland, New York, 1983.
  • [24] B. Schweizer, Associative functions and statistical triangle inequalities, Publ. Math. Debrecen 8 (1961), 169–185.
  • [25] B. Schweizer, Associative functions and abstract semigroups, Publ. Math. Debrecen 10 (1963), 69–81.
  • [26] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), no. 1, 313–334.
  • [27] T. Senapati, Y. B. Jun, and K. P. Shum, Cubic set structure applied in UP-algebras, Discrete Math. Algorithms Appl. 10 (2018), no. 4, 1850049.
  • [28] T. Senapati, G. Muhiuddin, and K. P. Shum, Representation of UP-algebras in interval-valued intuitionistic fuzzy environment, Ital. J. Pure Appl. Math. 38 (2017), 497–517.
  • [29] T. Senapati, M. Bhowmik, and M. Pal, Fuzzy B-subalgebras of B-algebra with respect to t-norm, J. Fuzzy Set Valued Anal. 2012 (2012), 11.
  • [30] B.-S. Shieh, Infinite fuzzy relation equations with continuous t-norms, Inf. Sci. 178 (2008), no. 8, 1961–1967.
  • [31] J. Somjanta, N. Thuekaew, P. Kumpeangkeaw, and A. Iampan, Fuzzy sets in UP-algebras, Ann. Fuzzy Math. Inform. 12 (2016), no. 6, 739–756.
  • [32] M. Songsaeng and A. Iampan, N -fuzzy UP-algebras and its level subsets, J. Algebra Relat. Top. 6 (2018), no. 1, 1–24.
  • [33] S. Sripaeng, K. Tanamoon, and A. Iampan, On anti Q-fuzzy UP-ideals and anti Q-fuzzy UP-subalgebras of UP-algebras, J. Inf. Optim. Sci. 39 (2018), no. 5, 1095–1127.
  • [34] K. Tanamoon, S. Sripaeng, and A. Iampan, Q-fuzzy sets in UP-algebras, Songklanakarin J. Sci. Technol. 40 (2018), no. 1, 9–29.
  • [35] L. A. Zadeh, Fuzzy sets, Inf. Cont. 8 (1965), 338–353.
There are 35 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Pakpimon Burandate This is me

Sawittree Thongarsa This is me

Aiyared Iampan 0000-0002-0475-3320

Publication Date October 15, 2019
Submission Date April 21, 2019
Acceptance Date September 20, 2019
Published in Issue Year 2019 Volume: 7 Issue: 2

Cite

APA Burandate, P., Thongarsa, S., & Iampan, A. (2019). Fuzzy Sets in UP-algebras with Respect to A Triangular Norm. Konuralp Journal of Mathematics, 7(2), 410-432.
AMA Burandate P, Thongarsa S, Iampan A. Fuzzy Sets in UP-algebras with Respect to A Triangular Norm. Konuralp J. Math. October 2019;7(2):410-432.
Chicago Burandate, Pakpimon, Sawittree Thongarsa, and Aiyared Iampan. “Fuzzy Sets in UP-Algebras With Respect to A Triangular Norm”. Konuralp Journal of Mathematics 7, no. 2 (October 2019): 410-32.
EndNote Burandate P, Thongarsa S, Iampan A (October 1, 2019) Fuzzy Sets in UP-algebras with Respect to A Triangular Norm. Konuralp Journal of Mathematics 7 2 410–432.
IEEE P. Burandate, S. Thongarsa, and A. Iampan, “Fuzzy Sets in UP-algebras with Respect to A Triangular Norm”, Konuralp J. Math., vol. 7, no. 2, pp. 410–432, 2019.
ISNAD Burandate, Pakpimon et al. “Fuzzy Sets in UP-Algebras With Respect to A Triangular Norm”. Konuralp Journal of Mathematics 7/2 (October 2019), 410-432.
JAMA Burandate P, Thongarsa S, Iampan A. Fuzzy Sets in UP-algebras with Respect to A Triangular Norm. Konuralp J. Math. 2019;7:410–432.
MLA Burandate, Pakpimon et al. “Fuzzy Sets in UP-Algebras With Respect to A Triangular Norm”. Konuralp Journal of Mathematics, vol. 7, no. 2, 2019, pp. 410-32.
Vancouver Burandate P, Thongarsa S, Iampan A. Fuzzy Sets in UP-algebras with Respect to A Triangular Norm. Konuralp J. Math. 2019;7(2):410-32.
Creative Commons License
The published articles in KJM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.