Kybernetika 58 no. 6, 996-1015, 2022

L-fuzzy ideal degrees in effect algebras

Xiaowei Wei and Fu-Gui ShiDOI: 10.14736/kyb-2022-6-0996

Abstract:

In this paper, considering $L$ being a completely distributive lattice, we first introduce the concept of $L$-fuzzy ideal degrees in an effect algebra $E$, in symbol $\mathfrak{D}_{ei}$. Further, we characterize $L$-fuzzy ideal degrees by cut sets. Then it is shown that an $L$-fuzzy subset $A$ in $E$ is an $L$-fuzzy ideal if and only if $\mathfrak{D}_{ei}(A)=\top,$ which can be seen as a generalization of fuzzy ideals. Later, we discuss the relations between $L$-fuzzy ideals and cut sets ($L_{\beta}$-nested sets and $L_{\alpha}$-nested sets). Finally, we obtain that the $L$-fuzzy ideal degree is an $(L,L)$-fuzzy convexity. The morphism between two effect algebras is an $(L,L)$-fuzzy convexity-preserving mapping.

Keywords:

effect algebra, $L$-fuzzy ideal degree, cut set, $(L, L)$-fuzzy convexity

Classification:

03B52, 03G27, 52A01

paper.pdf

References:

  1. B. A. Davey and H. A. Priestley: Introduction to Lattice and Order. Cambridge University Press, Cambridge 2002.   CrossRef
  2. Y. Y. Dong and J. Li: Fuzzy convex structures and prime fuzzy ideal space on residuated lattices. J. Nonlinear Convex Analysis 21 (2020), 12, 2725-2735.   CrossRef
  3. A. Dvurečenskij and S. Pulmannová: New trends in Quantum Structures. Springer-Science Business Media, B.V. 2000.   CrossRef
  4. P. Dwinger: Characterizations of the complete homomorphic images of a completely distributive lattice I. Mathematics 85 (1982), 4, 403-414.   DOI:10.1016/1385-7258(82)90034-8
  5. D. Foulis and M. K. Bennett: Effect algebras and unsharp quantum logice. Foundat. Physics 24 (1994), 1331-1352.   DOI:10.1007/BF02283036
  6. G. Gierz et al.: Continuous lattices and domains, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge 2003.   CrossRef
  7. J. A. Goguan: $L$-fuzzy sets. J. Math. Analysis Appl. 18 (1967), 145-174.   DOI:10.1016/0022-247X(67)90189-8
  8. U. Höhle and A. P. Šostak: Axiomatic foundations of fixed basis fuzzy topology. Math. Fuzzy Sets 3 (1999), 123-272.   DOI:10.1007/978-1-4615-5079-2\_5
  9. J. Li and F.-G. Shi: $L$-fuzzy convexity induced by $L$-convex fuzzy sublattice degree. Iranian J. Fuzzy Systems 14 (2017), 5, 83-102.   CrossRef
  10. D. L. Liu: Fuzzy filters and fuzzy ideals in pseudo-effect algebras. Computer Engrg. Appl. (2011), 50-52.   DOI:10.1111/j.1740-9713.2011.00484.x
  11. D. L. Liu and G. J. Wang: Fuzzy filters in effect algebras. Fuzzy Systems Math. 23 (2009), 6-17.   CrossRef
  12. C. Z. Luo: Fuzzy sets and nested sets. Fuzzy Math. 4 (1983), 113-126.   CrossRef
  13. Z. H. Ma: Note on ideals of effect algebras. Inform. Sci. 179 (2009), 505-507.   DOI:10.1016/j.ins.2008.07.018
  14. D. S. Malik and J. N. Mordeson: Fuzzy direct sums of fuzzy rings. Fuzzy Sets Systems 45 (1992), 83-91.   DOI:10.1016/0165-0114(92)90094-K
  15. F. Mehmood and F.-G. Shi: M-hazy vector spaces over M-hazy field. Mathematics 9 (2021), 1118.   DOI:10.3390/math9101118
  16. F. Mehmood, F.-G. Shi and K. Hayat: A new approach to the fuzzification of rings. J. Nonlinear Convex Analysis 21 (12) (2020), 2637-2646.   CrossRef
  17. J. Močkoř: $\alpha$-cuts and models of fuzzy logic. Int. J. General Systems 42 (2013), 67-78.   DOI:10.1080/03081079.2012.710438
  18. M. A. Öztürk, Y. B. Jun and H. Yazarli: A new view of fuzzy gamma rings. Hacettepe J. Math. Statist. 39 (2010), 3, 365-378.   CrossRef
  19. B. Pang: Convergence structures in $M$-fuzzifying convex spaces. Quaest. Math. 43 (2020), 11, 1541-1561.   DOI:10.2989/16073606.2019.1637379
  20. B. Pang: $L$-fuzzifying convex structures as $L$-convex structures. J. Nonlinear Convex Anal. 21 (2020), 12, 2831-2841.   CrossRef
  21. B. Pang: Bases and subbases in $(L,M)$-fuzzy convex spaces. Computat. Appl. Math. (2020), 39-41.   DOI:10.1007/s40314-020-1065-4
  22. B. Pang and F.-G. Shi: Strong inclusion orders between $L$-subsets and its applications in $L$-convex spaces. Quaest. Math. 41 (2018), 8, 1021-1043.   DOI:10.2989/16073606.2018.1436613
  23. D. Pei and T. Fan: On generalized fuzzy rough sets. Int. J. General Systems 38 (2009), 3, 255-271.   DOI:10.1080/03081070701553678
  24. M. V. Rosa: On fuzzy topology, fuzzy convexity spaces and fuzzy local convexity. Fuzzy Sets Systems 62 (1994), 97-100.   DOI:10.1016/0165-0114(94)90076-0
  25. C. Shen and F.-G. Shi: $L$-convex systems and the categorical isomorphism to Scott-hull operators. Iranian J. Fuzzy Systems 15 (2018), 2, 23-40.   CrossRef
  26. F.-G. Shi: Theory of $L_{\beta}$-nested set and $L_{\alpha}$-nested sets and applications. Fuzzy Systems Math. (1995), 65-72.   CrossRef
  27. F.-G. Shi and X. Xin: $L$-fuzzy subgroup degrees and $L$-fuzzy normal subgroup degrees. J. Advanced Res. Pure Math. 3 (2011), 4, 92-108.   DOI:10.5373/jarpm.762.020211
  28. F.-G. Shi and Z. Y. Xiu: A new approach to the fuzzification of convex structures. J. Appl. Math. 3 (2014), 1-12.   DOI:10.1155/2014/249183
  29. F.-G. Shi and Z. Y. Xiu: $(L,M)$-fuzzy convex structures. J. Nonlinear Sci. Appl. 10 (2017), 3655-3669.   DOI:10.22436/jnsa.010.07.25
  30. Y. Shi and H. L. Huang: A characterization of strong $Q$-concave spaces. J. Nonlinear Convex Anal. 21 (2020), 12, 2771-2781.   CrossRef
  31. M. L. J. van de Vel: Theory of Vonvex Structures. North Holland, N. Y. 1993.   CrossRef
  32. G. J. Wang: Theory of topological molecular lattices. Fuzzy Sets Systems 47 (1992), 351-376.   DOI:10.1016/0165-0114(92)90301-J
  33. K. Wang and F.-G. Shi: $M$-fuzzifying topological convex spaces. Iranian J. Fuzzy Systems 15, (2018), 6, 159-174.   CrossRef
  34. X. W. Wei, B. Pang and J. S. Mi: Axiomatic characterizations of $L$-valued rough sets using a single axiom. Inform, Sci. 580 (2021), 283-310.   DOI:10.1016/j.ins.2021.08.078
  35. X. W. Wei, B. Pang and J. S. Mi: Axiomatic characterizations of $L$-fuzzy rough sets by $L$-fuzzy unions and $L$-fuzzy intersections. Int. J. General Systems 51 (2022), 3, 277-312.   DOI:10.1080/03081079.2021.2024178
  36. X. W. Wei and B. Wang: Fuzzy (restricted) hull operators and fuzzy convex structures on $L$-sets. J. Nonlinear Convex Anal. 21 (2020), 12, 2805-2815.   CrossRef
  37. Y. F. Wen, Y. Zhong and F.-G. Shi: $L$-fuzzy convexity induced by $L$-convex degree on vector spaces. J. Intell. Fuzzy Systems 33 (2017), 4031-4041.   DOI:10.3233/JIFS-17912
  38. D. R. P. Williams, K. B. Latha and E. Chandrasekeran: Fuzzy bi-$\Gamma$-ideals in $\Gamma$-semigroups. Hacettepe J. Math. Statist. 38 (2009), 1, 1-15.   CrossRef
  39. J. Wu: Ideals, filters and supports in pseudo-effect algebras. Int. J. Theoret. Physics 43 (2004), 349-358.   DOI:10.1023/B:IJTP.0000028868.66790.0c
  40. H. Yang and E. Q. Li: A new approach to interval operators in $L$-convex spaces. J. Nonlinear Convex Anal. 21 (2020), 12, 2705-2714.   CrossRef
  41. Q. W. Zhang: Ideals and filters in dual effect algebras. Hennan Sci. 34 (2016), 8, 1211-1214.   DOI:10.1038/nbt1216-1214
  42. Q. W. Zhang: Note on ideals in effect algebras. Hennan Sci. 35 (2017), 10, 1567-1569.   DOI:10.1016/j.ajem.2017.04.002
  43. F. F. Zhao and H. L. Huang: The relationships among $L$-ordered hull operators, restricted $L$-hull operators and strong $L$-fuzzy convex structures. J. Nonlinear Convex Anal. 21 (2020), 12, 2817-2829.   CrossRef
  44. Y. Zhong and F.-G. Shi: Characterizations of $(L,M)$-fuzzy topological degrees. Iranian J. Fuzzy Syst. 15 (2018), 4, 129-149.   CrossRef
  45. X. W. Zhou and F.-G. Shi: Some new results on six types mappings between $L$-convex spaces. Filomat. 34 (2020), 4767-4781.   DOI:10.2298/FIL2014767Z