| Title:
|
$L$-fuzzy ideal degrees in effect algebras (English) |
| Author:
|
Wei, Xiaowei |
| Author:
|
Shi, Fu-Gui |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
58 |
| Issue:
|
6 |
| Year:
|
2022 |
| Pages:
|
996-1015 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
effect algebra |
| Keyword:
|
$L$-fuzzy ideal degree |
| Keyword:
|
cut set |
| Keyword:
|
$(L,L)$-fuzzy convexity |
| MSC:
|
03B52 |
| MSC:
|
03G27 |
| MSC:
|
52A01 |
| idZBL:
|
Zbl 07655868 |
| idMR:
|
MR4548225 |
| DOI:
|
10.14736/kyb-2022-6-0996 |
| . |
| Date available:
|
2023-02-10T13:55:31Z |
| Last updated:
|
2023-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151540 |
| . |
| Reference:
|
[1] Davey, B. A., Priestley, H. A.: Introduction to Lattice and Order..Cambridge University Press, Cambridge 2002. MR 1902334 |
| Reference:
|
[2] Dong, Y. Y., Li, J.: Fuzzy convex structures and prime fuzzy ideal space on residuated lattices..J. Nonlinear Convex Analysis 21 (2020), 12, 2725-2735. MR 4194714 |
| Reference:
|
[3] Dvurečenskij, A., Pulmannová, S.: New trends in Quantum Structures..Springer-Science Business Media, B.V. 2000. MR 1861369 |
| Reference:
|
[4] Dwinger, P.: Characterizations of the complete homomorphic images of a completely distributive lattice I..Mathematics 85 (1982), 4, 403-414. MR 0683528, |
| Reference:
|
[5] Foulis, D., Bennett, M. K.: Effect algebras and unsharp quantum logice..Foundat. Physics 24 (1994), 1331-1352. MR 1304942, |
| Reference:
|
[6] al., G. Gierz et: Continuous lattices and domains, Encyclopedia of Mathematics and its Applications..Cambridge University Press, Cambridge 2003. MR 1975381 |
| Reference:
|
[7] Goguan, J. A.: $L$-fuzzy sets..J. Math. Analysis Appl. 18 (1967), 145-174. MR 0224391, |
| Reference:
|
[8] Höhle, U., Šostak, A. P.: Axiomatic foundations of fixed basis fuzzy topology..Math. Fuzzy Sets 3 (1999), 123-272. MR 1788903, |
| Reference:
|
[9] Li, J., Shi, F.-G.: $L$-fuzzy convexity induced by $L$-convex fuzzy sublattice degree..Iranian J. Fuzzy Systems 14 (2017), 5, 83-102. MR 3751405 |
| Reference:
|
[10] Liu, D. L.: Fuzzy filters and fuzzy ideals in pseudo-effect algebras..Computer Engrg. Appl. (2011), 50-52. MR 2933851, |
| Reference:
|
[11] Liu, D. L., Wang, G. J.: Fuzzy filters in effect algebras..Fuzzy Systems Math. 23 (2009), 6-17. MR 2547361 |
| Reference:
|
[12] Luo, C. Z.: Fuzzy sets and nested sets..Fuzzy Math. 4 (1983), 113-126. MR 0743512 |
| Reference:
|
[13] Ma, Z. H.: Note on ideals of effect algebras..Inform. Sci. 179 (2009), 505-507. MR 2490189, |
| Reference:
|
[14] Malik, D. S., Mordeson, J. N.: Fuzzy direct sums of fuzzy rings..Fuzzy Sets Systems 45 (1992), 83-91. MR 1148455, |
| Reference:
|
[15] Mehmood, F., Shi, F.-G.: M-hazy vector spaces over M-hazy field..Mathematics 9 (2021), 1118. |
| Reference:
|
[16] Mehmood, F., Shi, F.-G., Hayat, K.: A new approach to the fuzzification of rings..J. Nonlinear Convex Analysis 21 (12) (2020), 2637-2646. MR 4194706 |
| Reference:
|
[17] Močkoř, J.: $\alpha$-cuts and models of fuzzy logic..Int. J. General Systems 42 (2013), 67-78. MR 2990334, |
| Reference:
|
[18] Öztürk, M. A., Jun, Y. B., Yazarli, H.: A new view of fuzzy gamma rings..Hacettepe J. Math. Statist. 39 (2010), 3, 365-378. MR 2732632 |
| Reference:
|
[19] Pang, B.: Convergence structures in $M$-fuzzifying convex spaces..Quaest. Math. 43 (2020), 11, 1541-1561. MR 4181551, |
| Reference:
|
[20] Pang, B.: $L$-fuzzifying convex structures as $L$-convex structures..J. Nonlinear Convex Anal. 21 (2020), 12, 2831-2841. MR 4194723 |
| Reference:
|
[21] Pang, B.: Bases and subbases in $(L,M)$-fuzzy convex spaces..Computat. Appl. Math. (2020), 39-41. MR 4059965, |
| Reference:
|
[22] Pang, B., Shi, F.-G.: Strong inclusion orders between $L$-subsets and its applications in $L$-convex spaces..Quaest. Math. 41 (2018), 8, 1021-1043. MR 3885942, |
| Reference:
|
[23] Pei, D., Fan, T.: On generalized fuzzy rough sets..Int. J. General Systems 38 (2009), 3, 255-271. MR 2527845, |
| Reference:
|
[24] Rosa, M. V.: On fuzzy topology, fuzzy convexity spaces and fuzzy local convexity..Fuzzy Sets Systems 62 (1994), 97-100. MR 1259888, |
| Reference:
|
[25] Shen, C., Shi, F.-G.: $L$-convex systems and the categorical isomorphism to Scott-hull operators..Iranian J. Fuzzy Systems 15 (2018), 2, 23-40. MR 3840019 |
| Reference:
|
[26] Shi, F.-G.: Theory of $L_{\beta}$-nested set and $L_{\alpha}$-nested sets and applications..Fuzzy Systems Math. (1995), 65-72. MR 1384670 |
| Reference:
|
[27] Shi, F.-G., Xin, X.: $L$-fuzzy subgroup degrees and $L$-fuzzy normal subgroup degrees..J. Advanced Res. Pure Math. 3 (2011), 4, 92-108. MR 2859291, |
| Reference:
|
[28] Shi, F.-G., Xiu, Z. Y.: A new approach to the fuzzification of convex structures..J. Appl. Math. 3 (2014), 1-12. MR 3259199, |
| Reference:
|
[29] Shi, F.-G., Xiu, Z. Y.: $(L,M)$-fuzzy convex structures..J. Nonlinear Sci. Appl. 10 (2017), 3655-3669. MR 3680307, |
| Reference:
|
[30] Shi, Y., Huang, H. L.: A characterization of strong $Q$-concave spaces..J. Nonlinear Convex Anal. 21 (2020), 12, 2771-2781. MR 4194718 |
| Reference:
|
[31] Vel, M. L. J. van de: Theory of Vonvex Structures..North Holland, N. Y. 1993. MR 1234493 |
| Reference:
|
[32] Wang, G. J.: Theory of topological molecular lattices..Fuzzy Sets Systems 47 (1992), 351-376. MR 1166284, |
| Reference:
|
[33] Wang, K., Shi, F.-G.: $M$-fuzzifying topological convex spaces..Iranian J. Fuzzy Systems 15, (2018), 6, 159-174. MR 3931764 |
| Reference:
|
[34] Wei, X. W., Pang, B., Mi, J. S.: Axiomatic characterizations of $L$-valued rough sets using a single axiom..Inform, Sci. 580 (2021), 283-310. MR 4308034, |
| Reference:
|
[35] Wei, X. W., Pang, B., Mi, J. S.: Axiomatic characterizations of $L$-fuzzy rough sets by $L$-fuzzy unions and $L$-fuzzy intersections..Int. J. General Systems 51 (2022), 3, 277-312. MR 4407612, |
| Reference:
|
[36] Wei, X. W., Wang, B.: Fuzzy (restricted) hull operators and fuzzy convex structures on $L$-sets..J. Nonlinear Convex Anal. 21 (2020), 12, 2805-2815. MR 4194721 |
| Reference:
|
[37] Wen, Y. F., Zhong, Y., Shi, F.-G.: $L$-fuzzy convexity induced by $L$-convex degree on vector spaces..J. Intell. Fuzzy Systems 33 (2017), 4031-4041. MR 3751405, |
| Reference:
|
[38] Williams, D. R. P., Latha, K. B., Chandrasekeran, E.: Fuzzy bi-$\Gamma$-ideals in $\Gamma$-semigroups..Hacettepe J. Math. Statist. 38 (2009), 1, 1-15. MR 2530686 |
| Reference:
|
[39] Wu, J.: Ideals, filters and supports in pseudo-effect algebras..Int. J. Theoret. Physics 43 (2004), 349-358. MR 2080537, |
| Reference:
|
[40] Yang, H., Li, E. Q.: A new approach to interval operators in $L$-convex spaces..J. Nonlinear Convex Anal. 21 (2020), 12, 2705-2714. MR 4194712 |
| Reference:
|
[41] Zhang, Q. W.: Ideals and filters in dual effect algebras..Hennan Sci. 34 (2016), 8, 1211-1214. |
| Reference:
|
[42] Zhang, Q. W.: Note on ideals in effect algebras..Hennan Sci. 35 (2017), 10, 1567-1569. MR 2490189, |
| Reference:
|
[43] Zhao, F. F., Huang, H. L.: 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. MR 4194722 |
| Reference:
|
[44] Zhong, Y., Shi, F.-G.: Characterizations of $(L,M)$-fuzzy topological degrees..Iranian J. Fuzzy Syst. 15 (2018), 4, 129-149. MR 3823001 |
| Reference:
|
[45] Zhou, X. W., Shi, F.-G.: Some new results on six types mappings between $L$-convex spaces..Filomat. 34 (2020), 4767-4781. MR 4290887, |
| . |
