This work investigated categorical connections between $ L $-grill spaces and $ L $-pre-proximity spaces and established a Galois connection between the category of $ L $-pre-proximity spaces and the category of stratified $ L $-grill spaces within the framework of complete residuated lattices. Furthermore, it introduced the concept of $ L $-grill-proximity spaces, a new structure derived from $ L $-grills, and studied their fundamental properties as well as their categorical connections with $ L $-closure, $ L $-interior, and $ L $-co-topological spaces. Finally, it demonstrated the applicability of the proposed framework through an application to environmental pollution modeling by integrating multiple environmental indicators for quantitative air-quality assessment and decision-making.
Citation: Anwar J. Fawakhreh. Categorical relationships between $ L $-grill-proximity spaces and related $ L $-topological structures[J]. AIMS Mathematics, 2026, 11(7): 22380-22400. doi: 10.3934/math.2026905
This work investigated categorical connections between $ L $-grill spaces and $ L $-pre-proximity spaces and established a Galois connection between the category of $ L $-pre-proximity spaces and the category of stratified $ L $-grill spaces within the framework of complete residuated lattices. Furthermore, it introduced the concept of $ L $-grill-proximity spaces, a new structure derived from $ L $-grills, and studied their fundamental properties as well as their categorical connections with $ L $-closure, $ L $-interior, and $ L $-co-topological spaces. Finally, it demonstrated the applicability of the proposed framework through an application to environmental pollution modeling by integrating multiple environmental indicators for quantitative air-quality assessment and decision-making.
| [1] | J. A. Goguen, $L$-fuzzy sets, J. Math. Anal. Appl., 8 (1967), 145–174. https://doi.org/10.1016/0022-247X(67)90189-8 |
| [2] | L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X |
| [3] | U. Höhle, A. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, In: U. Höhle, S. E. Rodabaugh, Mathematics of fuzzy sets, The Handbooks of Fuzzy Sets Series, Springer, Boston, 3 (1999), 123–272. https://doi.org/10.1007/978-1-4615-5079-2_5 |
| [4] | A. A. Ramadan, Smooth topological spaces, Fuzzy Sets Syst., 48 (1992), 371–375. https://doi.org/10.1016/0165-0114(92)90352-5 |
| [5] | R. Bělohlávek, Fuzzy closure operators, J. Math. Anal. Appl., 262 (2001), 473–489. https://doi.org/10.1006/jmaa.2000.7456 |
| [6] | R. Bělohlávek, T. Funioková, Fuzzy interior operators, Int. J. Gen. Syst., 33 (2004), 415–430. https://doi.org/10.1080/03081070410001679724 |
| [7] |
J. Fang, Relationships between $L$-ordered convergence structures and strong $L$-topologies, Fuzzy Sets Syst., 161 (2010), 2923–2944. https://doi.org/10.1016/j.fss.2010.07.010 doi: 10.1016/j.fss.2010.07.010
|
| [8] | J. M. Oh, Y. C. Kim, $L$-Fuzzy closure operators and $L$-fuzzy cotopologies, J. Math. Comput. Sci., 9 (2019), 131–145. |
| [9] |
A. A. Ramadan, A. J. Fawakhreh, E. H. Elkordy, Novel categorical relations between $\mathcal{L}$-fuzzy co-topologies and $\mathcal{L}$-fuzzy ideals, AIMS Math., 9 (2024), 20572–20587. https://doi.org/10.3934/math.2024999 doi: 10.3934/math.2024999
|
| [10] |
F. G. Shi, $L$-fuzzy interiors and $L$-fuzzy closures, Fuzzy Sets Syst., 160 (2009), 1218–1232. https://doi.org/10.1016/j.fss.2008.09.002 doi: 10.1016/j.fss.2008.09.002
|
| [11] |
D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets Syst., 158 (2007), 349–366. https://doi.org/10.1016/j.fss.2006.10.001 doi: 10.1016/j.fss.2006.10.001
|
| [12] | A. K. Katsaras, Fuzzy proximity spaces, J. Math. Anal. Appl., 68 (1979), 100–110. https://doi.org/10.1016/0022-247X(79)90102-1 |
| [13] |
G. Artico, R. Moresco, Fuzzy proximities and totally bounded fuzzy uniformities, J. Math. Anal. Appl., 99 (1984), 320–337. https://doi.org/10.1016/0022-247X(84)90221-X doi: 10.1016/0022-247X(84)90221-X
|
| [14] |
D. Čimoka, A. Šostak, $L$-fuzzy syntopogenous structures, Part Ⅰ: Fundamentals and application to $L$-fuzzy topologies, $L$-fuzzy proximities and $L$-fuzzy uniformities, Fuzzy Sets Syst., 232 (2013), 74–97. https://doi.org/10.1016/j.fss.2013.04.009 doi: 10.1016/j.fss.2013.04.009
|
| [15] |
Y. C. Kim, K. C. Min, $L$-fuzzy proximities and $L$-fuzzy topologies, Inf. Sci., 173 (2005), 93–113. https://doi.org/10.1016/j.ins.2004.07.005 doi: 10.1016/j.ins.2004.07.005
|
| [16] |
M. Jose, S. C. Mathew, On the categorical connections of $L$-$G$-filter spaces and a Galois correspondence with $L$-fuzzy pre-proximity spaces, New Mat. Nat. Comput., 19 (2023), 33–49. https://doi.org/10.1142/S1793005722500478 doi: 10.1142/S1793005722500478
|
| [17] |
K. K. Azad, Fuzzy grills and a characterization of fuzzy proximity, J. Math. Anal. Appl., 79 (1981), 13–17. https://doi.org/10.1016/0022-247X(81)90003-2 doi: 10.1016/0022-247X(81)90003-2
|
| [18] | M. N. Mukherjee, S. Das, Fuzzy grills and induced fuzzy topology, Math. Vesnk, 62 (2010), 285–297. |
| [19] | B. Roy, M. N. Mukherjee, On a typical topology induced by a grill, Soochow J. Math., 33 (2007), 771. |
| [20] | A. Kandil, S. A. El-Sheikh, M. Abdelhakem, S. A. Hazza, On ideals and grills in topological spaces, South Asian J. Math., 5 (2015), 233–238. |
| [21] | A. R. Hosny Proximity structures and grills, Int. J. Eng. Res., 4 (2013), 427–450. |
| [22] |
A. A. Ramadan, M. A. Usama, A. A. Abd El-Latif, $L$-fuzzy pre-proximities, $L$-fuzzy filters and $L$-fuzzy grills, J. Egypt. Math. Soc., 28 (2020), 47. https://doi.org/10.1186/s42787-020-00105-4 doi: 10.1186/s42787-020-00105-4
|
| [23] |
A. A. Ramadan, A. J. Fawakhreh, On $L$-fuzzy preproximity spaces and $L$-fuzzy ideals, New Math. Nat. Comput., 21 (2025), 867–886. https://doi.org/10.1142/S1793005725500437 doi: 10.1142/S1793005725500437
|
| [24] |
A. A. Ramadan, A. J. Fawakhreh, Categories of $L$-primals, $L$-pre-proximities, and $L$-topologies, Axioms, 14 (2025), 541. https://doi.org/10.3390/axioms14070541 doi: 10.3390/axioms14070541
|
| [25] |
A. A. Ramadan, A. J. Fawakhreh, On the categories of $LF$-ideals, $LF$-grills, and $LF$-topological spaces, Axioms, 15 (2026), 68. https://doi.org/10.3390/axioms15010068 doi: 10.3390/axioms15010068
|
| [26] |
A. J. Fawakhreh, Connections of $L$-quasi-internal and $L$-quasi-enclosed spaces with $L$-grills, $L$-filters and $L$-neighborhood spaces, Axioms, 15 (2026), 462. https://doi.org/10.3390/axioms15060462 doi: 10.3390/axioms15060462
|
| [27] |
S. E. Abbas, M. A. Hebeshi, Construction of stratified $L$-fuzzy topological structures, Appl. Math. Lett., 24 (2011), 1062–1068. https://doi.org/10.1016/j.aml.2011.01.023 doi: 10.1016/j.aml.2011.01.023
|
| [28] |
I. Ibedou, S. E. Abbas, Fuzzy topological concepts via ideals and grills, Ann. Fuzzy Math. Inf., 15 (2018), 137–148. https://doi.org/10.30948/afmi.2018.15.2.137 doi: 10.30948/afmi.2018.15.2.137
|
| [29] | A. Kandil, S. A. El-Sheikh, E. Said, Generalized $G$-proximity spaces, South Asian J. Math., 7 (2017), 140–147. |
| [30] | R. Bělohlávek, Fuzzy relational systems: foundations and principles, IFSR International Series in Systems Science and Systems Engineering, Vol. 20, Springer New York, NY, 2002. https://doi.org/10.1007/978-1-4615-0633-1 |
| [31] | S. Gottwald, Many-valued logic and fuzzy set theory, In: U. Höhle, S. E. Rodabaugh, Mathematics of fuzzy sets, The Handbooks of Fuzzy Sets Series, Springer, Boston, 3 (1999), 5–89. https://doi.org/10.1007/978-1-4615-5079-2_2 |
| [32] | J. Adámek, H. Herrlich, G. E. Strecker, Abstract and concrete categories, John Wiley and Sons, Inc., 1990. |
Anwar J. Fawakhreh. Categorical relationships between $ L $-grill-proximity spaces and related $ L $-topological structures[J]. AIMS Mathematics, 2026, 11(7): 22380-22400. doi: 10.3934/math.2026905
DownLoad: