Soft set theory and topology

[EN] In this paper we study and discuss the soft set theory giving new definitions, examples, new classes of soft sets, and properties for mappings between different classes of soft sets. Furthermore, we investigate the theory of soft topological spaces and we present new definitions, characterizations, and properties concerning the soft closure, the soft interior, the soft boundary, the soft continuity, the soft open and closed maps, and the soft homeomorphism.Georgiou, DN.; Megaritis, AC. (2014). Soft set theory and topology. Applied General Topology. 15(1):93-109. doi:http://dx.doi.org/10.4995/agt.2014.2268.93109151Aktaş, H., & Çağman, N. (2007). Soft sets and soft groups. Information Sciences, 177(13), 2726-2735. doi:10.1016/j.ins.2006.12.008Ali, M. I., Feng, F., Liu, X., Min, W. K., & Shabir, M. (2009). On some new operations in soft set theory. Computers & Mathematics with Applications, 57(9), 1547-1553. doi:10.1016/j.camwa.2008.11.009Aygünoğlu, A., & Aygün, H. (2011). Some notes on soft topological spaces. Neural Computing and Applications, 21(S1), 113-119. doi:10.1007/s00521-011-0722-3Çağman, N., & Enginoğlu, S. (2010). Soft set theory and uni–int decision making. European Journal of Operational Research, 207(2), 848-855. doi:10.1016/j.ejor.2010.05.004Çağman, N., & Enginoğlu, S. (2010). Soft matrix theory and its decision making. Computers & Mathematics with Applications, 59(10), 3308-3314. doi:10.1016/j.camwa.2010.03.015Çağman, N., Karataş, S., & Enginoglu, S. (2011). Soft topology. Computers & Mathematics with Applications, 62(1), 351-358. doi:10.1016/j.camwa.2011.05.016Chen, D., Tsang, E. C. C., Yeung, D. S., & Wang, X. (2005). The parameterization reduction of soft sets and its applications. Computers & Mathematics with Applications, 49(5-6), 757-763. doi:10.1016/j.camwa.2004.10.036Feng, F., Jun, Y. B., & Zhao, X. (2008). Soft semirings. Computers & Mathematics with Applications, 56(10), 2621-2628. doi:10.1016/j.camwa.2008.05.011Hussain, S., & Ahmad, B. (2011). Some properties of soft topological spaces. Computers & Mathematics with Applications, 62(11), 4058-4067. doi:10.1016/j.camwa.2011.09.051O. Kazanci, S. Yilmaz and S. Yamak, Soft Sets and Soft BCH-Algebras, Hacettepe Journal of Mathematics and Statistics 39, no. 2 (2010), 205-217.KHARAL, A., & AHMAD, B. (2011). MAPPINGS ON SOFT CLASSES. New Mathematics and Natural Computation, 07(03), 471-481. doi:10.1142/s1793005711002025Maji, P. K., Roy, A. R., & Biswas, R. (2002). An application of soft sets in a decision making problem. Computers & Mathematics with Applications, 44(8-9), 1077-1083. doi:10.1016/s0898-1221(02)00216-xMaji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers & Mathematics with Applications, 45(4-5), 555-562. doi:10.1016/s0898-1221(03)00016-6P. K. Maji, R. Biswas and A. R. Roy, Fuzzy soft sets, J. Fuzzy Math. 9, no. 3 (2001), 589-602.MAJUMDAR, P., & SAMANTA, S. K. (2008). SIMILARITY MEASURE OF SOFT SETS. New Mathematics and Natural Computation, 04(01), 1-12. doi:10.1142/s1793005708000908Min, W. K. (2011). A note on soft topological spaces. Computers & Mathematics with Applications, 62(9), 3524-3528. doi:10.1016/j.camwa.2011.08.068Molodtsov, D. (1999). Soft set theory—First results. Computers & Mathematics with Applications, 37(4-5), 19-31. doi:10.1016/s0898-1221(99)00056-5D. A. Molodtsov, The description of a dependence with the help of soft sets, J. Comput. Sys. Sc. Int. 40, no. 6 (2001), 977-984.D. A. Molodtsov, The theory of soft sets (in Russian), URSS Publishers, Moscow, 2004.D. A. Molodtsov, V. Y. Leonov and D. V. Kovkov, Soft sets technique and its application, Nechetkie Sistemy i Myagkie Vychisleniya 1, no. 1 (2006), 8-39.D. Pei and D. Miao, From soft sets to information systems, In: X. Hu, Q. Liu, A. Skowron, T. Y. Lin, R. R. Yager, B. Zhang, eds., Proceedings of Granular Computing, IEEE, 2 (2005), 617-621.Shabir, M., & Naz, M. (2011). On soft topological spaces. Computers & Mathematics with Applications, 61(7), 1786-1799. doi:10.1016/j.camwa.2011.02.006Shao, Y., & Qin, K. (2011). The lattice structure of the soft groups. Procedia Engineering, 15, 3621-3625. doi:10.1016/j.proeng.2011.08.678I. Zorlutuna, M. Akdag, W. K. Min and S. Atmaca, Remarks on soft topological spaces, Annals of Fuzzy Mathematics and Informatics 3, no. 2 (2012), 171-185.Zou, Y., & Xiao, Z. (2008). Data analysis approaches of soft sets under incomplete information. Knowledge-Based Systems, 21(8), 941-945. doi:10.1016/j.knosys.2008.04.00

On the fixed point theory of soft metric spaces

[EN] The aim of this paper is to show that a soft metric induces a compatible metric on the collection of all soft points of the absolute soft set, when the set of parameters is a finite set. We then show that soft metric extensions of several important fixed point theorems for metric spaces can be directly deduced from comparable existing results. We also present some examples to validate and illustrate our approach.Salvador Romaguera thanks the support of Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.Abbas, M.; Murtaza, G.; Romaguera Bonilla, S. (2016). On the fixed point theory of soft metric spaces. Fixed Point Theory and Applications. 2016(17):1-11. https://doi.org/10.1186/s13663-016-0502-yS111201617Zadeh, LA: Fuzzy sets. Inf. Control 8, 103-112 (1965)Molodtsov, D: Soft set theory - first results. Comput. Math. Appl. 37, 19-31 (1999)Aktaş, H, Çağman, N: Soft sets and soft groups. Inf. Sci. 177, 2726-2735 (2007)Ali, MI, Feng, F, Liu, X, Min, WK, Shabir, M: On some new operations in soft set theory. Comput. Math. Appl. 57, 1547-1553 (2009)Feng, F, Liu, X, Leoreanu-Fotea, V, Jun, YB: Soft sets and soft rough sets. Inf. Sci. 181, 1125-1137 (2011)Jiang, Y, Tang, Y, Chen, Q, Wang, J, Tang, S: Extending soft sets with description logics. Comput. Math. Appl. 59, 2087-2096 (2009)Jun, YB: Soft BCK/BCI-algebras. Comput. Math. Appl. 56, 1408-1413 (2008)Jun, YB, Lee, KJ, Khan, A: Soft ordered semigroups. Math. Log. Q. 56, 42-50 (2010)Jun, YB, Lee, KJ, Park, CH: Soft set theory applied to ideals in d-algebras. Comput. Math. Appl. 57, 367-378 (2009)Jun, YB, Park, CH: Applications of soft sets in ideal theory of BCK/BCI-algebras. Inf. Sci. 178, 2466-2475 (2008)Kong, Z, Gao, L, Wang, L, Li, S: The normal parameter reduction of soft sets and its algorithm. Comput. Math. Appl. 56, 3029-3037 (2008)Majumdar, P, Samanta, SK: Generalized fuzzy soft sets. Comput. Math. Appl. 59, 1425-1432 (2010)Li, F: Notes on the soft operations. ARPN J. Syst. Softw. 1, 205-208 (2011)Maji, PK, Roy, AR, Biswas, R: An application of soft sets in a decision making problem. Comput. Math. Appl. 44, 1077-1083 (2002)Qin, K, Hong, Z: On soft equality. J. Comput. Appl. Math. 234, 1347-1355 (2010)Xiao, Z, Gong, K, Xia, S, Zou, Y: Exclusive disjunctive soft sets. Comput. Math. Appl. 59, 2128-2137 (2009)Xiao, Z, Gong, K, Zou, Y: A combined forecasting approach based on fuzzy soft sets. J. Comput. Appl. Math. 228, 326-333 (2009)Xu, W, Ma, J, Wang, S, Hao, G: Vague soft sets and their properties. Comput. Math. Appl. 59, 787-794 (2010)Yang, CF: A note on soft set theory. Comput. Math. Appl. 56, 1899-1900 (2008)Yang, X, Lin, TY, Yang, J, Li, Y, Yu, D: Combination of interval-valued fuzzy set and soft set. Comput. Math. Appl. 58, 521-527 (2009)Zhu, P, Wen, Q: Operations on soft sets revisited (2012). arXiv:1205.2857v1Feng, F, Jun, YB, Liu, XY, Li, LF: An adjustable approach to fuzzy soft set based decision making. J. Comput. Appl. Math. 234, 10-20 (2009)Feng, F, Jun, YB, Zhao, X: Soft semirings. Comput. Math. Appl. 56, 2621-2628 (2008)Feng, F, Liu, X: Soft rough sets with applications to demand analysis. In: Int. Workshop Intell. Syst. Appl. (ISA 2009), pp. 1-4. (2009)Herawan, T, Deris, MM: On multi-soft sets construction in information systems. In: Emerging Intelligent Computing Technology and Applications with Aspects of Artificial Intelligence, pp. 101-110. Springer, Berlin (2009)Herawan, T, Rose, ANM, Deris, MM: Soft set theoretic approach for dimensionality reduction. In: Database Theory and Application, pp. 171-178. Springer, Berlin (2009)Kim, YK, Min, WK: Full soft sets and full soft decision systems. J. Intell. Fuzzy Syst. 26, 925-933 (2014). doi: 10.3233/IFS-130783Mushrif, MM, Sengupta, S, Ray, AK: Texture classification using a novel, soft-set theory based classification algorithm. Lect. Notes Comput. Sci. 3851, 246-254 (2006)Roy, AR, Maji, PK: A fuzzy soft set theoretic approach to decision making problems. J. Comput. Appl. Math. 203, 412-418 (2007)Zhu, P, Wen, Q: Probabilistic soft sets. In: IEEE Conference on Granular Computing (GrC 2010), pp. 635-638 (2010)Zou, Y, Xiao, Z: Data analysis approaches of soft sets under incomplete information. Knowl.-Based Syst. 21, 941-945 (2008)Cagman, N, Karatas, S, Enginoglu, S: Soft topology. Comput. Math. Appl. 62, 351-358 (2011)Das, S, Samanta, SK: Soft real sets, soft real numbers and their properties. J. Fuzzy Math. 20, 551-576 (2012)Das, S, Samanta, SK: Soft metric. Ann. Fuzzy Math. Inform. 6, 77-94 (2013)Abbas, M, Murtaza, G, Romaguera, S: Soft contraction theorem. J. Nonlinear Convex Anal. 16, 423-435 (2015)Chen, CM, Lin, IJ: Fixed point theory of the soft Meir-Keeler type contractive mappings on a complete soft metric space. Fixed Point Theory Appl. 2015, 184 (2015)Feng, F, Li, CX, Davvaz, B, Ali, MI: Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput. 14, 8999-9911 (2010)Maji, PK, Biswas, R, Roy, AR: Soft set theory. Comput. Math. Appl. 45, 555-562 (2003)Wardowski, D: On a soft mapping and its fixed points. Fixed Point Theory Appl. 2013, 182 (2013)Kannan, R: Some results on fixed points II. Am. Math. Mon. 76, 405-408 (1969)Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326-329 (1969)Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976)Kirk, WA: Caristi’s fixed-point theorem and metric convexity. Colloq. Math. 36, 81-86 (1976

Interval valued (\in,\ivq)-fuzzy filters of pseudo BLBL-algebras

We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued (\in,\ivq)-fuzzy filters of pseudo $BL$-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo $BL$-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo $BL$-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed

Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(A\otimes B)$ with $Cu(A)\otimes_{Cu}Cu(B)$ for certain classes of C*-algebras. As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic. We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results corrected, in particular added 5.2.3-5.2.

What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions