Document Type: Research Paper


1 Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Korea

2 Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea


The concept of soft sets, introduced by Molodtsov [20] is a mathematical
tool for dealing with uncertainties, that is free from the difficulties
that have troubled the traditional theoretical approaches. In this paper, we
apply the notion of the soft sets of Molodtsov to the theory of Hilbert algebras.
The notion of soft Hilbert (abysmal and deductive) algebras, soft subalgebras,
soft abysms and soft deductive systems are introduced, and their basic properties
are investigated. The relations between soft Hilbert algebras, soft Hilbert
abysmal algebras and soft Hilbert deductive algebras are also derived.


[1] H. Akta¸s and N. C¸ aˇgman, Soft sets and soft groups, Inform. Sci., 177 (2007), 2726-2735.
[2] A. Borumand Saeid and Y. B. Jun, Redefined fuzzy subalgebras of BCK/BCI-algebras, Iranian.
J. Fuzzy Systems, 5(2) (2008), 63-70.
[3] R. A. Borzooei and Y. B. Jun, Intuitionistic fuzzy hyper BCK-ideals of hyper BCK-algerbas,
Iranian. J. Fuzzy Systems, 1(1) (2004), 65-78.
[4] D. Busneag, A note on deductive systems of a Hilbert algebra, Kobe J. Math., 2 (1985),
[5] D. Busneag, Hilbert algebras of fractions and maximal Hilbert algebras of quotients, Kobe J.
Math., 5 (1988), 161-172.
[6] I. Chajda, The lattice of deductive systems on Hilbert algebras, SEA Bull. Math., 26 (2002),
[7] I. Chajda, R. Halaˇs and Y. B. Jun, Annihilators and deductive systems in commutative
Hilbert algebras, Comment. Math. Univ. Carolinae, 43(3) (2002), 407-417.

[8] D. Chen, E. C. C. Tsang, D. S. Yeung and X. Wang, The parametrization reduction of soft
sets and its applications, Comput. Math. Appl., 49 (2005), 757-763.
[9] B. Davvaz, Roughness based on fuzzy ideals, Inform. Sci., 176 (2006), 2417-2437.
[10] B. Davvaz and P. Corsini On ( , )-fuzzy Hv-ideals of Hv-rings Iranian. J. Fuzzy Systems,
5(2) (2008), 35-48.
[11] A. Diego, Sur les alg´ebres de Hilbert, Collection de Logigue Math. Ser. A (Ed. Hermann,
Paris), 21 (1966), 1-52.
[12] Y. B. Jun, Deductive systems of Hilbert algebras, Math. Jpn., 43(1) (1996), 51-54.
[13] Y. B. Jun, Commutative Hilbert algebras, Soochow J. Math., 22(4) (1996), 477-484.
[14] Y. B. Jun, S. Y. Kim and E. H. Roh, The abysm of a Hilbert algebra, Sci. Math. Jpn., 65(1)
[15] Y. B. Jun, M. A. ¨Ozt¨urk and C. H. Park, Intuitionistic nil radicals of intuitionistic fuzzy
ideals and Euclidean intuitionistic fuzzy ideals in rings, Inform. Sci., 177 (2007), 4662-4677.
[16] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras,
Inform. Sci., 178 (2008), 2466-2475.
[17] D. V. Kovkov, V. M. Kolbanov and D. A. Molodtsov, Soft sets theory-based optimization, J.
Comput. Syc. Sci. Internat., 46(6) (2007), 872-880.
[18] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003),
[19] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making
problem, Comput. Math. Appl., 44 (2002), 1077-1083.
[20] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl., 37 (1999), 19-31.
[21] Z. Pawlak, Rough sets: theoretical aspects of reasoning about data, Kluwer Academic, Boston,
MA, (1991).
[22] Z. Pawlak and A. Skowron, Rudiments of rough sets, Inform. Sci., 177 (2007), 3-27.
[23] Z. Pawlak and A. Skowron, Rough sets: some extensions, Inform. Sci., 177 (2007), 28-40.
[24] Z. Pawlak and A. Skowron, Rough sets and Boolean reasoning, Inform. Sci., 177 (2007),
[25] L. Torkzadeh, M. Abbasi and M. M. Zahedi Some results of intuitionistic fuzzy weak dual
hyper K-ideals, Iranian J. Fuzzy Systems, 5(1) (2008), 65-78.
[26] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng., 50 (1962) 856-865.
[27] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.
[28] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)- an outline, Inform. Sci.,
172 (2005) 1-40.