# FUZZY EQUATIONAL CLASSES ARE FUZZY VARIETIES

Document Type : Research Paper

Authors

1 College for professional studies for teachers, Sabac, Serbia

2 College for professional studies for teachers,Sabac, Mega- trend University, Beograd, Serbia

3 Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia

Abstract

In the framework of fuzzy algebras with fuzzy equalities and a
complete lattice as a structure of membership values, we investigate fuzzy
equational classes. They consist of special fuzzy algebras ful lling the same
fuzzy identities, de ned with respect to fuzzy equalities. We introduce basic
notions and the corresponding operators of universal algebra: construction of
fuzzy subalgebras, homomorphisms and direct products. We prove that every
fuzzy equational class is closed under these three operators, which means that
such a class is a fuzzy variety.

Keywords

#### References

[1] R. Belohlavek, Fuzzy relational systems: foundations and principles, Kluwer Academic/
Plenum Publishers, New York, 2002.
[2] R. Belohlavek and V. Vychodil, Fuzzy equational logic, Studies in Fuzziness and Soft Computing,
Springer 2005, 186 (2005).
[3] R. Belohlavek and V. Vychodil, Algebras with fuzzy equalities, Fuzzy Sets and Systems, 157
(2006), 161-201.
[4] A. Borumand Saeid, Interval-valued fuzzy B-algebras, Iranian Journal of Fuzzy Systems, 3(2)
(2006), 63{73.
[5] R. A. Borzooei and M. Bakhshi, Some properties of T-fuzzy generalized subgroups, Iranian
Journal of Fuzzy Systems, 6(4) (2009), 73{87.
[6] B. Budimirovic, V. Budimirovic, B. Seselja and A. Tepavcevic, Compatible fuzzy equalities
and fuzzy identities, preprint.
[7] B. Budimirovic, V. Budimirovic and A. Tepavcevic, Fuzzy "-subgroups, Information Sciences,
180 (2010), 4006-4014.
[8] S. Burris and H. P. Sankappanavar, A course in universal algebra, Springer-Verlag, N. Y.,
1981.
[9] A. B. Chakraborty and S. S. Khare, Fuzzy homomorphism and algebraic structures, Fuzzy
Sets and Systems, 59 (1993), 211{221.
[10] P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl., 84 (1981), 264-269.
[11] M. Demirci, Vague groups, J. Math. Anal. Appl., 230 (1999), 142-156.
[12] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued equiva-
lence relations part I: fuzzy functions and their applications, part II: vague algebraic notions,
part III: constructions of vague algebraic notions and vague arithmetic operations, Int. J.
General Systems, 32(3) (2003), 123-155, 157-175, 177-201.
[13] A. Di Nola and G. Gerla, Lattice valued algebras, Stochastica, 11 (1987), 137-150.
[14] L. Filep, Study of fuzzy algebras and relations from a general viewpoint, Acta Math. Acad.
Paedagog. Nyhazi, 14 (1998), 49{55.
[15] U. Hohle, Quotients with respect to similarity relations, Fuzzy Sets and Systems, 27 (1988),
31-44.
[16] T. Kuraoka and N. Y. Suzuki, Lattice of fuzzy subalgebras in universal algebra, Algebra
universalis, 47 (2002), 223{237.
[17] X. Luo and J. Fang, Fuzzifying Closure Systems And Closure Operators, Iranian Journal of
Fuzzy Systems, 8(1) (2011), 77{94.
[18] J. N. Malik, D.S. Mordeson and N. Kuroki, Fuzzy Semigroups, Springer, 2003.
[19] V. Murali, Fuzzy congruence relations, Fuzzy Sets and Systems, 41(1991), 359{369.
[20] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 36 (1971), 512{517.
[21] B. Seselja, Characterization of fuzzy equivalence relations and of fuzzy congruence relations
on algebras, Review of Research Faculty of Science { University of Novi Sad, 11 (1981), 153{
160.
[22] B. Seselja, Homomorphisms of poset-valued algebras, Fuzzy Sets and Systems, 121 (2001),
333{340.
[23] B. Seselja and A. Tepavcevic, Fuzzy Boolean algebras, Automated Reasoning, IFIP Transactions
A-19, (1992), 83{88.
[24] B. Seselja and A. Tepavcevic, Partially ordered and relational valued algebras and congru-
ences, Review of Research, Faculty of Science, Mathematical Series, 23 (1993), 273{287.
[25] B. Seselja and A. Tepavcevic, On generalizations of fuzzy algebras and congruences, Fuzzy
Sets and Systems, 65 (1994), 85{94.
[26] B. Seselja and A. Tepavcevic, Fuzzy groups and collections of subgroups, Fuzzy Sets and
Systems, 83 (1996), 85{91.
[27] B. Seselja and A. Tepavcevic, A note on fuzzy groups, YUJOR, 7(1) (1997), 49{54.
[28] B. Seselja and A. Tepavcevic, Fuzzy identities, Proc. of the 2009 IEEE International Conference
on Fuzzy Systems, 1660{1664.
[29] A. P. Sostak, On a fuzzy topological structure, Frolk, Z., Soucek, V. and Vinarek, J. eds.:
Proceedings of the 13th Winter School on Abstract Analysis. Section of Topology, Palermo,
(1985), 89{103.
[30] R. T. Yeh and S. Y. Bang, Fuzzy relations, fuzzy graphs, and their application to clyster-
ing analysis, L. A. Zadeh, K. S. Fu, K. Tanaka, M. Shimura, eds.: Fuzzy Sets and Their
Applications to Cognitive and Decision Processes, Academic Press, Inc., (1975), 125{149.