Fuzzy universal algebras on $L$-sets

Document Type: Research Paper

Authors

Department of Mathematics, Ocean University of China, 238 Songling Road, 266100, Qingdao, P.R.China

Abstract

This paper attempts to generalize universal algebras on classical sets to $L$-sets when $L$ is a GL-quantale. Some basic notions of fuzzy universal algebra on an $L$-set are introduced, such as subalgebra, quotient algebra, homomorphism, congruence, and direct product etc. The properties of them are studied. $L$-valued power algebra is also introduced and it is shown there is an onto homomorphism from $P(A)/R^{+}$ to $P(A/R)$ for any congruence $R$ on $L$-set $A$.

Keywords


[1] C. Brink, Power structures and logic, Queastions Mathematicae, 9 (1986), 69–94.
[2] C. Brink, Power structures, Algebra Universalis, 30 (1993), 177–216.
[3] R. B˘elohl´avek, V. Vychodil, Fuzzy equational logic, Archive for Mathematical Logic, 41 (2002), 83–90.
[4] R. B˘elohl´avek, V. Vychodil, Algebras with fuzzy equalities, Fuzzy Sets and Systems, 157 (2006), 161–201.
[5] I. Bo˘snjak, R. Madar´az, S. Roz´alia, Good quotient relations and power algebras, Novi Sad Journal of Mathematics,
29 (1999), 71–84.
[6] I. Bo˘snjak, R. Madar´asz, Power algebras and generalized quotient algebras, Algebra Universalis, 45 (2001), 179–189.
[7] I. Boˇsnjak, R. Madar´asz, G. Vojvodi´c, Algebras of fuzzy sets, Fuzzy Sets and Systems, 160 (2009), 2979–2988.
[8] I. Boˇsnjak, R. Madar´asz, On the composition of fuzzy power relations, Fuzzy Sets and Systems, 271 (2015), 81–87.
[9] A. B. Chakraborty, S. S. Khare, Fuzzy homomorphism and algebraic structures, Fuzzy Sets and Systems, 59 (1993),
211–221.
[10] C. L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24 (1968), 182–190.
[11] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part I:
fuzzy functions and their applications, International Journal of General Systems, 32(2) (2003), 123–155.
[12] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part II:
vague algebraic notions, International Journal of General Systems, 32(2) (2003), 157–175.
[13] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part III:
constructions of vague algebraic notions and vague arithmetic operations, International Journal of General Systems,
32(2) (2003), 177–201.
[14] G. Georgescu, Fuzzy power structures, Archive for Mathematical Logic, 47 (2008), 233–261.
[15] G. Gr¨atzer, Universal Algebra, 2nd ed. Springer-Verlag, 1979.
[16] U. H¨ohle, Commutative, residuated l-monoids, in: U. H¨ohle, E.P. Klement(Eds.), Non-Classical Logics and Their
Applications to Fuzzy Subsets: A Handbook on the Mathematical Foundations of Fuzzy Set Theory, Kluwer Academic
Publishers, Dordrecht, 1995, 53–105.
[17] J. Ignjatovi´ca, M. ´ Ciri´ca, S. Bogdanovi´cb, Fuzzy homomorphisms of algebras, Fuzzy Sets and Systems, 160 (2009), 2345–2365.
[18] H. Lai, D. Zhang, Good fuzzy preorders on fuzzy power structures, Archive for Mathematical Logic, 49 (2010),
469–489.

[19] F. Li, Y. Yue, L-valued fuzzy rough sets, Iranian Journal of Fuzzy Systems, 16 (2019), 111-127.
[20] R. Madar´az, Remarks on power structures, Algebra Universalis, 34 (1995), 179–184.
[21] V. Murali, Fuzzy congruence relations, Fuzzy Sets and Systems, 41 (1991), 359–369.
[22] B. Pang, F. G,Shi, Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces,
Fuzzy Sets and Systems, 2018, in press, DOI: http://doi.org/10.1016/j.fss.2018. 05. 012.
[23] B. Pang, Y. Zhao, Characterizations of L-convex spaces, Iranian Journal of Fuzzy Systems, 13(4) (2016), 51–61.
[24] Q. Pu, D. Zhang, Preordered sets valued in a GL-monoid, Fuzzy Sets and Systems, 187 (2012), 1–32.
[25] M. A. Samhan, Fuzzy quotient algebras and fuzzy factor congruences, Fuzzy Sets and Systems, 73 (1995), 269–277.
[26] L. Shen, Adjunctions in Quantaloid-enriched Categories, Ph.D Thesis, Sichuan University 2014.
[27] Z. Xiu, B. Pang, Base axioms and subbase axioms in M-fuzzifying convex spaces, Iranian Journal of Fuzzy Systems,
15(2) (2018), 75–87.
[28] W. Yao, Y. She, L. Lu, Metric-based L-fuzzy rough sets: Approximation operators and de nable sets, Knowledge-
Based Systems, 163 (2019), 91-102.
[29] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353.