Abstract
In the present paper we present a new approach to the fuzzification of groups, which is defined by the hazy associative law (a new fuzzy associative law) on hazy binary operations. It is also called an M-hazy group. The basic idea is to generalize the binary operation of classical algebras to fuzzy binary operation. Some properties of M-hazy groups and M-hazy subgroups are discussed. Besides, we also study the homomorphisms between two M-hazy groups.
Keywords
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References
1.
K.
Blount
and
C.
Tsinakis
, The structure of residuated lattices , International Journal of Algebra & Computation 13 (04 ) (2003 ), 437 –461 .
2.
M.
Demirci
,
Fuzzy
functions
and their fundamental properties , Fuzzy Sets and Systems 106 (1999 ), 239 –246 .
3.
M.
Demirci
, Smooth groups , Fuzzy Sets & Systems 117 (3 ) (2001 ), 431 –437 .
4.
M.
Demirci
, Smooth subgroups and smooth homomorphisms , Fuzzy Sets & Systems 117 (3 ) (2001 ), 439 –446 .
5.
M.
Demirci
, Vague groups , Journal of Mathematical Analysis & Applications 230 (1999 ), 142 –156 .
6.
E.Q.
Li
and
F.G.
Shi
, Some properties of M-fuzzifying convexities induced by M-orders , Fuzzy Sets and Systems 350 (2018 ), 41 –54 .
7.
Q.
Liu
and
F.G.
Shi
, M-fuzzifying median algebras and its induced convexities , Journal of Intelligent & Fuzzy Systems (2018 ). DOI: 10.3233/JIFS–172271
8.
B.
Pang
and
Z.Y.
Xiu
, Lattice-valued interval operators and its induced lattice-valued convex structures , IEEE Transactions on Fuzzy Systems 26 (3 ) (2018 ), 1525 –1534 .
9.
A.
Rosenfeld
, Fuzzy groups , Journal of Mathematical Analysis & Applications 35 (3 ) (1971 ), 512 –517 .
10.
S.
Sezer
, Some properties of vague rings , International Journal of Algebra 4 (16 ) (2010 ), 751 –760 .
11.
S.
Sezer
, Vague groups and generalized vague subgroups on the basis of ([0; 1]; ≤; ∧) , Information Sciences 174 (1–2 ) (2005 ), 123 –142 .
12.
J.Z.
Shen
, Fuzzifying groups based on complete residuated lattice-valued logic , Information Sciences 75 (1-2 ) (1993 ), 165 –186 .
13.
F.G.
Shi
and
Z.Y.
Xiu
, A new approach to the fuzzification of convex structures , Journal of Applied Mathematics 2014 (5 ) (2014 ), 1 –12 .
14.
E.
Turunen
, Mathematics behind fuzzy logic , PhysicaVerlag, Heidelberg , 1999 .
15.
M.
van de Vel
, Theory of convex structures , North-Holland, New York , 1993 .
16.
M.
Ward
and
R.P.
Dilworth
, Residuated Lattices , Transactions of the American Mathematical Society 45 (1939 ), 335 –354 .
17.
X.Y.
Wu
,
E.Q.
Li
and
S.Z.
Bai
, Geometric properties of M-fuzzifying convex structures , Journal of Intelligent & Fuzzy Systems 32 (6 ) (2017 ), 4273 –4284 .
18.
C.Y.
Xu
, Smooth rings , International Conference on Machine Learning and Cybernetics IEEE , 2008 , pp. 571 –575 .
19.
X.H.
Yuan
and
Y.H.
Ren
, A new Kind of Fuzzy Group , Journal of Liaoning Normal University (Natural Science Edition) 25 (1 ) (2002 ), 3 –6 .
20.
W.
Zhang
,
J.
Wang
,
W.
Liu
and
J.
Fang
, An introduction to fuzzy mathematics , Xi’an Jiaotong University Press , Xi’an , 1991 .