SOME PROPERTIES OF FUZZY SETS ON FINITE TYPE OF KAC-MOODY ALGEBRAS C4 & D4
Keywords:
Core, Convexity, fuzzy set, Generlized Cartan Matrix, height, Kac-Moody algebra, non-degenerate form, normal, root basis,α– cut decomposition, α-level set, strong α-level set..Abstract
The Kac-Moody algebras has been attracting the attention of a lot of Mathematicians because of its
various connections and applications to different branches of Mathematics and Mathematical Physics since the
introduction of the subject in 1968, developed simultaneously and independently by Kac and Moody. On the
other hand, Fuzzy sets first originated in a seminar paper by Lotfi A.Zadeh in 1965. The theory on fuzzy sets
has been applied not only to all branches of Mathematics but also acts as a tool for solving challenging
problems in science & technology and social problems.
In this paper, the classical finite type of Kac- Moody algebras C4 and D4 are considered. Fuzzy sets are defined
on the cartesian product of the root basis of Kac-Moody algebras using an invariant, non degenerate,
symmetric bilinear form . . , . > < Some of the basic properties like support, core, normality, height,
cardinality, relative cardinality and convexity are studied; α – level sets and strong α – level sets are
computed. α– cut decomposition for these fuzzy sets, associated with C4 and D4 families of finite type of Kac
Moody algebras are computed.
