The minimum depth d B,A of a subring B ⊆ A introduced in the work of Boltje, Danz and Külshammer 2011 is studied and compared with the tower depth of a Frobenius extension. We show that d B,A < ∞ if A is a finite-dimensional algebra and B has finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts ...

A mathematical procedure used to convert an element in the universe of discourse into the membership value of the fuzzy set is termed as fuzzification. In this paper we propose a new technique Vague-Fuzzification to implement fuzzification using vague sets. The proposed technique is designed that is using two methods to implement the same. First method includes a positive ordered transforming f...

Various attempts have been made by researchers on the study of vagueness of data through intuitionistic fuzzy sets and vague sets, and also it was shown that vague sets are intuitionistic fuzzy sets. But there are algebraic and graphical differences between vague sets and intuitionistic fuzzy sets. In this paper an attempt is made to define the correlation coefficient of interval vague sets lyi...