In this paper we study a representation of a fuzzy subgroup $mu$ of a group $G$, as a product of indecomposable fuzzy subgroups called the components of $mu$.  This representation is unique up to the number of components and their isomorphic copies. In the crisp group theory, this is a well-known Theorem attributed to Remak, Krull, and Schmidt. We consider the lattice of fuzzy subgroups and som...

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials derive interesting relations. We also obtain new properties by making use of power series expansions q-trigonometric functions, q-exponential q-analogues the binomial theorem. By using Mathematica program, computational formulae graphical representation for aforementioned are obtained. a partial derivative o...

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C∗-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we es...

It is known that the Grauert-Riemenschneider vanishing theorem is not valid in characteristic p ([1]). Here we show that it may be restored in the presence of a suitable Frobenius splitting. The proof uses interchanging two projective limits, one involving iterated Frobenius maps, cf. [2] and [4], the other coming from Grothendieck’s theorem on formal functions. That leads to the following gene...

Quantum homogeneous vector bundles are introduced by a direct description of their sections in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies inherited from the quantum groups, and their sections furnish projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are invest...

Frobenius published two proofs of a theorem which characterizes irreducible and fully indecomposable matrices in an algebraic manner. It is shown that the second proof, which depends on the Frobenius-Konig theorem, yields a stronger form of the result than the first. Some curious features in Frobenius’s last paper are examined; these include his criticisms of a result due to D. K&rig and the la...