In this paper a new quantity for real tensors, the sign-real spectral radius, is defined and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.

We discuss the relationship between algebras, coalgebras, and Frobenius algebras. We describe a method of constructing a Frobenius algebra, given a finite-dimensional algebra, and we demonstrate the method with several concrete examples.

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 ...

It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define may not have unit. We develop the elementary theory of these structures and show, in particular, how to nuclei whose quotients quantales. This yields phase semantics representation theorem via Important examples arise from Raney’s notion tight Galois connection: endomap...

Commutative Hilbertian Frobenius algebras are those commutative semigroup objects in the monoidal category of Hilbert spaces, for which adjoint multiplication satisfies compatibility relation, that is, this “comultiplication” is a bimodule map. In note we show relation forces operators to be normal. We then prove these have strong Wedderburn decomposition where (ortho)complement Jacobson radica...

A G-Frobenius graph, as defined recently by Fang, Li, and Praeger, is a connected orbital graph of a Frobenius group G = K:H with Frobenius kernel K and Frobenius complement H. Γ is also shown to be a Cayley graph, Γ = Cay(K,S) for K and some subset S of the group G. On the other hand, a network N with a routing function R, written as (N,R), is an undirected graph N together with a routing R wh...