Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers Z, that is, for a sub-semigroup Λ of the non-negative integers (N,+), we define the order by n ≤Λ m if m− n ∈ Λ. When Λ is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere....

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commu...

In this paper, we study cyclic stabiliser codes over Fp of length dividing p + 1 for some positive integer t. We call these t-Frobenius codes or just Frobenius codes for short. We give methods to construct them and show that they have efficient decoding algorithms. An important subclass of stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we have stronger results: We ...

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.

A numerical set S with Frobenius number g is a set of integers with min(S) = 0 and max(Z − S) = g, and its atom monoid is A(S) = {n ∈ Z | n+ s ∈ S for all s ∈ S}. Let γg be the number of numerical sets S having A(S) = {0} ∪ (g,∞) divided by the total number of numerical sets with Frobenius number g. We show that the sequence {γg} is decreasing and converges to a number γ∞ ≈ .4844 (with accuracy...

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 ...