The theory of hyperstructures has been introduced byMarty in 1934 during the 8th Congress of the Scandinavian Mathematicians [4]. Marty introduced the notion of a hypergroup and since then many researchers have worked on this new topic of modern algebra and developed it. The notion of a hyperfield and a hyperring was studied first by Krasner [2] and then some authors followed him, for example, ...

Given a finite group G, let ram(G) (resp. ramt(G)) denote the minimal positive integer such that G can be realized as the Galois group of an extension of Q (resp. a tamely ramified extension of Q) ramified only at ram(G) (resp. ramt(G)) finite primes. The present paper is devoted to study ram(G) and ramt(G), for some solvable groups G. More precisely, we consider the case where G is either a fi...

1 Brown Representability First we introduce a portly abelian category A(S) for any preadditive category S. This will be a certain full subcategory of the category of all contravariant additive functors S −→ Ab. Modulo the fact that S is not assumed to be small, this is precisely what we call a right module over a ringoid in our Rings With Several Objects (RSO) notes. For background on portly ab...

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguishing knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial knots. Furthermore we prove that these invariants decide if a genus one knot ...

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the ...

In [3], the authors proved a number of results regarding Gröbner bases and initial ideals of those ideals J , in the free associative algebra K 〈X1, . . . , Xn〉, which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one notion of generic initial ideals in K 〈X1, . . . , Xn〉, and sh...

In this paper we present our results on the homology cobordism group Θ3Z of the oriented integral homology 3-spheres. We specially emphasize the role played in the subject by the gauge theory including Floer homology and invariants by Donaldson and Seiberg – Witten. A closed oriented 3-manifold Σ is said to be an integral homology sphere if it has the same integral homology as the 3-sphere S. T...

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not...

1. Elegant Reedy categories 1.1. Reedy categories. Recall that a Reedy structure on a category C is given by: a pair of identity-on-objects subcategories C− and C+ of C together with a degree function |–| : ob C → N such that: • For all non-identity σ : c→ d in C−, we have |c| > |d|; • For all non-identity δ : c→ d in C+, we have |c| < |d|. • Every map γ : c→ d of C admits a unique factorisatio...

Very few known cryptographic primitives are based on noncommutative algebra. Each new scheme is of substantial interest, because noncommutative constructions are secure agains many standard cryptographic attacks. On the other hand, cryptography does not provide security proofs that would allow to base the security of a cryptographic primitive on structural complexity assumptions. Thus, it is im...