We notice that for a Hopf algebra H , its antipode S is both an epimorphism and a monomorphism from H to H in the category of Hopf algebras over a field. Together with the existence of Hopf algebras with non-injective or non-surjective antipode, this proves the existence of non-surjective epimorphisms and non-injective monomorphisms in the category of Hopf algebras. Using Schauenburg’s free Hop...

The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$-set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$-set, there exists...

It is well-known that a countably injective module is Σ-injective. In Proc. Amer. Math. Soc. 316, 10 (2008), 3461-3466, Beidar, Jain and Srivastava extended it and showed that an injective module M is Σ-injective if and only if each essential extension of M(א0) is a direct sum of injective modules. This paper extends and simplifies this result further and shows that an injective module M is Σ-i...

Recently, we obtained a new characterization for an orthogonal system to be simple-minded in the stable module category of any representation-finite self-injective algebra. In this paper, apply result give explicit construction s

We prove a basic result about tensor products of a II j factor with a finite von Neumann algebra and use it to answer, affirmatively, a question asked by S. Popa about maximal injective factors.

Justification Awareness Models, JAMs, were proposed by S. Artemov as a tool for modelling epistemic scenarios like Russel’s Prime Minister example. It was demonstrated that the sharpness and the injective property of a model play essential role in the epistemic usage of JAMs. The problem to axiomatize these properties using the propositional justification language was left opened. We propose th...

Lossy trapdoor functions (Peikert and Waters, STOC ’08) are an intriguing and powerful cryptographic primitive. Their main applications are simple and black-box constructions of chosen-ciphertext secure encryption, as well as collision-resistant hash functions and oblivious transfer. An appealing property of lossy trapdoor functions is the ability to realize them from a variety of number-theore...

For a finite dimensional algebra A over an algebraically closed field, let T (A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if TA is a tilting module and B = EndTA, then T (A) is tame if and only if T (B) is tame. Introduction. Let k be an algebraically closed field. In this paper, an algebra A is always assumed to be associative, with an ide...

The notions of purity and equational compactness of universal algebras have been studied by Banaschewski and Nelson. Also, Banaschewski deals with these notions in the special case of G-sets for a group G. In this paper we study these and related concepts in the category PRO of projection algebras, that is in N∞-sets, for the monoid N∞ with the binary operation m.n = min{m, n}. We show that eve...