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This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding. [ABSTRACT FROM PUBLISHER]

Banaschewski defines and gives sufficient conditions on a category A and a subclass M of its monomorphisms under which M-injectivity properly behaves with respect to the notions such as M-absolute retract, M-essentialness, and the existence of M-injective hulls. In this article, taking A to be the category of acts over a semigroup S and Md to be the class of sequentially dense monomorphisms, introduced and studied by Ebrahimi, Giuli, Mahmoudi, Moghaddasi, and Shahbaz, we study the behaviour of Md-injectivity (also called sequentially dense injectivity). We show, among other things, that S2 = S is a sufficient, but not necessary, condition for the behaviour of s-injectivity. Finally, we give an explicit description of the s-injective hulls of acts when they exist. [ABSTRACT FROM AUTHOR]

For a class L of right modules we investigate the relations of monomorphic L-envelopes and epimorphic L-covers over pullback-pushout diagrams, and obtain some isomorphism equations between the envelopes of covers and the covers of envelopes. [ABSTRACT FROM AUTHOR]

In this paper we classify, up to exact isomorphism, the algebras in the title which contain a faithful element and in which every non-trivial table subset has dimension at least five. [ABSTRACT FROM AUTHOR]

Let D be a Prüfer domain, and denote by ± b ℑ( D ) the multiplicative group of all invertible fractional ideals of D , ordered by A ≤ B if and only if A ⊇ B . Denote by G i the value group of the valuation associated with the valuation ring D M i , where { M i } i ∈ I is the collection of all maximal ideals of D . In this note we prove that the natural map from ± b ℑ( D ) into ± b ∏ i ∈ I G i is an isomorphism onto the cardinal sum ± b ∐ i ∈ I G i if and only if D is h -local. As a corollary, the group of divisibility of an h -local Bézout domain is isomorphic to ± b ∐ i ∈ I G i , the notation being as above. [ABSTRACT FROM AUTHOR]

We describe the Lie isomorphisms on associative H*-algebras with zero annihilator. [ABSTRACT FROM AUTHOR]