a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...

A module $M$ is called $emph{H}$-cofinitely supplemented if for every cofinite submodule $E$ (i.e. $M/E$ is finitely generated) of $M$ there exists a direct summand $D$ of $M$ such that $M = E + X$ holds if and only if $M = D + X$, for every submodule $X$ of $M$. In this paper we study factors, direct summands and direct sums of $emph{H}$-cofinitely supplemented modules. Let $M$ be an $emph{H}...

It is proved that the vertex operator algebra V is isomorphic to the moonshine VOA V ♮ of Frenkel-Lepowsky-Meurman if it satisfies conditions (a,b,c,d) or (a,b,c,d). These conditions are: (a) V is the only irreducible module for itself and V is C2-cofinite; (a) dimVn ≤ dimV ♮ n for n ≥ 3; (b) the central charge is 24; (c) V1 = 0; (d) V2 (under the first product on V ) is isomorphic to the Gries...

In this paper the W -algebra W (2, 2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to a highest irreducible W (2, 2)-module or a tensor product of two irreducible Virasoro vertex operator algebras. Furthermore, any rational, C2-cofinite and simple vertex operator alg...

The strong shape category of topological spaces SSh can be defined using the coherent homotopy category CH, whose objects are inverse systems consisting of topological spaces, indexed by cofinite directed sets. In particular, if X, Y are spaces and q : Y → Y is a cofinite HPol-resolution of Y , then there is a bijection between the set SSh(X, Y ) of strong shape morphisms F : X → Y and the set ...

We introduce a simple, self-dual, rational, and $C_2$-cofinite vertex operator algebra of CFT-type associated with $\mathbb{Z}_k$-code for $k \ge 2$. Our argument is based on the $\mathbb{Z}_k$-symmetry among simple current modules parafermion $K(\mathfrak{sl}_2, k)$. show that it naturally realized as commutant certain subalgebra in lattice algebra. Furthermore, we construct all irreducible in...

Let Γ = (V,E) be a reflexive relation having a transitive group of automorphisms and let v ∈ V. Let F be a subset of V with F ∩Γ−(v) = {v}. (i) If F is finite, then |Γ(F ) \ F | ≥ |Γ(v)| − 1. (ii) If F is cofinite, then |Γ(F ) \ F | ≥ |Γ−(v)| − 1. In particular, let G be group, B be a finite subset of G and let F be a finite or a cofinite subset of G such that F ∩ B = {1}. Then |(FB) \ F | ≥ |B...

Abstract The rank 1 bosonic ghost vertex algebra, also known as the $$\beta \gamma $$ ? ? ghosts, symplectic bosons or Weyl is a simple example of conformal field theory which neither rational, nor $$C_2$$ C 2 -co...