we study algebraic properties of categories of merotopic, nearness, and filter algebras. we show that the category of filter torsion free abelian groups is an epireflective subcategory of the category of filter abelian groups. the forgetful functor from the category of filter rings to filter monoids is essentially algebraic and the forgetful functor from the category of filter groups to the cat...

This paper is devoted to deformation theory of graded Lie algebras over Z or Zl with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type An, by counting the mutation class of any quiver with underlying graph An. It will also follow that if T and T ′ are cluster-tilting objects in a cluster category C, then EndC(T ) is isomorphic to EndC(T ) if and only if T = τ T . 1. Cluster-tilted algebras The cluster categ...

In this article we prove that there exists an explicit bijection between nice $d$-pre-Calabi-Yau algebras and $d$-double Poisson differential graded algebras, where $d \in \mathbb{Z}$, extending a result proved by N. Iyudu M. Kontsevich. We also show correspondence is functorial in quite satisfactory way, giving rise to (partial) functor from the category of dg partial algebras. Finally, furthe...

In this paper we presented the semantics of database mappings in the relational DB category based on the power-view monad T and monadic algebras. The objects in this category are the database-instances (a database-instance is a set of n-ary relations, i.e., a set of relational tables as in standard RDBs). The morphisms in DB category are used in order to express the semantics of viewbased Globa...

lthough fuzzy set theory and  sheaf theory have been developed and studied independently,  ulrich hohle shows that a large part of fuzzy set  theory  is in fact a subfield of sheaf theory. many authors have studied mathematical structures, in particular, algebraic structures, in both  categories of these generalized (multi)sets. using hohle's idea, we show that for a (universal) algebra $a$, th...

The category of group-graded modules over an abelian group G is a monoidal category. For any bicharacter of G this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have n-ary multiplications between various graded components. They possess universal enveloping algeb...

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere’s presentation of metric spaces and Barr’s presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in ...

We propose a uniform, category-theoretic account of structural induction for inductively deened data types. The account is based on the understanding of inductively deened data types as initial algebras for certain kind of endofunctors T : B !B on a bicartesian/distributive category B. Regarding a predicate logic as a bration p : P!B over B , we consider a logical predicate lifting of T to the ...