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 generators of ...

We investigate the triangulated hull of orbit categories perfect derived category and bounded a ring concerning power suspension functor. It turns out that will correspond to full subcategory compact objects certain periodic complexes. This specializes Stai Zhao's result when is finite dimensional algebra with global dimension over field. As first application, if $A,B$ are flat algebras commuta...

Abstract Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to categories are gs-monoidal also known as CD categories. These omit a condition that implements normalization probability. Extending work Corradini and Gadducci, we c...

Let D1, D2, D be non empty sets and let x be an element of [: [:D1, D2 :], D :]. Then x1,1 is an element of D1. Then x1,2 is an element of D2. Let D1, D2 be non empty sets and let x be an element of [:D1, D2 :]. Then x2 is an element of D2. Next we state the proposition (1) Let C, D be category structures. Suppose the category structure of C = the category structure of D. If C is category-like,...