Motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of TOPOLOGICAL system introduced by S. Vickers, providing an algebraic and a coalgebraic category of the new structures. As a result, the nature of the category TopSys of S. Vickers gets clari ed, and a metatheorem is stated, claiming that (latticevalued) topology can be embedded into algebra.

In this paper we give some characterizations of TOPOLOGICAL extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative TOPOLOGICAL left invariant mean then T is TOPOLOGICAL left lumpy if and only if there is a multiplicative TOPOLOGICAL left invariant mean M on M(S)* such that M(XT)=1, where XT is the characteristic functional of T. Consequently if T is a TOPOLOGICAL left lumpy locally compact Borel subsemigroup of a locally compact semigroup S, then T is extremely TOPOLOGICAL left amenable if and only if S is.

In this paper, we study the separtion axioms T 0, T 1, T 2 and T 5/2 on TOPOLOGICAL and semiTOPOLOGICAL residuated lattices and we show that they are equivalent on TOPOLOGICAL residuated lattices. Then we prove that for every infinite cardinal number a, there exists at least one nontrivial Hausdor TOPOLOGICAL residuated lattice of cardinality a. In the follows, we obtain some conditions on (semi) TOPOLOGICAL residuated lattices under which this spaces will convert into regular and normal spaces.Finally by using of regularity and normality, we convert (semi) TOPOLOGICAL residuated lattices into metrizable TOPOLOGICAL residuated lattices.

In this paper we introduce the concept of TOPOLOGICAL number for locally convex TOPOLOGICAL spaces and prove some of its properties. It gives some criterions to study locally convex TOPOLOGICAL spaces in a discrete approach.

We consider a category C enriched over the segment [0; 1] whose hom-objects are real numbers from [0; 1]. For a suitably defined function ^v assigning to each formula ' some object of C, the hom-object C(^v('); ^v( )) represents the degree of derivability ofψ from φ. We reformulate completeness result for intuitionistic propositional logic, as well as H ajek's completeness results concerning the product, G odel and Lukasiewicz fuzzy logic in the context of enriched category theory.

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 cluster algebras. From quiver representations, we proceed to the cluster category, which yields a complete categorification of the cluster algebra and its combinatorial underpinnings.