The goal of this paper is to prove that if Q is a connected non-Dynkin quiver then the preprojective algebra ΠQ(k) of Q over any field k is Koszul, and has Hilbert series (1 − Ct+ t2)−1, where C is the adjacency matrix of the double Q̄ of Q. We also prove a similar result for the partial preprojective algebra ΠQ,J(k) of any connected quiver Q, where J ⊂ I is a subset of the set I of vertices of ...

This paper is a contribution to the development of model theory of fuzzy logic in narrow sense. We consider a formal system Ev L of fuzzy logic that has evaluated syntax, i.e. axioms need not be fully convincing and so, they form a fuzzy set only. Consequently, formulas are provable in some general degree. A generalization of Gödel’s completeness theorem does hold in Ev L. The truth values form...

Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and A T ; a 1 through a n and h 1 through h m are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N (A) is the orthogonal complement of R(A T). These four s...

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality betwe...