We show that a large number of equations are preserved by DedekindMacNeille completions when applied to subdirectly irreducible FL-algebras/residuated lattices. These equations are identified in a systematic way, based on proof-theoretic ideas and techniques in substructural logics. It follows that a large class of varieties of Heyting algebras and FL-algebras admits completions.

Normal affine algebraic varieties in characteristic 0 are uniquely determined (up to isomorphism) by the Lie algebra of derivations of their coordinate ring. This was shown by Siebert [Si] and, independently, by Hauser and the third author [HM]. In both papers the assumption of normality is essential. There are nonisomorphic non-normal varieties with isomorphic Lie algebras. The third author [M...

The aim of this work is to discuss the concepts of degeneration, deformation and rigidity of Hopf algebras and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the ndimensional rigid Hopf algebras and the irreducible components for n = p, p is prime number, and for n < 14. This paper is organized as follows : first we present the basi...

both greatly acknowledge the hospitality of the RSISE.

Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories ...

We consider the loci of invertible linear maps f : ℂn → (ℂn)* together with pairs flags (E•, F•) in such that various restrictions Fj $$ {E}_i^{\ast } have specified ranks. Identifying an map its graph viewed as a point Grassmannian, we show closures these cohomology classes represented by back-stable Schubert polynomials Lam, Lee, and Shimozono. As special case, recover result Knutson, Speyer ...

Representable relation algebras are characterised by a certain property of networks. Representations of relation algebras can be further classiied as homogeneous or universal. These properties are also characterised by properties of networks properties (amalgamation and joint embedding properties). For the homogeneous, universal case a method of constructing interval algebras from point algebra...

Some new algebraic structures related to the coloured Yang-Baxter equation, and termed coloured Hopf algebras, are reviewed. Coloured quantum universal enveloping algebras of Lie algebras are defined in this context. An extension to the coloured graded Yang-Baxter equation and coloured Hopf superalgebras is also presented. The coloured two-parameter quantum universal enveloping algebra of gl(1/...