We define the concepts of representable and abstract sequential Q-algebra, which are generalizations of the (relational) Q-algebras in [10]. Just as in that paper, we then prove that the two concepts coincide. In the following section we recall the concept of observation space and note that all complex algebras of observation spaces are representable sequential algebras. Finally we give an unco...

We consider the families L of propositional superintuitionistic logics (s.i.l.) andNE(K) of normal modal logics (n.m.l.). It is well known that there is a duality between L and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also NE(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig’s int...

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish th...

For a given variety Var of algebras we define the variety Var of dialgebras. This construction turns to be closely related with varieties of pseudo-algebras: every Var-dialgebra can be embedded into an appropriate pseudo-algebra of the variety Var. In particular, Leibniz algebras are exactly Lie dialgebras, and every Leibniz algebra can be embedded into current Lie conformal algebra.

This note is to study a variety of graded algebras that arise from the induced representations for semisimple algebraic groups and quantum groups. These algebras will play an important role in a study of the cohomology groups of line bundles over the flag varieties. This short note concentrates on the calculation of the Gelfand-Kirillov dimensions of these algebras.

A proof of Poincaré-Birkhoff-Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich S-Lie algebras. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type An and Mp,q,ǫ(n, K), which is a nonstandard quantum deformation of GL(n). In particular, we get, for both algebras, a unified proof of the...

In this paper we establish a link between satisfiability of universal sentences with respect to varieties of distributive lattices with operators and satisfiability with respect to certain classes of relational structures. We use these results for giving a method for translation to clause form of universal sentences in such varieties, and then use results from automated theorem proving to obtai...

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F (x1, ..., xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X = {x1, ..., xn}, EndF be the semigroup of endomorphisms of F , and Aut EndF be the group of automorphisms of the semigroup EndF ...

In this paper we introduce a new setting, based on partial algebras, for studying constructions of finitely generated free algebras. We give sufficient conditions under which the finitely generated free algebras for a variety V may be described as the colimit of a chain of finite partial algebras obtained by repeated application of a functor. In particular, our method encompasses the constructi...