Let τ be an invertible skew pairing on (B,H), where B and H are Hopf algebras in a symmetric monoidal category C with (co)equalizers. Assume that H is quasitriangular. Then we obtain a new algebra structure such that B is a Hopf algebra in the braided category HYD and there exists a Hopf algebra isomorphism w : B∞H → B τH in C, where B∞H is a Hopf algebra with (co)algebra structure the smash (c...

Let HH be an algebra with a distinguished element εH∈H∗εH∈H∗ and C,DC,D two coalgebras. Based on the construction of Brzeziński’s crossed coproduct, under some suitable conditions, we introduce coassociative coalgebra C×GTHβR×DC×TGHRβ×D which is more general two-sided coproduct structure including smash coproduct. Necessary sufficient conditions for equipped tensor product C⊗H⊗DC⊗H⊗D to bialgeb...

In the theory of dynamical Yang-Baxter equation, with any Hopf algebra H and a certain H-module and H-comodule algebra L (base algebra) one associates a monoidal category. Given an algebra A in that category, one can construct an associative algebra A⋊L, which is a generalization of the ordinary smash product when A is an ordinary H-algebra. We study this ”dynamical smash product” and its modul...

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω1. Here, I show that assuming the Proper Forcing Axiom (PFA), every proper Scott set is the standard system of a model of PA. I define that a Scott set X is proper if it is arithmetically closed and the quotie...

Theoretical models of recursion schemes have been well studied under the names well-founded coalgebras, recursive coalgebras, corecursive algebras, and Elgot algebras. Much of this work focuses on conditions ensuring unique or canonical solutions, e.g. when the coalgebra is well-founded. If the coalgebra is not well-founded, then there can be multiple solutions. The standard semantics of recurs...

In this paper, we introduce an algebra H from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra H and the quantum affine algebra Uq1/2(ŝl2), where q denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice...

Dana Scott’s model of λ-calculus was based on a limit construction which started from an algebra of a suitable endofunctor F and continued by iterating F . We demonstrate that this is a special case of the concept we call coalgebra relatively terminal w.r.t. the given algebra A. This means a coalgebra together with a universal coalgebra-toalgebra morphism into A. We prove that by iterating F co...

Let H be a semisimple Hopf algebra over an algebraically closed field k. It is assumed that either char k = 0 or char k > dimH . Semisimplicity of H means that H is a semisimple left H-module. In that case H has finite dimension. Throughout the paper we shall keep to notations from [M]. For example H stands for the dual Hopf algebra with the natural pairing 〈−,−〉 : H ⊗ H → k. The algebra H is a...

Let us consider the algebra H-B-M , where is a distributive lattice, is a H-algebra (Heyting algebra), is a B-algebra (Brouwer algebra), < A, 0, 1,∨,∧,⇒, −̇ > is HB-algebra (semi-Boolean algebra) [10], is a de Morgan algebra, < A, 0, 1,∨,∧,⇒,∼> is a symmetrical Heyting algebra [5] and, respectively is a symmetr...

1 Introduction. In 8] we showed how several issues of importance in logic programming can prootably be viewed by casting them as questions concerned with topologies on spaces of interpretations and topological continuity of operators associated with logic programs. In doing this, we were building on earlier work of Batarekh and Subrahmanian 1, 2, 3] and extending and generalising their ideas. F...