On Minimal Coalgebras

We define an out-degree for F-coalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all F-coalgebras, this class has a terminal object, which for many problems can stand in for the terminal F-coalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimization of coalgebras is functorial, that products of finitely many minimal coalgebras exist and are given by their largest common subcoalgebra, that minimal subcoalgebras have no inner endomorphisms and show how minimal subcoalgebras can be constructed from Moore-automata. Since the elements of minimal subcoalgebras must correspond uniquely to the formulae of any logic characterizing observational equivalence, we give in the last section a straightforward and self-contained account of the coalgebraic logic of D. Pattinson and L. Schröder, which we believe is simpler and more direct than the original exposition.     

When is a Coalgebra a Generator?

Let C be a coalgebra over a field k. The aim of this paper is to study the following problem : (P) If C is a k-coalgebra such that C is a generator for the category of left comodules, is C a left quasi-co-Frobenius coalgebra ? The converse always holds. We show that if C has a finite coradical series, the answer is positive.     

Balanced Bilinear Forms on Matrix and Matrix-Like Coalgebras

In this short note, we determine all balanced bilinear forms for a matrix coalgebra and for one type of matrix-like coalgebra.     

Constructing New Quasitriangular Turaev Group Coalgebras

In this article, we construct a lot of new examples of Hopf group coalgebras by considering Majid's bicrossproduct Hopf algebra in Turaev category. Furthermore, we find the sufficient and necessary conditions for such Majid's bicrossproduct Hopf algebra to admit quasitriangular structures in the sense of Turaev group coalgebras.     

The coefficient coalgebra of a symmetrized tensor space

The coefficient coalgebra of r-fold tensor space and its dual, the Schur algebra, are generalized in such a way that the role of the symmetric group Σ _r is played by an arbitrary subgroup of Σ _r . The dimension of the coefficient coalgebra of a symmetrized tensor space is computed and the dual of this coalgebra is shown to be isomorphic to the analogue of the Schur algebra.     

(*)-Serial coalgebras

In this paper, we introduce the notion of (*)-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of (*)-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be (*)-serial.     

弱Doi-Hopf系统的逆系统

首先引入了弱Doi—Hopf系统的逆系统的概念,然后证明了任意一个弱Doi—Hopf系统都存在逆,并且还得到了一个余环同构．     

On Gorenstein coalgebras

We investigate the comodule representation category over the Morita-Takeuchi context coalgebra Γ and study the Gorensteinness of Γ. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra Γ and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic.     

Coproducts and the cd-Index

The linear span of isomorphism classes of posets, P, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space P to the algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cd-index of the Boolean algebra and the cubical lattice. Moreover, these operations also have interpretations for certain classes of permutations, including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly, we show how to compute the ab-index of the Cartesian product of two posets given the ab-indexes of each poset.