Gorenstein Coalgebras

We study a class of semiperfect coalgebras which generalizes Quasi-co-Frobenius coalgebras (from the point of view of the projective dimension). This class of coalgebras allows us to study a relative concept of injective comodule in terms of the cotensor functor, generalizing a well-known result about coflat comodules.     

Functors for Coalgebras

Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor which associates a set X with the set (X) of all filters on X. It turns out that this functor preserves weak pullbacks, yet does not preserve weak generalized pullbacks. Since topological spaces can be considered as -coalgebras, in fact they constitute a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1-generated -coalgebras need not exist. Received August 24, 1998; accepted in final form October 12, 1998.     

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.     

Algebras Versus Coalgebras

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× – on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of ( F , G )-dimodules associated to two functors between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.      

Homology of Newtonian Coalgebras

Given a Newtonian coalgebra we associate to it a chain complex. The homology groups of this Newtonian chain complex are computed for two important Newtonian coalgebras arising in the study of flag vectors of polytopes:R a, b and Rc, d. The homology of Ra, b corresponds to the homology of the boundary of then -crosspolytope. In contrast, the homology of Rc, d depends on the characteristic of the underlying ring R. In the case the ring has characteristic 2, the homology is computed via cubical complexes arising from distributive lattices. This paper ends with a characterization of the integer homology ofZ c, d.     

Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

Universality of Categories of Coalgebras

Under GCH, a set functor F does not preserve finite unions of non-empty sets if and only if the category Coalg F of all F-coalgebras is universal. Independently of GCH, we show that for any non-accessible functor F preserving intersections, the category Coalg F has a large discrete full subcategory, and we give an example of a category of F-coalgebras that is not universal, yet has a large discrete full subcategory.     

余代数上的Green等价

本文在余代数上定义了五类等价关系,它们是Ｇｒｅｅｎ等价Ｇ,Ｄ,Ｌ,Ｒ,Ｈ．然 后给出了这些等价关系一些基本性质和结构特点．在每个Ｇｒｅｅｎ等价的等价类集上构 造了一种偏序并给出了偏序上半格和格结构的刻划．用Ｇ－,Ｌ－,Ｒ－类分别给出了子余 代数、左余理想、右余理想的结构刻划．进一步地,本文研究了张量积上的Ｇｒｅｅｎ等 价以及余代数同态的Ｇｒｅｅｎ保持性和提升性．作为应用,本文得到了两个不可约余代 数的张量积仍为不可约余代数的一个条件;证明了不可约余代数在Ｇ－保持余代数同 态下的同态像是不可约的．     

Quasi-co-Frobenius Coalgebras. II

Abstract

We prove new characterizations of Quasi-co-Frobenius (QcF) coalgebras and co-Frobenius coalgebras. Among them, we prove that a coalgebra is QcF if and only if C generates every left and every right C-comodule. We also prove that every QcF coalgebra is Morita-Takeuchi equivalent to a co-Frobenius coalgebra.     

Coalgebras and Bialgebras in Combinatorics

The following material is discussed in this paper: Incidence Coalgebras for PO sets; Reduced Boolean Coalgebras; Divided Powers Coalgebra; Dirichlet Coalgebra; Eulerian Coalgebra; Faà di Bruno Bialgebra; Incidence Coalgebras for Categories; The Umbral Calculus; Infinitesimal Coalgebras; Creation and Annihilation Operators; Point Lattice Coalgebras; Restricted Placements; Cleavages; and Hereditary Bialgebras.     

Hereditary and Formally Smooth Coalgebras

We define formally smooth coalgebras and we study their relation with hereditary coalgebras. The main result of the paper establishes that a coalgebra with separable coradical is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra , where C₀ is the coradical of C and N is a certain (C₀,C₀)–bicomodule. Presented by A. Verschoren Mathematics Subject Classifications (2000) primary: 16W30; secondary: 18G20. D. Ştefan: This author thanks the members of the Algebra Department of the University of Granada for their warm hospitality. He is especially grateful to Pascual Jara for the kind invitation to visit University of Granada. This research was partially supported by CERES, Contract 4-147.     

Flat Comodules and Perfect Coalgebras

Stenström introduced the notion of flat object in a locally finitely presented Grothendieck category 𝒜. In this article we investigate this notion in the particular case of the category 𝒜 = C-Comod of left C-comodules, where C is a coalgebra over a field K. Several characterizations of flat left C-comodules are given and coalgebras having enough flat left C-comodules are studied. It is shown how far these coalgebras are from being left semiperfect. As a consequence, we give new characterizations of a left semiperfect coalgebra in terms of flat comodules. Left perfect coalgebras are introduced and characterized in analogy with Bass's Theorem P. Coalgebras whose injective left C-comodules are flat are discussed and related to quasi-coFrobenius coalgebras.     

余代数和Morita-Takeuchi关系

本文证明了若余代数Ｃ和Ｄ是Ｍｏｒｉｔａ Ｔａｋｅｕｃｈｉ等价的，则Ｃ的子余代数格和Ｄ的子余代数格同构．设ＭΓ，ＮΓ是拟有限右Γ 余模，则（ｈ－Γ（Ｍ，Ｍ），ｈ－Γ（Ｎ，Ｎ）；ｈ－Γ（Ｎ，Ｍ），ｈ－Γ（Ｍ，Ｎ）；ｆ，ｇ）有Ｍｏｒｉｔａ Ｔａｋｅｕｃｈｉ关系．并给出了此Ｍ Ｔ关系是Ｍ Ｔ等价条件的及由已知的Ｍ Ｔ关系构造新的Ｍ Ｔ关系的方法．