Abstract:
Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’) ^2+(b’) ^2-(c’)) ^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’).

Abstract:
In this paper we construct a coalgebra for an intrusion detection system todescribe the behaviour of a packet stream together with selected actions in the case ofintrusions. We start with an extension of the notion of the many-typed signature to thegeneralised signature and we construct the category of packets as a basic structure of ourapproach. A defined endofunctor captures the expected behaviour of the packet stream. Theconstructed coalgebra enables the description of the behaviour of the packet streamtogether with the reaction to intrusions.

Abstract:
Kleene algebra with tests (KAT) is an equational system that combines Kleene and Boolean algebras. One can model basic programming constructs and assertions in KAT, which allowed for its application in compiler optimization, program transformation and dataflow analysis. To provide semantics for KAT expressions, Kozen first introduced emph{automata on guarded strings}, showing that the regular sets of guarded strings plays the same role in KAT as regular languages play in Kleene algebra. Recently, Kozen described an elegant algorithm, based on ``derivatives'', to construct a deterministic automaton that accepts the guarded strings denoted by a KAT expression. This algorithm generalizes Brzozowski's algorithm for regular expressions and inherits its inefficiency arising from the explicit computation of derivatives. In the context of classical regular expressions, many efficient algorithms to compile expressions to automata have been proposed. One of those algorithms was devised by Berry and Sethi in the 80's (we shall refer to it as Berry-Sethi construction/algorithm, but in the literature it is also referred to as position or Glushkov automata algorithm). In this paper, we show how the Berry-Sethi algorithm can be used to compile a $KAT$ expression to an automaton on guarded strings. Moreover, we propose a new automata model for KAT expressions and adapt the construction of Berry and Sethi to this new model.

Abstract:
Behavior of running program can be described by evaluating a coalgebraic structure over a collection of algebraic terms on state space. Coalgebras are defined by polynomial endofunctors. We formulate substantiation of coalgebras in categories. We use approach via Kleisli categories together with monads beside the approach via topoi and comonads as dual structures to monads.

Abstract:
Let P be a partially ordered set (poset) locally finite and kQ the path coalgebra over a field k associated to P. In this paper we investigate finiteness properties of this coalgebra by using an injective morphism of coalgebras from incidence coalgebra kS of P to the path coalgebra kQ. We deduce that kQ is left semiperfect only if kS have the same property, and that kQ is cosemisimple when the order relation on P is the equality. Finally we characterize the coradical filtration of the path coalgebra.

Abstract:
Unlike the widespread applications of algebraic methods in computer science, coalgebraic methods, as the dual concepts of algebras, have not been noticed by computer scientists till the mid 1990s. In algebraic methods, the constructive elements of data types are studied, while in coalgebraic methods, the observable behaviors of systems are investigated. Coalgebraic methods have distinct advantages in mathematic study of state-based systems since them enable the depth research on those systems?properties such as behavior equivalence, nondeterminism and so on. Coalgebraic methods have also been applied in many research fields, for example, automata theory, semantics of concurrency, specifications of object-oriented software etc. The recent progress of coalgebraic methods, including its basic concepts, categorical foundations, logical foundations and its applications, is summarized for raising the attention of the relative researchers.

Abstract:
摘要： 设C是域上的一个右半完全余代数,证明了每个右C-余模有一个Gorenstein内射包。且如果余代数C作为右C-余模的投射维数有限,给出了Gorenstein内射右C-余模的一个等价刻画。 Abstract: Let C be a right semiperfect coalgebra over a field. It is shown that every right C-comodule has a Gorenstein injective envelope. Moreover, if the projective dimension of C as a right C-comodule is finite, an equivalent characterization of Gorenstein injective right C-comodules is given

Abstract:
本文研究了可计算余模和Hom-可计算余代数的局部化问题.利用局部化方法,得到了可计算余模和Hom-可计算余代数的等价条件,推广了余代数上局部化理论的发展. In this paper, the localization problems of computable comodules and Homcomputable coalgebras are studied. By applying some localization techniques, the equivalent conditions for computable comodules and Hom-computable coalgebras are obtained, which extend the developing of localization theory of coalgebras

Abstract:
In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ～L, defined as x ～L y if and only if xz ∈ L exactly when yz ∈ L, ？z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras.