余代数上的Green等价

Green's Equivalences on Coalgebras

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．

In this paper, five equivalence relations on a coalgebra are defined, which are the Green's equivalence, i.e. G, D, C, R, H. About these equivalences, some properties are shown and some structures are characterized. A partial order relation is introduced on each of the sets C/L, C/R, C/H, C/G and the structures of semilattices and lattices are studied for the partial orders. The structures of subcoalgebras (resp. left, right coideals) is characterized as consisting of G-(resp. L-, R-)classes. The Green's equivalences are researched on tensor product. The Green's preserving and lifting properties of coalgebra morphisms are discussed. As applications, we obtain a condition such that the tensor product of two irreducible coalgebras is irreducible and show that the image of an irreducible coalgebra is irreducible under a G-preserving coalgebra morphism,