Fang LI
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,