The Remanchy Twist Structure Of Homogeneous Iterated Residual Algebra

In this paper, we define "cotwist" to connect Hom-coassociate coalgebra (C,△,β) with coassociate coalgebra (C,δ). we say (C,Δ,β) is a cotwist, or (C,δ) is a anticotwist, if Δ(c)=δβ(c), (?)c ∈C. The paper’s aim is to proof:when Hom-coassociate coalgebra (C, Δ,β) fulfill some conditions, we can define comulitplicaion δ such that (C,δ) is a coassociate coalgebra and (C, Δ,β) is a anticotwist. The paper has been divided as four sections.The first and the second stctions are introduction and preliminary respectively.In the third section, we introduce "weak counitary" conditon, we proof four calcu-lation lemmas in Hom-coassociate coalgebra (C,Δ,β), and use them to proof:a weak counitary Hom-coassociate coalgebra (C,Δ,β) with sujective β can be represented as a cotwist.In the fourth section, we introduce "strong degenerate", we proof the Hom-coassociate coalgebra (C,Δ,β) with injective β fufill the "weaker coassociate conditon ", then we use some knowledge in linear algebra to proof:If a Hom-coassociate coalgebra (C,Δ,β) with injective β is not strong degenerate, it is a cotwist. In the end of the paper, we proof:If the Hom-coassociate coalgebra (C,Δ,β) has finite dimensions, the conclusion in section 4 include the conclusion in section 3.