Dialgebras And Associative Algebras

In this paper, we present two kinds of dialgebras:dendriform dialgebras and asso-ciative dialgebras. Mainly, we consider the relation of dialgebras and associative algebras and the structure of associative dialgebras.For any dendriform dialgebra, one can construct an associative algebra (see [15]). In this paper, we consider that some kinds of Rota-Baxter type algebras could have dendri-form structures and give a concrete example about a free dendriform dialgebras in the set of isomorphic classes of binary trees BT. Simultaneously, we consider the Hopf structures in the set of isomorphic classes of binary trees BT.In data structures, binary trees have an important application-binary searching trees in which the nodes are labeled with elements of a set. By a binary searching tree, we can judge that an element is in a set or not (see [1]). In this paper, we consider the Hopf structures of binary trees. Specifically, in the section 2.2, we define its coproduct through of rooted trees (see [9]) and construct a Hopf structure HPB-From [16], we still know, to the set of isomorphic classes of strictly binary trees PB, HPB+=⊕·≠t∈PB kt is a free dendriform dialgebra and HPB=⊕t∈PB kt is a Hopf algebra. Thus, in the lager range the set of isomorphic classes of binary trees BT, we consider that whether kBT+=⊕·≠t∈BT kt is a free dendriform dialgebra or not and whether kBT==⊕t∈BT kt is a Hopf algebra or not. We give a definite answer. In the last of section 2.2, we consider the relation between HBT and kBT.From ([15]), we know, for any associative dialgebra D, DAs is an associative algebra. Here DAs is the quotient of D by the ideal generated by the elements x (?) y - x (?) y, for all x,y∈D. We have known the structure of associative algebras very well. So in this paper, we want to consider the structure of associative dialgebras. For example, for associative dialgebras, we gives the definitions of four kinds of annihilators, of nilpotent and nil and of of Jacobson radical. We consider the Jacobson semisimple of associative dialgebras. In the last, we point that Artinian associative dialgebras with a bar-unit e are Noetherian.In some situation, we find that associative dialgebras and associative algebras are similar.