Semi-Associative 3-Algebras

In the paper,we define a new algebraic structure which is called semi-associative 3-algebra.The main contents are as following:Over a field IF of characteristic zero,the semi-associative 3-algebra is introduced.And it is proved that if A is a non-abelian semi-associative 3-algebra with dim A = m?3,then there exist linearly independent vectors ei,ej,ek such that {ei,ej,ek}?0.Meanwhile,it is proved that when dim A?6,A1(?)Z(A).The derivations of semi-associative 3-algebras are studied,and the definitions of the left multiplication L(x,y)and the right multiplication R(x,y)are provided.We prove that L(x,y)and R(x,y)are not derivations,however,S(x,y)=L(x,y)-R(x,y)are derivations for all x,y ? A.Furthermore,the representations of semi-associative 3-algebras are discussed,and according to the structure of double modules,the new semi-associative 3-algebras can be constructed.The relationships between 3-Lie algebras and semi-associative 3-algebras are discussed,from the semi-associative 3-algebra(A,{,,}),3-Lie algebra(A,[,,]C)is constructed.At last of the paper,the classification of low dimensional semi-associative 3-algebras is provided.