On assosymmetric rings

Free assosymmetric algebras as modules of groups

“…P 1 ∼ = S (1) ; P 2 ∼ = S (2) ⊕ S (1,1) ; P 3 ∼ = 2 * S (3) ⊕ 2 * S (2,1) ⊕ S (1,1,1) ; P 4 ∼ = 3 * S (4) ⊕ 4 * S (3,1) ⊕ 2 * S (2,2) ⊕ 3 * S (2,1,1) ⊕ S (1,1,1,1) ; P 5 ∼ = 4 * S (5) ⊕6 * S (4,1) ⊕6 * S (3,2) ⊕6 * S (3,1,1) ⊕5 * S (2,2,1,1) ⊕4 * S (2,1,1,1,1) ⊕S (1,1,1,1,1) . Let V be a vector space with dimension m. Let F (V ) be free assosymmetric algebra generated by basis elements of V and H n (V ) be homogeneous part of F (V ) of degree n. Definition 2.6.…”

“…A nonassociative algebra (R, •) is called assosymmetric, if for any a, b, c ∈ R the following identities are hold The factor algebra F (X) = K{X}/T (R) is called free assosymmetric algebra. Assosymmetric algebras was studied in [19], [25], [2], [17], [7]. Basis of free assosymmetric algebras was constricted in [17].…”

Assosymmetric algebras under jordan product

“…for any a, b, c, d, e ∈ A. Assosymmetric algebras was studied in ( [2], [6], [7], [10], [12], [13], [14]). Free base of assosymmetric algebras was found in [6].…”

“…Here we give some preliminary results that we will use in proof of our theorems. In proof of our lemmas we use results of [2] . This paper needs restriction p = 2, 3.…”

The structure of assosymmetric algebras

“…(For example, see [4,5,13,15,16,18,[21][22][23][24][25], etc.) If a left-symmetric algebra has also the right-symmetry condition, it has been called an assosymmetric algebra [3,9,12,19] or a bi-symmetric algebra [4]. A left-symmetric algebra can be viewed geometrically as a locally simple transitive affine actions on an affine space and vice versa.…”