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By Orlando E. Villamayor

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P. Shestakov’s result (see [51)]that any Akivis algebra is linear. An Akivis algebra A is linear if A can be embedded in some non-associative algebra B with above operations. 3. ([25]) Let ( A ,+, [ , I , (, ,)) be an Akivis algebra with a linearly ordered basis {eil i E I } . Let m n where ( Y ~ , P , T ;E. ~ k. ,e, by {eiej} and m n {eiejek}, respectively. Let U ( A ) = M ( { e i } l l eiej-ejei = {eiej}, (eiej)ek-ei(ejek) = {eiejek}, i , j ,k E be the universal enveloping algebra of A . Let S = {eiej - ejei - { e i e j } (i > j ) , (eiej)ek - ei(ejek) - {eiejek} (i,j,k E I ) , ei(ejek) - ej(eiek) - {eiej}ek - {ejeiek} + {eiejek} (i > j , k 2 j ) } .

Z i k is the projection of u,and [u] 5 [u]a monomial order on X*. -polynomial f may have several leading monomials o f f . We call f a strong polynomial if f is unique. -polynomials, and a strong I’-Grobner-Shirshov basis. -polynomials that is closed under compositions. 4. ) a strong r-Grobner-Shirshou basis. Then (a) Iff E I d ( S ) , then f = aSb, where f is a leading monomial o f f , s E S , a,b r-words. -words} is a linear basis of k(x;qs). -algebras with strong r-Grobner-Shirshov bases.

Let X be a set, r a group, r(z), r'(x) isomorphic subgroups, x E X . -algebra. xikyk, z i ~ X , y€i I ' , k > O , which are equivalent under transformations yx + xy' above. 1 5 [u], 42 where [u]= zil . * z i k is the projection of u,and [u] 5 [u]a monomial order on X*. -polynomial f may have several leading monomials o f f . We call f a strong polynomial if f is unique. -polynomials, and a strong I’-Grobner-Shirshov basis. -polynomials that is closed under compositions. 4. ) a strong r-Grobner-Shirshou basis.

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Algebra Lineal OEA 5 by Orlando E. Villamayor


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